Stone and Flat Topologies on the Minimal Prime Spectrum of a Commutator Lattice

Abstract

In previous work we have studied minimal prime spectra, as well as extensions of universal algebras whose term condition commutator behaves like the modular commutator in the sense that it is commutative and distributive with respect to arbitrary joins, while modularity does not even need to be enforced on their congruence lattices, let alone on those of the members of the variety they generate. Commutator lattices, defined by Czelakowski in 2008, are commutative multiplicative lattices having as prototype the algebraic structure of the congruence lattice of a such an algebra. Considering the prime elements with respect to the commutator operation, we obtain algebraic characterizations for minimal primes, then study the Stone and flat topologies on the set of minimal primes in a commutator lattice. We also prove abstract versions of congruence extension properties, actually of the general case of arbitrary morphisms instead of algebra embeddings, by means of complete join–semilattice morphisms between commutator lattices. We thus obtain abstractions for our results on congruence lattices and generalizations for results on frames and quantales, but also further cases in which these results hold. Furthermore, we investigate the lattice structures of these topologies as sublattices of the power sets of the sets of (minimal) primes.

1. Introduction

The prime spectrum of congruences of an algebra A, denoted $\mathrm{Spec}(A)$, consists of its prime congruences w.r.t. the term condition commutator ${[·,·]}_A$. If ${[·,·]}_A$ is commutative and distributive w.r.t. arbitrary joins, in particular if A belongs to a congruence–modular variety, then $\mathrm{Spec}(A)$ can be endowed with the Stone or spectral topology, by generalizing the construction of the Zariski topology from rings [1,2].

Using the Stone topology on the prime spectrum of congruences, we have generalized properties of ring extensions in [3,4] and constructed the reticulation of a universal algebra in [5]. In ref. [6] we have investigated the topology induced on the antichain $\mathrm{Min}(A)$ of the minimal prime congruences of an algebra A, called the minimal prime spectrum of congruences of A, by the Stone topology of $\mathrm{Spec}(A)$, along with the flat or inverse topology on $\mathrm{Min}(A)$, and used these topologies to study universal algebra extensions that generalize certain types of ring extensions, investigated in [7,8,9].

In ref. [10], we have investigated certain algebraic properties of commutator lattices; these are abstractions for congruence lattices, consisting of complete lattices endowed with binary operations called commutators that satisfy commutativity and distributivity with respect to arbitrary joins. A more general version of this notion is that of a multiplicative lattice [11,12].

Our results from [5,6], along with other properties from the papers cited above, can be obtained in this abstract case of commutator lattices. We begin this investigation in the current paper, with an algebraic and topological study of the minimal prime spectrum of a commutator lattice, consisting of its minimal prime elements with respect to the commutator operation. We obtain characterizations for minimal prime elements, study the Stone and flat topologies on the minimal prime spectra of commutator lattices, then obtain results on complete join–semilattice morphisms between commutator lattices which serve as abstractions for congruence extension properties along with their generalizations to arbitrary morphisms, using properties of commutator lattices from [10] and proving in this abstract case results similar to the ones we have obtained in [5,6]. This abstract approach proves fruitful, as it reveals further cases in which these results hold.

In Section 2 we recall some lattice–theoretical notions and establish the basic notations that we use throughout the paper.

In Section 3 we recall the definition and some properties of commutator lattices, introduce some more notations, recall properties of congruence lattices for arbitrary algebras, but also for members of (semidegenerate) congruence–modular varieties, introduce the Stone topology on the set of the prime elements with respect to the commutator, as well as the flat topology in the particular case of compact commutator lattices, study prime, radical and compact elements, as well as quotients of commutator lattices through their elements, which serve as abstractions for congruence lattices of quotient algebras. In this section and the following ones, we are interested in the distinction between results that need properties of commutator lattices which congruence lattices satisfy, such as algebraicity, and results which hold in more general or different contexts.

In Section 4 we begin our study of minimal prime elements with respect to the commutator. We introduce in this section Conditions 1 and 2, which are key for our main results; these translate into this abstract framework Conditions 1.(iv) and 1.(v) from [6], respectively, the first of which describes the congruence lattice of an algebra that admits a reticulation [5], while the second one has arisen in our purely lattice–theoretical approach to these problems. The reticulation can be used directly to transfer properties, for instance from rings, as we do in Proposition 11. Under one or the other of Conditions 1 and 2, commutator lattices satisfy characterizations for their minimal primes (Proposition 7) which are needed for some of the following results. These conditions in themselves reveal interesting algebraic properties of lattices fulfilling one or both of the subsequent characterizations for minimal primes; for instance, the lattice of filters of the Boolean algebra A in [6] (Example 1) satisfies Condition 1, but not Condition 2, the latter of which translates into the fact that one of its prime principal ideals includes a prime ideal which is not principal.

In Section 5 we study the Stone and the flat topologies on the set of minimal primes of a commutator lattice in terms of their lattice structures when ordered by set inclusion, their relation to each other and regarding compactness and separability. We conclude this section with a several examples of finite (thus compact, in particular algebraic) commutator lattices which satisfy the properties needed for our results, such as their top elements being neutral with respect to their commutators and their bottom elements being radical, as well as Conditions 1 and 2. Some of these examples have nonassociative commutators, which ensures that they can not be congruence lattices of commonly studied algebras such as lattice–ordered or other congruence–distributive algebras, whose commutators also happen to be associative since they equal the intersection, or congruence–modular structures such as groups or rings, which also happen to have associative commutators, unlike modular commutators in general. We determine the Stone and flat topologies on the sets of primes and of minimal primes of these two commutator lattices, with their lattice structures as bounded sublattices of the power sets of the sets of primes and minimal primes, respectively.

In Section 6 we investigate an abstraction for congruence extensions, by means of complete morphisms of join–semilattices between commutator lattices. Since we do not yet have a description of those complete join–semilattice morphisms in the particular case of extensions of algebras, we actually obtain an abstraction for the generalization from algebra embeddings to arbitrary morphisms between the algebras whose congruence lattices we apply our results in this section to. We introduce, for such complete join–semilattice morphisms the notions of admissibility and Min–admissibility, as well as Condition 3, which holds in the particular case of congruence lattices of members of congruence–modular varieties, but is also satisfied by admissible complete join–semilattice morphisms such that commutators of elements from their image are radical elements, in particular by such morphisms whose codomains are algebraic frames endowed with the commutator equalling their meet. We illustrate these notions for several lattice morphisms between the commutator lattices from the example in Section 5. We obtain conditions for the right adjoint of such a complete join–semilattice morphism to be continuous or a homeomorphism between the Stone and flat topologies on the set of minimal primes of its codomain and that of its domain and apply this result to congruence lattices, thus relating these topologies on the congruence lattices of the codomain and domain of an arbitrary morphism of similar algebras.

We conclude with some remarks on our current work and directions for future research in Section 7.

2. Preliminaries

See refs. [13,14,15,16] for more details on the following notions from universal algebra, refs. [17,18,19,20] for the lattice–theoretical ones, refs. [13,16,21,22] for commutators and refs. [4,13,23,24,25,26] for the Stone topologies. For minimal prime spectra of frames see [27,28].

We denote by $\mathbb{N}$ the set of the natural numbers and by ${\mathbb{N}}^{*}=\mathbb{N}\setminus \left\{0\right\}$. For any set M and any $S\subseteq M$, $i_{S,M}:S\to M$ will be the inclusion map, $i{d}_M:=i_{M,M}$ the identity map, $\mathcal{P}(M)$ the set of the subsets of M, ${\Delta }_M=\left\{(x,x)\right|x\in M\}$ and ${\nabla }_M=M^2$. Any sets, in particular congruences, ideals and filters, will always be ordered by set inclusion. Unless mentioned otherwise, the usual notations will be used for the order of any poset and the operations of any lattice. $Min(P)$ and $Max(P)$ will denote the set of the minimal elements and that of the maximal elements of a poset P, respectively. ${\mathcal{M}}_3$, respectively ${\mathcal{N}}_5$ will be the 5–element modular nondistributive, respectively nonmodular lattice and, for any $n\in {\mathbb{N}}^{*}$, ${\mathcal{C}}_n$ will denote the n–element chain. ⊕ will denote the ordinal (also called glued) sum of bounded lattices.

Let L be an arbitrary lattice. $\mathrm{Cp}(L)$, $\mathrm{Mi}(L)$ and $\mathrm{Smi}(L)$ will denote the sets of the compact, the meet–irreducible and the strictly meet–irreducible elements of L, respectively. If L has a 1, then $1\in \mathrm{Mi}(L)\setminus \mathrm{Smi}(L)$, since ${\displaystyle 1=\bigwedge \varnothing =\bigwedge \{x\in L|1<x\}}$.

Recall that L is said to be compact, respectively algebraic, if and only if each of its elements is compact, respectively a join of compact elements. If L is compact, then the join of any nonempty subset U of L equals the join of a finite subset of U.

Remember that L is called a frame if and only if L is complete and the meet in L is distributive with respect to arbitrary joins.

Let ★ be an arbitrary binary operation on L. If L has a 1, then we denote by ${\mathrm{Max}}_L=Max(L\setminus \left\{1\right\},\le )$, by ${\mathrm{Spec}}_{(L,★)}$ the set of the prime elements of L with respect to ★:

$${\mathrm{Spec}}_{(L,★)}=\{p\in L\setminus \left\{1\right\}|(\forall a,b\in L)(a★b\le p\Rightarrow (a\le porb\le p))\}$$

and by ${\mathrm{Min}}_{(L,★)}=Min({\mathrm{Spec}}_{(L,★)},\le )$.

Notice that ${\mathrm{Max}}_L$ is the set of the coatoms of L and, if L is distributive and has a 1, then ${\mathrm{Spec}}_{(L,\wedge )}=\mathrm{Mi}(L)\setminus \left\{1\right\}\supseteq \mathrm{Smi}(L)\supseteq {\mathrm{Max}}_L$.

$\mathrm{Id}(L)$ and $\mathrm{Filt}(L)$ will be the lattices of the ideals and filters of L, respectively; $\mathrm{PId}(L)$ and $\mathrm{PFilt}(L)$ will be their sublattices of the principal ideals and principal filters of L, respectively. Recall that ${\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{L}),\cap )}$ and ${\mathrm{Spec}}_{(\mathrm{Filt}(\mathrm{L}),\cap )}$ are the sets of prime ideals and the prime filters of L, respectively.

For any $U\subseteq L$ and any $a\in L$, ${(U]}_L$ and ${[U)}_L$ will be the ideal and the filter of L generated by U, respectively, and we denote by ${(a]}_L={(\left\{a\right\}]}_L$ and ${[a)}_L={[\left\{a\right\})}_L$.

For any $x\in L$, we denote by:

• $V_{(L,★)}(x)={[x)}_L\cap {\mathrm{Spec}}_{(L,★)}$,
• $D_{(L,★)}(x)={\mathrm{Spec}}_{(L,★)}\setminus V_{(L,★)}(x)={\mathrm{Spec}}_{(L,★)}\setminus {[x)}_L$,
• ${\rho }_{(L,★)}(x)=\bigwedge V_{(L,★)}(x)=\bigwedge \{p\in {\mathrm{Spec}}_{(L,★)}|x\le p\}$,
• $R(L,★)=\left\{{\rho }_{(L,★)}(x)\right|x\in L\}$.

Remark 1.

Clearly:

• for all $x\in L$, $V_{(L,★)}({\rho }_{(L,★)}(x))=V_{(L,★)}(x)$, thus $D_{(L,★)}({\rho }_{(L,★)}(x))=D_{(L,★)}(x)$ and ${\rho }_{(L,★)}({\rho }_{(L,★)}(x))={\rho }_{(L,★)}(x)$, thus:
• ${\mathrm{Spec}}_{(L,★)}\subseteq R(L,★)=\{x\in L|{\rho }_{(L,★)}(x)=x\}$.

If L has a 0, then, for all $a,b\in L$ and any $U\subseteq L$:

• ${\mathrm{Ann}}_{(L,★)}(a)$ and ${\mathrm{Ann}}_{(L,★)}(U)$ will be the (left) annihilator of a and U in L with respect to ★, respectively:

  $${\mathrm{Ann}}_{(L,★)}(a)=\{x\in L|x★a=0\}\mathrm{and}{\mathrm{Ann}}_{(L,★)}(U)=\bigcap _{u\in U}{\mathrm{Ann}}_{(L,★)}(u),$$

  which are ideals of L if ★ is distributive with respect to the join in the first argument (thus order–preserving in the first argument) and principal ideals of L if ★ is distributive with respect to arbitrary joins in the first argument;
• we consider a binary operation ${\to }_{(L,★)}$ and a unary operation ^⊥(L,★) on L defined by:

  $$a{\to }_{(L,★)}b=\bigvee \{x\in L|x★a\le b\}\mathrm{and}{a}^{\perp (L,★)}=a\to 0=\bigvee {\mathrm{Ann}}_{(L,★)}(a)$$

  .

Remark 2.

Let $a,b\in L$, arbitrary.

By the definitions of these operations, $(a{\to }_{(L,★)}b)★a\le b$, in particular $a^{\perp (L,★)}★a=0$.

If ★ is distributive with respect to arbitrary joins in the first argument, then $a{\to }_{(L,★)}b=max\{x\in L|x★a\le b\}$ and $a^{\perp (L,★)}=max{\mathrm{Ann}}_{(L,★)}(a)$, thus ${\mathrm{Ann}}_{(L,★)}(a)={(a^{\perp ★})]}_L$ and, for all $x\in L$:

$x★a\le b$ if and only if $x\le a{\to }_{(L,★)}b$;

in particular: $x★a=0$ if and only if $x\le a^{\perp (L,★)}$.

3. Commutator Lattices, the Topological Structure of Their Prime Spectra and Their Residuated Structure

Commutator lattices are complete lattices endowed with an additional binary operation which is commutative, smaller than its arguments and completely distributive with respect to the join:

Definition 1.

([10,23,29,30]). A commutator lattice is an algebra $(L,\wedge ,\vee ,[·,·],0,1)$ (that we also denote, simply, by $(L,[·,·])$) such that $(L,\wedge ,\vee ,0,1)$ is a complete bounded lattice and $[·,·]$ is a binary operation on L, called commutator, which satisfies the following for all $x,y\in L$ and any family ${(y_i)}_{i\in I}\subseteq L$:

$$[x,y]=[y,x]\le x\wedge y \mathit{and} [x,\underset{i\in I}{\bigvee }{y}_i]=\underset{i\in I}{\bigvee }[x,y_i]$$

.

Note that the commutator $[·,·]$ is distributive with respect to the join in each argument and thus order–preserving in each argument. For any lattice L: $(L,\wedge )$ is a commutator lattice (with the commutator equalling the meet) if and only if L is a frame.

Throughout the rest of this section, $(L,[·,·])$ will be an arbitrary commutator lattice.

3.1. Basics on Commutator Lattices

Let A be an arbitrary member of a variety $\mathcal{V}$ and ${[·,·]}_A$ be its term condition commutator [31]. If ${[·,·]}_A$ is commutative and distributive with respect to arbitrary joins, in particular if $\mathcal{V}$ is congruence–modular, then the congruences of A form an algebraic commutator lattice $(\mathrm{Con}(A),\cap ,\wedge ,{[·,·]}_A,{\Delta }_A,{\nabla }_A)$ [5,6,10]. See in [6] results from the current paper for this particular case of commutator lattices of congruences.

Recall that A is called a semiprime algebra if and only if ${\rho }_{(\mathrm{Con}(A),{[·,·]}_A)}({\Delta }_A)={\Delta }_A$.

Remember, also, that:

• if $\mathcal{V}$ is semidegenerate, then the one–class congruence of A is finitely generated: ${\nabla }_A\in \mathrm{Cp}(\mathrm{Con}(A))$;
• if $\mathcal{V}$ is congruence–modular and semidegenerate, then A is semiprime and ${[\theta ,{\nabla }_A]}_A=\theta$ for all $\theta \in \mathrm{Con}(A)$;
• if $\mathcal{V}$ is congruence–distributive, then the commutator ${[·,·]}_A$ of A equals the intersection.

Let’s get back to arbitrary commutator lattices. We call:

• the elements of ${\mathrm{Spec}}_{(L,[·,·])}$ prime elements of $(L,[·,·])$,
• those of ${\mathrm{Min}}_{(L,[·,·])}$ minimal prime elements of $(L,[·,·])$,
• the sets ${\mathrm{Spec}}_{(L,[·,·])}$ and ${\mathrm{Min}}_{(L,[·,·])}$ the prime spectrum and minimal prime spectrum of $(L,[·,·])$, respectively,
• ${\rho }_{(L,[·,·])}(x)$ the radical of x in $(L,[·,·])$,
• the elements of $R(L,[·,·])$ radical elements of $(L,[·,·])$,
• ${\equiv }_{(L,[·,·])}$ the radical equivalence of $(L,[·,·])$,
• the operations ${\to }_{(L,[·,·])}$ and ${·}^{\perp (L,[·,·])}$ the implication and polar in $(L,[·,·])$, respectively.

Remark 3.

By [10] (Proposition 5.9. (i)):

• ${\equiv }_{(L,[·,·])}\in \mathrm{Con}(L,[·,·])$, i.e., ${\equiv }_{(L,[·,·])}$ is a lattice congruence of L that preserves the commutator operation;
• ${\equiv }_{(L,[·,·])}$ preserves arbitrary joins and satisfies $[a,b]{\equiv }_{(L,[·,·])}a\wedge b$ for all $a,b\in L$;
• $R(L,[·,·])=\{max(x/{\equiv }_{(L,[·,·])})|x\in L\}=\{x\in L|x=max(x/{\equiv }_{(L,[·,·])})\}$;
• $0/{\equiv }_{(L,[·,·])}={(\rho (0)]}_L$;
• for all $x\in L$, ${\rho }_{(L,[·,·])}(x)=max(x/{\equiv }_{(L,[·,·])})=max({\rho }_{(L,[·,·])}(x)/{\equiv }_{(L,[·,·])})=min({[x)}_L$$\cap R(L,[·,·]))$.

By the first two statements above, the quotient commutator lattice of $(L,[·,·])$ through ${\equiv }_{(L,[·,·])}$ is the frame $(L/{\equiv }_{(L,[·,·])},\wedge )$, whose commutator operation equals the meet.

Also, by the items above:

• $0\in R(L,[·,·])$ if and only if ${\rho }_{(L,[·,·])}(0)=0$ if and only if $0/{\equiv }_{(L,[·,·])}=\left\{0\right\}$
• if and only if, for any $x\in L$, ${\rho }_{(L,[·,·])}(x)=0⟺x=0$
• if and only if, for all $a,b\in L$, $[a,b]=0$ is equivalent to $a\wedge b=0$
• if and only if, for all $a\in L$, ${\mathrm{Ann}}_{(L,[·,·])}(a)={\mathrm{Ann}}_{(L,\wedge )}(a)$
• if and only if, for all $U\subseteq L$, ${\mathrm{Ann}}_{(L,[·,·])}(U)={\mathrm{Ann}}_{(L,\wedge )}(U)$.

Also, ${\equiv }_{(L,[·,·])}\cap R{(L,[·,·])}^2={\Delta }_{R(L,[·,·])}$, thus: $R(L,[·,·])=L$ if and only if ${\equiv }_{(L,[·,·])}={\Delta }_L$. See [10] (Remarks 5.10 and 5.11, Proposition 5.15. (i), Lemma 5.18).

If $[·,·]=\wedge$, so that L is a frame, in particular it is distributive, then ${\mathrm{Spec}}_{(L,[·,·])}={\mathrm{Spec}}_{(L,\wedge )}=\mathrm{Mi}(L)$. Since all elements of an algebraic lattice are meets of strictly meet–irreducible elements, it follows that, if L is algebraic and $[·,·]=\wedge$, then $R(L,[·,·])=R(L,\wedge )=L$.

By [10] (Proposition 5.15. (ii), (iii)), if $R(L,[·,·])=L$, then $[·,·]=\wedge$, hence, if L is algebraic, then $R(L,[·,·])=L$ if and only if $[·,·]=\wedge$.

By [10] (Lemma 5.7), if $1\in \mathrm{Cp}(L)$ and $[1,1]=1$, then $1/{\equiv }_{(L,[·,·])}=\left\{1\right\}$.

Lemma 1.

Let $x\in L\setminus \left\{1\right\}$. If L is algebraic, then: $x\in {\mathrm{Spec}}_{(L,[·,·])}$ if and only if, for any $a,b\in \mathrm{Cp}(L)$, if $[a,b]\le x$, then $a\le x$ or $b\le x$.

Proof. 

The left–to–right implication is clear. For the converse, assume that all $a,b\in \mathrm{Cp}(L)$ such that $[a,b]\le x$ satisfy $a\le x$ or $b\le x$ and assume by absurdum that x is not prime, so that there exist $u,v\in L$ with $[u,v]\le x$, but $u\nleqq x$ and $v\nleqq x$. Since L is algebraic, $u=\bigvee ({(u]}_L\cap \mathrm{Cp}(L))$ and $v=\bigvee ({(v]}_L\cap \mathrm{Cp}(L))$. Then there exist $c,d\in \mathrm{Cp}(L)$ such that $c\nleqq x$ and $d\nleqq x$, but $c\le u$ and $d\le v$, so that $[c,d]\le [u,v]\le x$, which contradicts the hypothesis of this implication. □

3.2. The Stone Topology on the Prime Spectrum, Along with the Flat Topology in the Compact Case

We call an $r\in L$ a semiprime element of $(L,[·,·])$ if and only if, for all $a\in L$, if $[a,a]\le r$, then $a\le r$.

Lemma 2.

(1) If $[1,1]=1$, then ${\mathrm{Max}}_L\subseteq {\mathrm{Spec}}_{(L,[·,·])}$.

(2) 

An element of L is radical in $(L,[·,·])$ if and only if it is semiprime.

(3) 

${\mathrm{Spec}}_{(L,[·,·])}=(\mathrm{Mi}(L)\cap R(L,[·,·]))\setminus \left\{1\right\}$.

Proof. 

(1) This is [10] (Lemma 5.2. (iii)).

(2) This is [11] (Lemma 4.7).

(3) By (2) and [10] (Lemma 5.2. (ii)), according to which the prime elements of $(L,[·,·])$ are exactly the semiprime members of $\mathrm{Mi}(L)\setminus \left\{1\right\}$. □

Recall, also, from [10] (Lemma 5.2. (iv)) that, if $1\in \mathrm{Cp}(L)$, then, for any $x\in L\setminus \left\{1\right\}$, there exists an $m\in {\mathrm{Max}}_L$ such that $x\le m$.

We denote by ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}=\left\{D_{(L,[·,·])}(x)\right|x\in L\}$.

Proposition 1.

${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ is a topology on ${\mathrm{Spec}}_{(L,[·,·])}$, which satisfies, for all $a,b\in L$ and any family ${(a_i)}_{i\in I}\subseteq L$:

(1) 

${\rho }_{(L,[·,·])}({\rho }_{(L,[·,·])}(a))={\rho }_{(L,[·,·])}(a)$, $D_{(L,[·,·])}({\rho }_{(L,[·,·])}(a))=D_{(L,[·,·])}(a)$ and

$V_{(L,[·,·])}({\rho }_{(L,[·,·])}(a))=V_{(L,[·,·])}(a)$;

(2) 

$D_{(L,[·,·])}(a)\subseteq D_{(L,[·,·])}(b)$ if and only if $V_{(L,[·,·])}(a)\supseteq V_{(L,[·,·])}(b)$ if and only if ${\rho }_{(L,[·,·])}(a)$$\le {\rho }_{(L,[·,·])}(b)$;

$D_{(L,[·,·])}(a)=D_{(L,[·,·])}(b)$ if and only if $V_{(L,[·,·])}(a)=V_{(L,[·,·])}(b)$ if and only if ${\rho }_{(L,[·,·])}(a)$$={\rho }_{(L,[·,·])}(b)$;

(3) 

$a\le b$ implies ${\rho }_{(L,[·,·])}(a)\le {\rho }_{(L,[·,·])}(b)$;

$a\le {\rho }_{(L,[·,·])}(a)$; ${\rho }_{(L,[·,·])}(a)=0$ implies $a=0$;

$D_{(L,[·,·])}(1)={\mathrm{Spec}}_{(L,[·,·])}=V_{(L,[·,·])}(0)$ and $D_{(L,[·,·])}(0)=\varnothing =V_{(L,[·,·])}(1)$;

(4) 

if $0\in R(L,[·,·])$, then: $D_{(L,[·,·])}(a)=\varnothing$ if and only if $V_{(L,[·,·])}(a)={\mathrm{Spec}}_{(L,[·,·])}$ if and only if ${\rho }_{(L,[·,·])}(a)=0$ if and only if $a=0$;

(5) 

if $1/{\equiv }_{(L,[·,·])}=\left\{1\right\}$, in particular if $1\in \mathrm{Cp}(L)$ and $[1,1]=1$, then: $D_{(L,[·,·])}(a)={\mathrm{Spec}}_{(L,[·,·])}$ if and only if $V_{(L,[·,·])}(a)=\varnothing$ if and only if ${\rho }_{(L,[·,·])}(a)=1$ if and only if $a=1$;

(6) 

${\rho }_{(L,[·,·])}([a,b])={\rho }_{(L,[·,·])}(a\wedge b)={\rho }_{(L,[·,·])}(a)\wedge {\rho }_{(L,[·,·])}(b)$;

$D_{(L,[·,·])}([a,b])=D_{(L,[·,·])}(a\wedge b)=D_{(L,[·,·])}(a)\cap D_{(L,[·,·])}(b)$;

$V_{(L,[·,·])}([a,b])=V_{(L,[·,·])}(a\wedge b)=V_{(L,[·,·])}(a)\cup V_{(L,[·,·])}(b)$;

${\displaystyle D_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)=\bigcup _{i\in I}{D}_{(L,[·,·])}(a_i)}$ and ${\displaystyle V_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)=\bigcap _{i\in I}{V}_{(L,[·,·])}(a_i)}$;

${\displaystyle {\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)={\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }{\rho }_{(L,[·,·])}(a_i))}$;

(7) 

if the lattice L is algebraic, then ${\displaystyle V_{(L,[·,·])}(a)=\bigcap _{x\in {(a]}_L\cap \mathrm{Cp}(L)}{V}_{(L,[·,·])}(x)}$ and ${\displaystyle D_{(L,[·,·])}(a)=\bigcup _{x\in {(a]}_L\cap \mathrm{Cp}(L)}{D}_{(L,[·,·])}(x)}$, therefore the topology ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·]}$ has $\left\{D_{(L,[·,·])}(x)\right|x\in \mathrm{Cp}(L)\}$ as a basis.

Proof. 

(1) By Remark 1.

(2), (3) Clear.

(4) Assume that $0\in R(L,[·,·])$, so that $0/{\equiv }_{(L,[·,·])}=\left\{0\right\}$ and ${\rho }_{(L,[·,·])}(0)=0$, thus, by (2), (3): $D_{(L,[·,·])}(a)=\varnothing =D_{(L,[·,·])}(0)$ if and only if $V_{(L,[·,·])}(a)={\mathrm{Spec}}_{(L,[·,·])}=V_{(L,[·,·])}(0)$ if and only if ${\rho }_{(L,[·,·])}(a)={\rho }_{(L,[·,·])}(0)$ if and only if ${\rho }_{(L,[·,·])}(a)=0$; also, ${\rho }_{(L,[·,·])}(a)={\rho }_{(L,[·,·])}(0)$ if and only if $a{\equiv }_{(L,[·,·])}0$ if and only if $a\in 0/{\equiv }_{(L,[·,·])}$ if and only if $a\in \left\{0\right\}$ if and only if $a=0$.

(5) Assume that $1/{\equiv }_{(L,[·,·])}=\left\{1\right\}$, which holds if $1\in \mathrm{Cp}(L)$ and $[1,1]=1$. Then, by (2), (3): $D_{(L,[·,·])}(a)={\mathrm{Spec}}_{(L,[·,·])}=D_{(L,[·,·])}(1)$ if and only if $V_{(L,[·,·])}(a)=\varnothing =V_{(L,[·,·])}(1)$ if and only if ${\rho }_{(L,[·,·])}(a)={\rho }_{(L,[·,·])}(1)$ if and only if ${\rho }_{(L,[·,·])}(a)=1$; also, ${\rho }_{(L,[·,·])}(a)={\rho }_{(L,[·,·])}(1)$ if and only if $a{\equiv }_{(L,[·,·])}1$ if and only if $a\in 1/{\equiv }_{(L,[·,·])}$ if and only if $a\in \left\{1\right\}$ if and only if $a=1$.

(6) We have $[a,b]{\equiv }_{(L,[·,·])}a\wedge b$, which means that ${\rho }_{(L,[·,·])}([a,b])={\rho }_{(L,[·,·])}(a\wedge b)$, thus $V_{(L,[·,·])}([a,b])=V_{(L,[·,·])}(a\wedge b)$ and $D_{(L,[·,·])}([a,b])=D_{(L,[·,·])}(a\wedge b)$ by (2). For any $p\in {\mathrm{Spec}}_{(L,[·,·])}$, we have, since $[a,b]\le a\wedge b$: $p\in V_{(L,[·,·])}([a,b])$ if and only if $[a,b]\le p$ if and only if $a\le p$ or $b\le p$ if and only if $p\in V_{(L,[·,·])}(a)$ or $p\in V_{(L,[·,·])}(b)$ if and only if $p\in V_{(L,[·,·])}(a)\cup V_{(L,[·,·])}(b)$. Hence $V_{(L,[·,·])}([a,b])=V_{(L,[·,·])}(a)\cup V_{(L,[·,·])}(b)$, thus $D_{(L,[·,·])}([a,b])={\mathrm{Spec}}_{(L,[·,·])}\setminus (V_{(L,[·,·])}(a)\cup V_{(L,[·,·])}(b))=({\mathrm{Spec}}_{(L,[·,·])}\setminus V_{(L,[·,·])}(a))\cap ({\mathrm{Spec}}_{(L,[·,·])}\setminus V_{(L,[·,·])}(b))=D_{(L,[·,·])}(a)\cap D_{(L,[·,·])}(b)$.

For any $p\in {\mathrm{Spec}}_{(L,[·,·])}$: ${\displaystyle p\in V_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)}$ if and only if ${\displaystyle \underset{i\in I}{\bigvee }{a}_i\le p}$ if and only if, for all $i\in I$, $a_i\le p$ if and only if, for all $i\in I$, $p\in V_{(L,[·,·])}(a_i)$ if and only if ${\displaystyle p\in \bigcap _{i\in I}{V}_{(L,[·,·])}(a_i)}$. Hence ${\displaystyle V_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)=\bigcap _{i\in I}{V}_{(L,[·,·])}(a_i)}$, thus ${\displaystyle D_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)={\mathrm{Spec}}_{(L,[·,·])}\setminus V_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)={\mathrm{Spec}}_{(L,[·,·])}\setminus (\bigcap _{i\in I}{V}_{(L,[·,·])}(a_i))=\bigcup _{i\in I}({\mathrm{Spec}}_{(L,[·,·])}\setminus V_{(L,[·,·])}(a_i))=\bigcup _{i\in I}{D}_{(L,[·,·])}(a_i)}$.

(7) As shown by (3) and (6), ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}=\left\{D_{(L,[·,·])}(x)\right|x\in L\}$ is a topology on ${\mathrm{Spec}}_{(L,[·,·])}$.

Now assume that L is algebraic. Then ${\displaystyle a=\bigvee ({(a]}_L\cap \mathrm{Cp}(L))=\underset{x\in {(a]}_L\cap \mathrm{Cp}(L)}{\bigvee }x}$, hence, by (6), ${\displaystyle V_{(L,[·,·])}(a)=\bigcap _{x\in {(a]}_L\cap \mathrm{Cp}(L)}{V}_{(L,[·,·])}(x)}$ and ${\displaystyle D_{(L,[·,·])}(a)=\bigcup _{x\in {(a]}_L\cap \mathrm{Cp}(L)}{D}_{(L,[·,·])}(x)}$. Again by (6), for any $x,y\in \mathrm{Cp}(L)$, we have $D_{(L,[·,·])}(x)\cap D_{(L,[·,·])}(y)=D_{(L,[·,·])}(x\vee y)$ and $x\vee y\in \mathrm{Cp}(L)$. Hence the topology ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ has $\left\{D_{(L,[·,·])}(x)\right|x\in \mathrm{Cp}(L)\}$ as a basis. □

We call ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ the Stone topology or the spectral topology on ${\mathrm{Spec}}_{(L,[·,·])}$.

Remark 4.

The Stone topology ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ on ${\mathrm{Spec}}_{(L,[·,·])}$ induces the topology ${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}=\{D_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}|x\in L\}$ on ${\mathrm{Min}}_{(L,[·,·])}$, having $\{V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}|x\in L\}$ as the family of closed sets and, if L is algebraic, then the set $\{D_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}|x\in \mathrm{Cp}(L)\}$ as a basis. We call ${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$ the Stone or spectral topology on ${\mathrm{Min}}_{(L,[·,·])}$.

If ${\mathrm{Max}}_L\subseteq {\mathrm{Spec}}_{(L,[·,·])}$, in particular if $[1,1]=1$ (see Lemma 2. (1)), then the Stone topology ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ on ${\mathrm{Spec}}_{(L,[·,·])}$ induces the Stone or spectral topology on ${\mathrm{Max}}_L$: ${\mathcal{S}}_{\mathrm{Max},L}=\{D_{(L,[·,·])}(x)\cap {\mathrm{Max}}_L|x\in L\}=\{{\mathrm{Max}}_L\setminus {[x)}_L|x\in L\}$, which has $\{D_{(L,[·,·])}(x)\cap {\mathrm{Max}}_L|x\in \mathrm{Cp}(L)\}=\{{\mathrm{Max}}_L\setminus {[x)}_L|x\in \mathrm{Cp}(L)\}$ as a basis if L is algebraic.

Remark 5.

If L is compact, then:

• the Stone topology on ${\mathrm{Spec}}_{(L,[·,·])}$ coincides with its basis $\left\{D_{(L,[·,·])}(x)\right|x\in \mathrm{Cp}(L)=L\}$;
• ${\mathcal{F}}_{\mathrm{Spec},(L,[·,·])}:=\left\{V_{(L,[·,·])}(x)\right|x\in \mathrm{Cp}(L)=L\}$ is also a topology on ${\mathrm{Spec}}_{(L,[·,·])}$.

We call ${\mathcal{F}}_{\mathrm{Spec},(L,[·,·])}$ the flat topology or the inverse topology on ${\mathrm{Spec}}_{(L,[·,·])}$. Of course, it satisfies the properties in Remark 4, as well.

Remark 6.

For any $a,b\in L$ and any $r\in R(L,[·,·])$, we have:

• $a/{\equiv }_{(L,[·,·])}\le b/{\equiv }_{(L,[·,·])}$ if and only if ${\rho }_{(L,[·,·])}(a)\le {\rho }_{(L,[·,·])}(b)$;
• $a/{\equiv }_{(L,[·,·])}\le r/{\equiv }_{(L,[·,·])}$ if and only if $a\le r$; hence ${(r/{\equiv }_{(L,[·,·])}]}_{L/{\equiv }_{(L,[·,·])}}={(r]}_L/{\equiv }_{(L,[·,·])}$.

Indeed, $a/{\equiv }_{(L,[·,·])}\le b/{\equiv }_{(L,[·,·])}$ if and only if $a/{\equiv }_{(L,[·,·])}\wedge b/{\equiv }_{(L,[·,·])}=a/{\equiv }_{(L,[·,·])}$ if and only if $(a\wedge b)/{\equiv }_{(L,[·,·])}=a/{\equiv }_{(L,[·,·])}$ if and only if ${\rho }_{(L,[·,·])}(a\wedge b)={\rho }_{(L,[·,·])}(a)$ if and only if ${\rho }_{(L,[·,·])}(a)\wedge {\rho }_{(L,[·,·])}(b)={\rho }_{(L,[·,·])}(a)$ if and only if ${\rho }_{(L,[·,·])}(a)\le {\rho }_{(L,[·,·])}(b)$.

$a\le r$ obviously implies $a/{\equiv }_{(L,[·,·])}\le r/{\equiv }_{(L,[·,·])}$, which implies $a\le {\rho }_{(L,[·,·])}(a)\le {\rho }_{(L,[·,·])}(r)=r$ and thus $a\le r$ by the above.

Following [5], we denote, for any family ${(a_i)}_{i\in I}\subseteq L$, by ${\displaystyle \stackrel{•}{{\displaystyle \underset{i\in I}{\bigvee }}}{a}_i:={\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)}$.

Proposition 2.

$(R(L,[·,·]),\wedge ,\stackrel{•}{\vee },{\rho }_{(L,[·,·])}(0),1)$ is a frame, ordered by the lattice order ≤ of L, isomorphic to $L/{\equiv }_{(L,[·,·])}$.

Proof. 

By Remark 1, for all $a\in L$, we have: $a\in R(L,[·,·])$ if and only if ${\rho }_{(L,[·,·])}(a)=a$ if and only if $a={\rho }_{(L,[·,·])}(x)$ for some $x\in L$.

Let $a,b\in R(L,[·,·])$ and ${(a_i)}_{i\in I}\subseteq R(L,[·,·])$, arbitrary.

Then, by Proposition 1. (6), $a\wedge b={\rho }_{(L,[·,·])}(a)\wedge {\rho }_{(L,[·,·])}(b)={\rho }_{(L,[·,·])}(a\wedge b)\in R(L,[·,·])$, hence $R(L,[·,·])$ is a meet–subsemilattice of L. Clearly, ${\rho }_{(L,[·,·])}(0)$ and ${\rho }_{(L,[·,·])}(1)=1$ are its bottom and top element, respectively.

${\displaystyle \stackrel{•}{{\displaystyle \underset{i\in I}{\bigvee }}}{a}_i={\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)\in R(L,[·,·])}$. If $I=\varnothing$, then ${\displaystyle \stackrel{•}{{\displaystyle \underset{i\in I}{\bigvee }}}{a}_i={\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)={\rho }_{(L,[·,·])}(0)}$.

We will apply Proposition 1. (3) in what follows.

If $I\ne \varnothing$, then ${\displaystyle a_j\le \stackrel{•}{{\displaystyle \underset{i\in I}{\bigvee }}}{a}_i}$ for all $j\in I$. If, furthermore, $a_i\le b$ for all $i\in I$, then ${\displaystyle \underset{i\in I}{\bigvee }{a}_i\le b}$, thus ${\displaystyle \stackrel{•}{{\displaystyle \underset{i\in I}{\bigvee }}}{a}_i={\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)\le {\rho }_{(L,[·,·])}(b)=b}$. Hence ${\displaystyle \stackrel{•}{{\displaystyle \underset{i\in I}{\bigvee }}}{a}_i}$ is the supremum of in ${(a_i)}_{i\in I}$ in $R(L,[·,·])$. Thus $(R(L,[·,·]),\wedge ,\stackrel{•}{\vee })$ is a complete lattice.

By Proposition 1. (6), ${\displaystyle a\wedge (\stackrel{•}{{\displaystyle \underset{i\in I}{\bigvee }}}{a}_i)={\rho }_{(L,[·,·])}(a)\wedge {\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)={\rho }_{(L,[·,·])}([a,\underset{i\in I}{\bigvee }{a}_i])={\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }[a,a_i])={\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }{\rho }_{(L,[·,·])}([a,a_i]))={\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }{\rho }_{(L,[·,·])}(a\wedge a_i))=}$${\displaystyle {\rho }_{(L,[·,·])}(\underset{i\in I}{\bigvee }(a\wedge a_i))=\stackrel{•}{{\displaystyle \underset{i\in I}{\bigvee }}}(a\wedge a_i)}$. Hence $(R(L,[·,·]),\wedge ,\stackrel{•}{\vee })$ is a frame, which also follows from the lattice isomorphism we establish next.

Let $\phi :L/{\equiv }_{(L,[·,·])}\to R(L,[·,·])$, defined by $\phi (x/{\equiv }_{(L,[·,·])})={\rho }_{(L,[·,·])}(x)$ for all $x\in L$.

By Remark 6, for all $x,y\in L$, $x/{\equiv }_{(L,[·,·])}=y/{\equiv }_{(L,[·,·])}$ if and only if ${\rho }_{(L,[·,·])}(x)={\rho }_{(L,[·,·])}(y)$ if and only if $\phi (x/{\equiv }_{(L,[·,·])})=\phi (y/{\equiv }_{(L,[·,·])})$, thus $\phi$ is well defined and injective. By Remark 1, $\phi$ is also surjective.

Again by Remark 6, for all $x,y\in L$, $x/{\equiv }_{(L,[·,·])}\le y/{\equiv }_{(L,[·,·])}$ if and only if ${\rho }_{(L,[·,·])}(x)\le {\rho }_{(L,[·,·])}(y)$ if and only if $\phi (x/{\equiv }_{(L,[·,·])})\le \phi (y/{\equiv }_{(L,[·,·])})$. Therefore $\phi$ is an order isomorphism. □

Proposition 3.

${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ is a bounded sublattice of the Boolean algebra $\mathcal{P}({\mathrm{Spec}}_{(L,[·,·])})$ closed with respect to arbitrary unions, thus it is a frame, and it is isomorphic to $L/{\equiv }_{(L,[·,·])}$.

Its set $\left\{V_{(L,[·,·])}(x)\right|x\in L\}$ of closed sets, that is ${\mathcal{F}}_{\mathrm{Spec},(L,[·,·])}$ in the particular case when L is compact, is a bounded sublattice of $\mathcal{P}({\mathrm{Spec}}_{(L,[·,·])})$ closed with respect to arbitrary intersections, dually isomorphic to ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$.

Proof. 

By Proposition 1. (3) & (6), ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ is a bounded sublattice of $\mathcal{P}({\mathrm{Spec}}_{(L,[·,·])})$ and it is closed with respect to arbitrary unions.

Let $\psi :{\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}\to R(L,[·,·])$, defined by $\psi (D_{(L,[·,·])}(x))={\rho }_{(L,[·,·])}(x)$ for all $x\in L$.

By Proposition 1. (2), for all $x,y\in L$, $D_{(L,[·,·])}(x)=D_{(L,[·,·])}(y)$ if and only if ${\rho }_{(L,[·,·])}(x)={\rho }_{(L,[·,·])}(y)$ if and only if $\psi (D_{(L,[·,·])}(x))=\psi (D_{(L,[·,·])}(y))$, hence $\psi$ is well defined and injective. By Remark 1, it is also surjective.

Again by Proposition 1. (2), for all $x,y\in L$, $D_{(L,[·,·])}(x)\subseteq D_{(L,[·,·])}(y)$ if and only if ${\rho }_{(L,[·,·])}(x)\le {\rho }_{(L,[·,·])}(y)$ if and only if $\psi (D_{(L,[·,·])}(x))\le \psi (D_{(L,[·,·])}(y))$. Therefore $\psi$ is an order isomorphism.

Hence the lattice ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ is isomorphic to $R(L,[·,·])$ and thus to $L/{\equiv }_{(L,[·,·])}$ by Proposition 2.

The map $D_{(L,[·,·])}(x)\mapsto V_{(L,[·,·])}(x)$ for all $x\in L$ is the restriction to ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ of the dual Boolean automorphism $S\mapsto {\mathrm{Spec}}_{(L,[·,·])}\setminus S$ of $\mathcal{P}({\mathrm{Spec}}_{(L,[·,·])})$, thus it is an order isomorphism. □

3.3. On the Residuated Structure of a Commutator Lattice

Throughout the rest of this section, we denote ${\to }_{(L,[·,·])}$, ${·}^{\perp (L,[·,·])}$, ${\equiv }_{(L,[·,·])}$ and ${\rho }_{(L,[·,·])}$ simply by →, ${·}^{\perp }$, ≡ and $\rho$, respectively.

Since the commutator is distributive with respect to arbitrary joins in each argument, the properties in Remark 2 hold for $★=[·,·]$. Hence, in the particular case when $[·,·]$ is associative, $(L,\wedge ,\vee ,[·,·],\to ,0,1)$ is a (bounded commutative integral) residuated lattice, with the negation equalling ${·}^{\perp }$.

Lemma 3.

If $\rho (0)=0$, then $x^{\perp }\in R(L,[·,·])$ for any $x\in L$.

Proof. 

Let $a,b\in L$ such that $[a,a]\le b^{\perp }$. Then, by Remark 3 and the fact that $\rho (0)=0$, $\left[\right[a,a],b]=0$, which is equivalent to $\rho ([[a,a],b])=0$, that is $\rho (a\wedge b)=0$, that is $\rho ([a,b])=0$, which means that $[a,b]=0$, which in turn is equivalent to $a\le b^{\perp }$. Hence $b^{\perp }$ is a semiprime and thus a radical element of L by Lemma 2. (2). □

Lemma 4.

For any $a,b,c\in L$:

(1) 

$b\le a\to b$;

(2) 

$(a\vee c)\to (b\vee c)=a\to (b\vee c)$.

Proof. 

(1) $[b,a]\le b\wedge a\le b$, thus $b\le a\to b$ by Remark 2.

(2) We will apply Remark 2. For all $x\in L$, we have, since $[x,c]\le c\le b\vee c$: $x\le (a\vee c)\to (b\vee c)$ if and only if $[x,a\vee c]\le b\vee c$ if and only if $[x,a]\vee [x,c]\le b\vee c$ if and only if $[x,a]\le b\vee c$ if and only if $x\le a\to (b\vee c)$. By taking $x=(a\vee c)\to (b\vee c)$ and then $x=a\to (b\vee c)$ in the previous equivalences, we get: $(a\vee c)\to (b\vee c)=a\to (b\vee c)$. □

Let $a\in L$, arbitrary, and let us denote, for all $x,y\in {[a)}_L$, by ${[x,y]}_a:=[x,y]\vee a$. Then $({[a)}_L,{[·,·]}_a)$ is a commutator lattice.

Note that, if A is an algebra and $\theta \in \mathrm{Con}(A)$ is such that the commutator ${[·,·]}_{A/\theta }$ of the quotient algebra $A/\theta$ satisfies:

${[(\alpha \vee \theta )/\theta ,(\beta \vee \theta )/\theta ]}_{A/\theta }=({[\alpha ,\beta ]}_A\vee \theta )/\theta$ for all $\alpha ,\beta \in \mathrm{Con}(A)$ or, equivalently:

${[\alpha /\theta ,\beta /\theta ]}_{A/\theta }=({[\alpha ,\beta ]}_A\vee \theta )/\theta$ for all $\alpha ,\beta \in {[\theta )}_{\mathrm{Con}(A)}$,

then the lattice isomorphism $\alpha \mapsto \alpha /\theta$ between the lattice reducts of the commutator lattices $({[\theta )}_{\mathrm{Con}(A)},{[·,·]}_{\theta })$ and $(\mathrm{Con}(A/\theta ),{[·,·]}_{A/\theta })$ also preserves the commutators.

In particular, if A is a member of a congruence–modular variety, then any congruence $\theta$ of A is such that the commutator of the quotient algebra $A/\theta$ is defined as above and hence the commutator lattices $(\mathrm{Con}(A/\theta ),{[·,·]}_{A/\theta })$ and $({[\theta )}_{\mathrm{Con}(A)},{[·,·]}_{\theta })$ are isomorphic.

This is why, for any $a\in L$, the commutator lattice $({[a)}_L,{[·,·]}_a)$ is called the quotient of the commutator lattice $(L,[·,·])$ through a.

For all $x,y\in {[a)}_L$:

$x{\to }_{({[a)}_L,{[·,·]}_a)}y=max\{u\in {[a)}_L|{[u,x]}_a\le y\}=max\{u\in {[a)}_L|[u,x]\vee a\le y\}=max\{u\in {[a)}_L|[u,x]\le y\}$ and

$x^{\perp ({[a)}_L,{[·,·]}_a)}=x{\to }_{({[a)}_L,{[·,·]}_a)}a=max\{u\in {[a)}_L|{[u,x]}_a\le a\}=max\{u\in {[a)}_L|{[u,x]}_a=a\}=max\{u\in {[a)}_L|[u,x]\vee a=a\}=max\{u\in {[a)}_L|[u,x]\le a\}$.

Remark 7.

For any $a\in L$ and any $x,y\in {[a)}_L$:

• $x{\to }_{({[a)}_L,{[·,·]}_a)}y=x\to y$, that is ${\to }_{({[a)}_L,{[·,·]}_a)}=\to ={\to }_{(L,[·,·])}$: the implication in $({[a)}_L,{[·,·]}_a)$ coincides to that in $(L,[·,·])$ and hence:
• $x^{\perp ({[a)}_L,{[·,·]}_a)}=x{\to }_{({[a)}_L,{[·,·]}_a)}a=x\to a$.

Indeed, by Lemma 4. (1), $a\le y\le x\to y=max\{u\in L|[u,x]\le y\}$, so $max\{u\in L|[u,x]\le y\}\in {[a)}_L$, thus:

$x\to y=max\{u\in L|[u,x]\le y\}=max\{u\in L|[u,x]\le y\}\in {[a)}_L=x{\to }_{({[a)}_L,{[·,·]}_a)}y$.

Remark 8.

For any $a,x,y\in L$:

• $(x\vee a){\to }_{({[a)}_L,{[·,·]}_a)}(y\vee a)=(x\vee a)\to (y\vee a)=x\to (y\vee a)$;
• ${(x\vee a)}^{\perp ({[a)}_L,{[·,·]}_a)}=x\to a$.

Indeed, by Remark 7 and the fact that $[u,a]\le a\le y\vee a$ for all $u\in L$, we have:

$(x\vee a){\to }_{({[a)}_L,{[·,·]}_a)}(y\vee a)=(x\vee a)\to (y\vee a)=max\{u\in L|[u,x\vee a]\le y\vee a\}=max\{u\in L|[u,x]\vee [u,a]\le y\vee a\}=max\{u\in L|[u,x]\le y\vee a\}=x\to (y\vee a)$.

By Remark 7, it follows that ${(x\vee a)}^{\perp ({[a)}_L,{[·,·]}_a)}=(x\vee a)\to a=x\to a$.

For any $a\in L$ and any $x\in {[a)}_L$:

${\rho }_{({[a)}_L,{[·,·]}_a)}(x)=\bigcap ({[x)}_L\cap {\mathrm{Spec}}_{({[a)}_L,{[·,·]}_a)})$ and

${\equiv }_{({[a)}_L,{[·,·]}_a)}=\left\{(u,v)\right|u,v\in {[a)}_L,{\rho }_{({[a)}_L,{[·,·]}_a)}(u)={\rho }_{({[a)}_L,{[·,·]}_a)}(v)\}$.

Remark 9.

For any $a\in L$:

• ${\mathrm{Spec}}_{({[a)}_L,{[·,·]}_a)}={[a)}_L\cap {\mathrm{Spec}}_{(L,[·,·])}$,
• for any $x\in {[a)}_L$, ${\rho }_{({[a)}_L,{[·,·]}_a)}(x)=\rho (x)$;
• $R({[a)}_L,{[·,·]}_a)={[a)}_L\cap R(L,[·,·])$;
• ${\equiv }_{({[a)}_L,{[·,·]}_a)}={[a)}_L^2\cap \equiv$.

Indeed, it can be easily verified, with the definition, that an element of ${[a)}_L$ is prime with respect to ${[·,·]}_a$ if and only if it is prime with respect to $[·,·]$.

Hence the radical in $({[a)}_L,{[·,·]}_a)$ coincides to the radical in $(L,[·,·])$, since, for any $x\in {[a)}_L$: ${\rho }_{({[a)}_L,{[·,·]}_a)}(x)=\bigcap ({[x)}_L\cap {[a)}_L\cap {\mathrm{Spec}}_{(L,[·,·])})=\bigcap ({[x)}_L\cap {\mathrm{Spec}}_{(L,[·,·])})=\rho (x)$.

Thus $R({[a)}_L,[·,·])=\left\{{\rho }_{({[a)}_L,{[·,·]}_a)}(x)\right|x\in {[a)}_L\}=\left\{\rho (x)\right|x\in {[a)}_L\}=\{x\in {[a)}_L|\rho (x)=x\}=\{x\in {[a)}_L|x\in R(L,[·,·])\}={[a)}_L\cap R(L,[·,·])$.

Hence ${\equiv }_{({[a)}_L,{[·,·]}_a)}=\left\{(u,v)\right|u,v\in {[a)}_L,{\rho }_{({[a)}_L,{[·,·]}_a)}(u)={\rho }_{({[a)}_L,{[·,·]}_a)}(v)\}=\left\{(u,v)\right|u,v$$\in {[a)}_L,\rho (u)=\rho (v)\}=\{(u,v)\in {[a)}_L^2|(u,v)\in \equiv \}={[a)}_L^2\cap \equiv$.

Remark 10.

For any $a\in L$:

$\mathrm{Cp}(L)\cap {[a)}_L\subseteq \{c\vee a|c\in \mathrm{Cp}(L)\}\subseteq \mathrm{Cp}({[a)}_L)$.

Indeed, any $c\in \mathrm{Cp}(L)\cap {[a)}_L$ satisfies $c=c\vee a$, thus $c\in \{x\vee a|x\in \mathrm{Cp}(L)\}$.

Now let $b\in \mathrm{Cp}(L)$ and let us prove that $b\vee a\in \mathrm{Cp}({[a)}_L)$.

Let ${(x_i)}_{i\in I}$ be a nonempty family of elements of ${[a)}_L$ such that ${\displaystyle b\vee a\le \underset{i\in I}{\bigvee }{x}_i}$.

Then ${\displaystyle b\le \underset{i\in I}{\bigvee }{x}_i}$ and thus, since $b\in \mathrm{Cp}(L)$, there exist $n\in {\mathbb{N}}^{*}$ and $i_1,\dots ,i_n\in I$ such that ${\displaystyle b\le \underset{j=1}{\overset{n}{\bigvee }}{x}_{i_j}}$. But $a\le x_{i_j}$ for each $j\in \overline{1,n}$, hence ${\displaystyle b\vee a\le \underset{j=1}{\overset{n}{\bigvee }}{x}_{i_j}}$. Therefore $b\vee a$ is compact in the sublattice ${[a)}_L$ of L.

Remark 11.

Assume that L is algebraic, $1\in \mathrm{Cp}(L)$, $\mathrm{Cp}(L)$ is closed with respect to the commutator and, for all $x\in L$, $[x,1]=x$.

Then, by [32] (Corollary 9), for any $a\in L$, $\mathrm{Cp}({[a)}_L)=\{c\vee a|c\in \mathrm{Cp}(L)\}$.

In particular, if $a\in \mathrm{Cp}(L)$, then $\mathrm{Cp}({[a)}_L)=\{c\vee a|c\in \mathrm{Cp}(L)\}\subseteq \mathrm{Cp}(L)\cap {[a)}_L\subseteq \mathrm{Cp}({[a)}_L)$, thus $\mathrm{Cp}({[a)}_L)=\mathrm{Cp}(L)\cap {[a)}_L$.

Lemma 5.

Let $a,b\in L$. Then:

(1) 

$0^{\perp }=1$ and, if $[x,1]=x$ for all $x\in L$, then $1^{\perp }=0$;

(2) 

$a\le b$ implies $b^{\perp }\le a^{\perp }$; $b^{\perp }\le a^{\perp }$ if and only if $a^{\perp \perp }\le b^{\perp \perp }$, so $a^{\perp }=b^{\perp }$ if and only if $a^{\perp \perp }=b^{\perp \perp }$;

(3) 

$a\le a^{\perp \perp }$ and $a^{\perp \perp \perp }=a^{\perp }$;

(4) 

${(a\vee b)}^{\perp }=a^{\perp }\wedge b^{\perp }={(a^{\perp }\wedge b^{\perp })}^{\perp \perp }$;

(5) 

if $\rho (0)=0$, then ${[a,b]}^{\perp }={(a\wedge b)}^{\perp }$ and ${(a\wedge b)}^{\perp \perp }=a^{\perp \perp }\wedge b^{\perp \perp }$;

(6) 

if $\rho (0)=0$, then: $a^{\perp }\le b^{\perp }$ if and only if ${[a,b]}^{\perp }=b^{\perp }$;

(7) 

if $\rho (0)=0$, then: $a\le a^{\perp }$ if and only if $a=0$.

Proof. 

Similar to the proof of [6] (Lemma 6). □

4. The Minimal Prime Spectrum

Throughout this section, $(L,[·,·])$ will be an arbitrary commutator lattice and we will denote ${\to }_{(L,[·,·])}$, ${\perp }^{(L,[·,·])}$, ${\rho }_{(L,[·,·])}$ and ${\equiv }_{(L,[·,·])}$ by →, ^⊥, $\rho$ and ≡, respectively.

Let ${(p_i)}_{i\in I}\subseteq {\mathrm{Spec}}_{(L,[·,·])}$ be a nonempty totally ordered family of prime elements of $(L,[·,·])$ and ${\displaystyle p:=\underset{i\in I}{\bigwedge }{p}_i}$. Then $p\ne 1$ and $p\in R(L,[·,·])$.

Let $a,b\in L$ such that $p=a\wedge b$, thus $p\le a$, $p\le b$ and, for each $j\in I$, ${\displaystyle p_j\ge \underset{i\in I}{\bigwedge }{p}_i=p=a\wedge b\ge [a,b]}$, hence $a\le p_j$ or $b\le p_j$ since $p_j$ is prime.

If $a\le p_i$ for all $i\in I$, then $a\le p$, hence $p=a$.

If there exists $j\in I$ such that $a\nleqq p_j$, then let $S:=\{i\in I|p_i\le p_j\}$ and $G:=\{i\in I|p_j\le p_i\}$, so that $S\cap G=\left\{j\right\}$ and ${\displaystyle \underset{i\in G}{\bigwedge }{p}_i=p_j}$. Since ${(p_i)}_{i\in I}$ is totally ordered, we have $I=S\cup G$, hence ${\displaystyle p=\underset{i\in S}{\bigwedge }{p}_i\cap \underset{i\in G}{\bigwedge }{p}_i=\underset{i\in S}{\bigwedge }{p}_i\cap p_j=\underset{i\in S}{\bigwedge }{p}_i}$. We have $a\nleqq p_i$ for all $i\in S$, therefore $b\le p_i$ for all $i\in S$, thus ${\displaystyle b\le \underset{i\in S}{\bigwedge }{p}_i=p}$, hence $p=b$.

Therefore $p\in \mathrm{Mi}(L)$, hence $p\in {\mathrm{Spec}}_{(L,[·,·])}$ by Lemma 2. (3).

Thus ${\mathrm{Spec}}_{(L,[·,·])}$ is inductively ordered and clearly the same holds for $V_{(L,[·,·])}(x)={[x)}_L\cap {\mathrm{Spec}}_{(L,[·,·])}$ for any $x\in L$, therefore, by Zorn’s Lemma:

• for any $p\in {\mathrm{Spec}}_{(L,[·,·])}$, there exists an $m\in {\mathrm{Min}}_{(L,[·,·])}$ such that $m\le p$, hence $\rho (0)=\bigcap {\mathrm{Spec}}_{(L,[·,·])}=\bigcap {\mathrm{Min}}_{(L,[·,·])}$;
• moreover, for any $x\in L$ and any $p\in V_{(L,[·,·])}(x)={[x)}_L\cap {\mathrm{Spec}}_{(L,[·,·])}$, there exists an $m\in Min(V_{(L,[·,·])}(x))=Min({[x)}_L\cap {\mathrm{Spec}}_{(L,[·,·])})$ such that $m\le p$, hence:

Remark 12.

For any $x\in L$, we have:

• $\rho (x)=\bigcap Min(V_{(L,[·,·])}(x))=\bigcap Min({[x)}_L\cap {\mathrm{Spec}}_{(L,[·,·])})$;
• $D_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing$ if and only if $V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}={\mathrm{Min}}_{(L,[·,·])}$ if and only if ${[x)}_L\cap {\mathrm{Min}}_{(L,[·,·])}={\mathrm{Min}}_{(L,[·,·])}$ if and only if ${\mathrm{Min}}_{(L,[·,·])}\subseteq {[x)}_L$ if and only if $x\le \bigcap {\mathrm{Min}}_{(L,[·,·])}$ if and only if $x\le \rho (0)$ if and only if $\rho (x)=\rho (0)$;
• $D_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}={\mathrm{Min}}_{(L,[·,·])}$ if and only if $V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing$;
  $V_{(L,[·,·])}(x)=\varnothing$ if and only if $\rho (x)=1$, which holds if $x=1$; recall that, if $1\in \mathrm{Cp}(L)$ and $[1,1]=1$, then $1/\equiv =\left\{1\right\}$, so: $\rho (x)=1$ if and only if $x=1$;
  clearly, $V_{(L,[·,·])}(x)=\varnothing$ implies $V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing$; the converse implication holds if and only if ${\mathrm{Min}}_{(L,[·,·])}={\mathrm{Spec}}_{(L,[·,·])}$ if and only if ${\mathrm{Spec}}_{(L,[·,·])}$ is an antichain.

Indeed, ${\mathrm{Spec}}_{(L,[·,·])}$ is an antichain if and only if ${\mathrm{Min}}_{(L,[·,·])}={\mathrm{Spec}}_{(L,[·,·])}$, case in which $V_{(L,[·,·])}(x)=V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}$.

Now, if $V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing$ implies $V_{(L,[·,·])}(x)=\varnothing$, then let us assume by absurdum that ${\mathrm{Min}}_{(L,[·,·])}\ne {\mathrm{Spec}}_{(L,[·,·])}$, that is ${\mathrm{Spec}}_{(L,[·,·])}⊈{\mathrm{Min}}_{(L,[·,·])}$, so that there exists a $p\in {\mathrm{Spec}}_{(L,[·,·])}\setminus {\mathrm{Min}}_{(L,[·,·])}$. But then $V_{(L,[·,·])}(p)\cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing$, while $V_{(L,[·,·])}(p)\ne \varnothing$ since $p\in V_{(L,[·,·])}(p)$; a contradiction.

4.1. Two Conditions on Commutator Lattices

Let us consider the following conditions on $(L,[·,·])$ as an arbitrary commutator lattice:

Condition 1.

L is algebraic, $\mathrm{Cp}(L)$ is closed with respect to the commutator, $1\in \mathrm{Cp}(L)$ and $1/\equiv =\left\{1\right\}$.

If L is compact and $1/\equiv =\left\{1\right\}$, then $(L,[·,·])$ clearly satisfies Condition 1.

As mentioned in Section 3, by [10] (Lemma 5.7), if L is algebraic, $\mathrm{Cp}(L)$ is closed with respect to the commutator, $1\in \mathrm{Cp}(L)$ and $[1,1]=1$, then L satisfies Condition 1.

Thus if L is compact and $[1,1]=1$, then $(L,[·,·])$ satisfies Condition 1.

Condition 2.

All principal ideals of $L/\equiv$ generated by minimal prime elements are minimal prime ideals, that is: for any $p\in {\mathrm{Min}}_{(L/\equiv ,\wedge )}$, we have ${(p]}_{L/\equiv }\in {\mathrm{Min}}_{(\mathrm{Id}(L/\equiv ),\cap )}$.

As we’ve mentioned in [6], in any lattice, a principal ideal is prime if and only if its generator is meet–prime, hence any minimal prime principal ideal is generated by a minimal meet–prime element. So the converse of the implication in Condition 2 always holds, thus Condition 2 is equivalent to:

• for any $p\in L/\equiv$, $p\in {\mathrm{Min}}_{(L/\equiv ,\wedge )}$ if and only if ${(p]}_{L/\equiv }\in {\mathrm{Min}}_{(\mathrm{Id}(L/\equiv ),\cap )}$.

Note that $(L,[·,·])$ satisfies Condition 2 if all prime ideals of $L/\equiv$ are principal, in particular if all ideals of $L/\equiv$ are principal, that is if $L/\equiv$ is compact, which means that $L/\equiv =\mathrm{Cp}(L/\equiv )$, in particular if $L/\equiv =\mathrm{Cp}(L)/\equiv$, in particular if $L=\mathrm{Cp}(L)$, that is if L is compact. Thus $(L,[·,·])$ satisfies Conditions 1 and 2 if L is compact and $1/\equiv =\left\{1\right\}$, in particular if L is compact and $[1,1]=1$.

4.2. m–systems

A subset S of $\mathrm{Cp}(L)$ is called an m–system in $(L,[·,·])$ if and only if, for any $a,b\in S$, there exists a $c\in S$ such that $c\le [a,b]$. For instance, if $0\in S\subseteq \mathrm{Cp}(L)$, then S is an m–system; also, if $1\in \mathrm{Cp}(L)$ and $[1,1]=1$, then $\left\{1\right\}$ is an m–system in $(L,[·,·])$.

Lemma 6.

([11] (Lemma $4.2$)). If L is algebraic and $p\in L\setminus \left\{1\right\}$, then: $p\in {\mathrm{Spec}}_{(L,[·,·])}$ if and only if $\mathrm{Cp}(L)\setminus {(p]}_L$ is an m–system in $(L,[·,·])$.

Lemma 7.

Let S be a nonempty m–system in $(L,[·,·])$ and let $a\in L$ such that $S\cap {(a]}_L=\varnothing$.

(1) 

If L is algebraic, then $Max\{x\in L|a\le x,S\cap {(x]}_L=\varnothing \}\subseteq {\mathrm{Spec}}_{(L,[·,·])}$, in particular, for the case $a=0$, $Max\{x\in L|S\cap {(x]}_L=\varnothing \}\subseteq {\mathrm{Spec}}_{(L,[·,·])}$.

(2) 

If $1\in \mathrm{Cp}(L)$, then the set $Max\{x\in L|a\le x,S\cap {(x]}_L=\varnothing \}$ is nonempty, in particular the set $Max\{x\in L|S\cap {(x]}_L=\varnothing \}$ is nonempty.

Proof. 

(1) This is [11] (Proposition $4.8$).

(2) Let ${(x_i)}_{i\in I}$ be a nonempty chain in ${[a)}_L$ such that $S\cap {(x_i]}_L=\varnothing$ for every $i\in I$. Then ${\displaystyle \underset{i\in I}{\bigvee }{x}_i\in {[a)}_L}$ and ${\displaystyle S\cap {(\underset{i\in I}{\bigvee }{x}_i]}_L=\varnothing }$. Indeed, assuming by absurdum that there exists some ${\displaystyle c\in S\cap {(\underset{i\in I}{\bigvee }{x}_i]}_L}$, it follows that $c\in S\subseteq \mathrm{Cp}(L)$ and ${\displaystyle c\le \underset{i\in I}{\bigvee }{x}_i}$, hence there exist $n\in {\mathbb{N}}^{*}$ and $i_1,\dots ,i_n\in I$ such that ${\displaystyle c\le \underset{j=1}{\overset{n}{\bigvee }}{x}_{i_j}=x_{i_k}}$ for some $k\in \overline{1,n}$. Thus $c\in S\cap {(x_{i_k}]}_L=\varnothing$, a contradiction.

Hence the subset $\{x\in L|a\le x,S\cap {(x]}_L=\varnothing \}$ of L is inductively ordered, thus it has maximal elements by Zorn’s lemma. □

Proposition 4.

If L is algebraic and $1\in \mathrm{Cp}(L)$, then, for any $a\in L$ and any $p\in V_{(L,[·,·])}(a)$, the following are equivalent:

(1) 

$p\in Min(V_{(L,[·,·])}(a))$;

(2) 

$\mathrm{Cp}(L)\setminus {(p]}_L$ is a maximal element of the set of m–systems of $(L,[·,·])$ which are disjoint from ${(a]}_L$.

Proof. 

Since $1\in \mathrm{Cp}(L)$ and $p\in V_{(L,[·,·])}(a)$, that is $p\in {\mathrm{Spec}}_{(L,[·,·])}$ and $a\le p$, we have $1\in \mathrm{Cp}(L)\setminus {(p]}_L\subseteq \mathrm{Cp}(L)\setminus {(a]}_L$, so $(\mathrm{Cp}(L)\setminus {(p]}_L)\cap {(a]}_L=\varnothing$, and, by Lemma 6, $\mathrm{Cp}(L)\setminus {(p]}_L$ is an m–system.

Note that, since any m–system S is included in $\mathrm{Cp}(L)$, S is disjoint from ${(a]}_L$ if and only if $S\subseteq \mathrm{Cp}(L)\setminus {(a]}_L$.

(1)⇒(2): By Zorn’s Lemma, there exists a maximal element M of the set of m–systems of $(L,[·,·])$ which include $\mathrm{Cp}(L)\setminus {(p]}_L$ and are disjoint from ${(a]}_L$, so $\mathrm{Cp}(L)\setminus {(p]}_L\subseteq M$ and clearly M is a maximal element of the set of m–systems of $(L,[·,·])$ which are disjoint from ${(a]}_L$.

By Lemma 7. (1) & (2), there is $q\in Max\{x\in L|a\le x,M\cap {(x]}_L=\varnothing \}\subseteq {\mathrm{Spec}}_{(L,[·,·])}$, so that $q\in V_{(L,[·,·])}(a)$ and $M\cap {(q]}_L=\varnothing$, thus $\mathrm{Cp}(L)\setminus {(p]}_L\subseteq M\subseteq \mathrm{Cp}(L)\setminus {(q]}_L$, hence $\mathrm{Cp}(L)\setminus (\mathrm{Cp}(L)\cap {(p]}_L)=\mathrm{Cp}(L)\setminus {(p]}_L\subseteq \mathrm{Cp}(L)\setminus {(q]}_L=\mathrm{Cp}(L)\setminus (\mathrm{Cp}(L)\cap {(q]}_L)$, therefore $\mathrm{Cp}(L)\cap {(q]}_L\subseteq \mathrm{Cp}(L)\cap {(p]}_L$, thus, since L is algebraic, ${\displaystyle q=\bigvee (\mathrm{Cp}(L)\cap {(q]}_L)\le \bigvee (\mathrm{Cp}(L)\cap {(p]}_L)=p}$, hence $p=q$, so ${(p]}_L={(q]}_L$, thus $\mathrm{Cp}(L)\setminus {(p]}_L=\mathrm{Cp}(L)\setminus {(q]}_L$, therefore $\mathrm{Cp}(L)\setminus {(p]}_L=M=\mathrm{Cp}(L)\setminus {(q]}_L$, so $\mathrm{Cp}(L)\setminus {(p]}_L$ is a maximal element of the set of m–systems of $(L,[·,·])$ which are disjoint from ${(a]}_L$.

(2)⇒(1): Let $r\in Min(V_{(L,[·,·])}(a))$ with $r\le p$.

By Lemma 6, $\mathrm{Cp}(L)\setminus {(r]}_L$ is an m–system, disjoint from ${(a]}_L$ since $(\mathrm{Cp}(L)\setminus {(r]}_L)\cap {(a]}_L\subseteq (\mathrm{Cp}(L)\setminus {(r]}_L)\cap {(r]}_L=\varnothing$, and $\mathrm{Cp}(L)\setminus {(p]}_L\subseteq \mathrm{Cp}(L)\setminus {(r]}_L$. By the hypothesis of this implication, it follows that $\mathrm{Cp}(L)\setminus {(p]}_L=\mathrm{Cp}(L)\setminus {(r]}_L$, that is $\mathrm{Cp}(L)\setminus (\mathrm{Cp}(L)\cap {(p]}_L)=\mathrm{Cp}(L)\setminus (\mathrm{Cp}(L)\cap {(r]}_L)$, therefore $\mathrm{Cp}(L)\cap {(p]}_L=\mathrm{Cp}(L)\cap {(r]}_L$, thus, since L is algebraic, ${\displaystyle p=\bigvee (\mathrm{Cp}(L)\cap {(p]}_L)=\bigvee (\mathrm{Cp}(L)\cap {(r]}_L)=r\in Min(V_{(L,[·,·])}(a))}$. □

Corollary 1.

If L is algebraic and $1\in \mathrm{Cp}(L)$, then, for any $p\in {\mathrm{Spec}}_{(L,[·,·])}$, the following are equivalent:

• $p\in {\mathrm{Min}}_{(L,[·,·])}$;
• $\mathrm{Cp}(L)\setminus {(p]}_L$ is a maximal element of the set of m–systems of $(L,[·,·])$ which do not contain 0.

Proof. 

By Proposition 4 for $a=0$. □

4.3. Preparatives for Characterizing Minimal Primes

We will need the following characterization of minimal prime ideals in bounded distributive lattices.

Lemma 8.

([33]). For any bounded distributive lattice D and any $P\in {\mathrm{Spec}}_{(\mathrm{Id}(D),\cap )}$, the following are equivalent:

• $P\in {\mathrm{Min}}_{(\mathrm{Id}(D),\cap )}$;
• for any $x\in P$, ${\mathrm{Ann}}_{(D,\wedge )}(x)⊈P$.

For what follows, recall from Lemma 2. (3) that the prime elements of $(L,[·,·])$ are the meet–irreducible radical elements in $L\setminus \left\{1\right\}$ and that all annihilators in $(L,[·,·])$ are principal lattice ideals of L. Since $L/\equiv$ is a frame, in particular distributive, the prime elements of the commutator lattice $(L/\equiv ,\wedge )$ are exactly its meet–irreducible elements.

Lemma 9.

If $\rho (0)=0$, then:

(1) 

for any $U\subseteq L$, ${\mathrm{Ann}}_{L/\equiv }(U/\equiv )={\mathrm{Ann}}_{(L,[·,·])}(U)/\equiv$;

(2) 

${\mathrm{Spec}}_{(L/\equiv ,\wedge )}=\{p/\equiv |p\in {\mathrm{Spec}}_{(L,[·,·])}\}$;

(3) 

for all $r\in R(L,[·,·])$, $r/\equiv \cap R(L,[·,·])=\left\{r\right\}$ and $r=max(r/\equiv )$;

(4) 

$p\mapsto p/\equiv$ is an order isomorphism from ${\mathrm{Spec}}_{(L,[·,·])}$ to ${\mathrm{Spec}}_{(L/\equiv ,\wedge )}$;

(5) 

$r\mapsto r/\equiv$ is an order isomorphism from $R(L,[·,·])$ to $R(L/\equiv ,\wedge )$.

Proof. 

(1) By [10] (Lemma $4.2$).

(2) By [10] (Proposition $6.2$).

(3) By [10] (Remark $5.11$).

(4) By (2), (3) and the fact that ${\mathrm{Spec}}_{(L,[·,·])}\subseteq R(L,[·,·])$ and ${\mathrm{Spec}}_{(L/\equiv ,\wedge )}\subseteq R(L/\equiv ,\wedge )$.

(5) By (2) and the definition of radical elements, $R(L/\equiv ,\wedge )=\{r/\equiv |r\in R(L,[·,·])\}$. Moreover, by (3), for any $r\in L$: $r\in R(L,[·,·])$ if and only if $r/\equiv \in R(L/\equiv ,\wedge )$. □

We will follow the reasoning from [5].

Since $[a,b]\equiv a\wedge b$ and thus $[a,b]/\equiv =a/\equiv \wedge b/\equiv$ for all $a,b\in L$, it follows that, for any subset S of L which is closed with respect to the join and the commutator operation, $S/\equiv$ is a sublattice of $L/\equiv$ and thus $S/\equiv$ is a distributive lattice.

Hence, if $\mathrm{Cp}(L)$ is closed with respect to the commutator operation, then $\mathrm{Cp}(L)/\equiv$ is a sublattice of $L/\equiv$ and thus a distributive lattice with 0, since $0\in \mathrm{Cp}(L)$. So, if $1\in \mathrm{Cp}(L)$ and $\mathrm{Cp}(L)$ is closed with respect to the commutator operation, then $\mathrm{Cp}(L)/\equiv$ is a bounded sublattice of $L/\equiv$ and thus a bounded distributive lattice.

Let us consider the maps:

• ${·}^{*}:L\to \mathcal{P}(\mathrm{Cp}(L)/\equiv )$, for all $u\in L$, $u^{*}:=(\mathrm{Cp}(L)\cap {(u]}_L)/\equiv$;
• ${·}_{*}:\mathcal{P}(\mathrm{Cp}(L)/\equiv )\to L$, for all $S\subseteq \mathrm{Cp}(L)/\equiv$, $S_{*}:=\bigvee \{a\in \mathrm{Cp}(L)|a/\equiv \in S\}$.

Remark 13.

The maps ${·}^{*}$ and ${·}_{*}$ are order–preserving, since, for all $u,v\in L$ and all $X,Y\in \mathcal{P}(\mathrm{Cp}(L)/\equiv )$:

$u\le v$ implies ${(u]}_L\subseteq {(v]}_L$, thus $\mathrm{Cp}(L)\cap {(u]}_L\subseteq \mathrm{Cp}(L)\cap {(v]}_L$, so $u^{*}\subseteq v^{*}$;

$X\subseteq Y$ implies that $\{a\in \mathrm{Cp}(L)|a/\equiv \in X\}\subseteq \{b\in \mathrm{Cp}(L)|b/\equiv \in Y\}$, thus $X_{*}=\bigvee \{a\in \mathrm{Cp}(L)|a/\equiv \in X\}\le \bigvee \{b\in \mathrm{Cp}(L)|b/\equiv \in Y\}=Y_{*}$.

If $\mathrm{Cp}(L)$ is closed with respect to the commutator, then the restriction ${·}_{*}{\mid }_{\mathrm{Id}(\mathrm{Cp}(L)/\equiv )}:\mathrm{Id}(\mathrm{Cp}(L)/\equiv )\to L$ will be denoted by ${·}_{*}$, as well. Note that, in this case, for every $I\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$, $0\in \{a\in \mathrm{Cp}(L)|a/\equiv \in I\}$.

Remark 14.

Let $u\in L$. Then ${(u]}_L/\equiv \subseteq {(u/\equiv ]}_{L/\equiv }$ since, for any $a\in L$, $a\le u$ implies $a/\equiv \le u/\equiv$. Thus $u^{*}=(\mathrm{Cp}(L)\cap {(u]}_L)/\equiv =\mathrm{Cp}(L)/\equiv \cap {(u]}_L/\equiv \subseteq \mathrm{Cp}(L)/\equiv \cap {(u/\equiv ]}_{L/\equiv }$.

Lemma 10.

If $\mathrm{Cp}(L)$ is closed with respect to the commutator, then:

• for any $u\in L$, $u^{*}\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$;
• for any $c\in \mathrm{Cp}(L)$, $c^{*}=\mathrm{Cp}(L)/\equiv \cap {(c/\equiv ]}_{L/\equiv }={(c/\equiv ]}_{\mathrm{Cp}(L)/\equiv }\in \mathrm{PId}(\mathrm{Cp}(L)/\equiv )$.

Proof. 

Since $0\in \mathrm{Cp}(L)\cap {(u]}_L$, we have $0/\equiv \in u^{*}$, so $u^{*}$ is nonempty.

Let $x,y\in u^{*}$, so that $x=a/\equiv$ and $y=b/\equiv$ for some $a,b\in \mathrm{Cp}(L)\cap {(u]}_L$. Then $a\vee b\in \mathrm{Cp}(L)\cap {(u]}_L$, thus $x\vee y=(a\vee b)/\equiv \in u^{*}$.

Now let $y\in u^{*}$ and $x\in \mathrm{Cp}(L)/\equiv$ such that $x\le y$. Then $x=a/\equiv$ and $y=b/\equiv$ for some $a\in \mathrm{Cp}(L)$ and $b\in \mathrm{Cp}(L)\cap {(u]}_L$. Hence $x=x\wedge y=a/\equiv \wedge b/\equiv =(a\wedge b)/\equiv =[a,b]/\equiv \in (\mathrm{Cp}(L)\cap {(u]}_L)/\equiv =u^{*}$ since $[a,b]\in \mathrm{Cp}(L)$ and $[a,b]\le b\le u$, so $[a,b]\in {(u]}_L$.

Therefore $u^{*}\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$.

By Remark 14, $c^{*}\subseteq \mathrm{Cp}(L)/\equiv \cap {(c/\equiv ]}_{L/\equiv }={(c/\equiv ]}_{\mathrm{Cp}(L)/\equiv }$ since $c/\equiv \in \mathrm{Cp}(L)/\equiv$, which is a sublattice of $L/\equiv$.

Now let $x\in {(c/\equiv ]}_{\mathrm{Cp}(L)/\equiv }$, so that $x=a/\equiv$ for some $a\in \mathrm{Cp}(L)$ such that $a/\equiv \le c/\equiv$. Then $x=x\wedge c/\equiv =a/\equiv \wedge c/\equiv =(a\wedge c)/\equiv =[a,c]/\equiv \in (\mathrm{Cp}(L)\cap {(c]}_L)/\equiv =c^{*}$ since $[a,c]\in \mathrm{Cp}(L)$ and $[a,c]\le c$, so $[a,c]\in {(c]}_L$. Thus ${(c/\equiv ]}_{\mathrm{Cp}(L)/\equiv }\subseteq c^{*}$. □

If $\mathrm{Cp}(L)$ is closed with respect to the commutator, then the corestriction ${·}^{*}:L\to \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$ will be denoted by ${·}^{*}$, as well.

Lemma 11.

Let $c\in \mathrm{Cp}(L)$ and $S\subseteq \mathrm{Cp}(L)/\equiv$.

• If $c/\equiv \in S$, then $c\le S_{*}$.
• If S is nonempty and closed with respect to the join and to lower bounds, then $c/\equiv \in S$ if and only if $c\le S_{*}$.
  In particular, if $\mathrm{Cp}(L)$ is closed with respect to the commutator and $I\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$, then $c/\equiv \in I$ if and only if $c\le I_{*}$.

Proof. 

If $c/\equiv \in S$, then $c\in \{a\in \mathrm{Cp}(L)|a/\equiv \in S\}$, thus $c\le \bigvee \{a\in \mathrm{Cp}(L)|a/\equiv \in S\}=S_{*}$.

Now assume that S is nonempty and closed with respect to the join and to lower bounds and that $c\le S_{*}=\bigvee \{a\in \mathrm{Cp}(L)|a/\equiv \in S\}$. Since $c\in \mathrm{Cp}(L)$, it follows that there exist $n\in {\mathbb{N}}^{*}$ and $a_1,\dots ,a_n\in \mathrm{Cp}(L)$ such that $a_1/\equiv ,\dots ,a_n/\equiv \in S$ and $c\le a_1\vee \dots \vee a_n$. Then $c/\equiv \le (a_1\vee \dots \vee a_n)/\equiv =a_1/\equiv \vee \dots \vee a_n/\equiv \in S$, hence $c/\equiv \in S$. □

Lemma 12.

(1) If L is algebraic, then $u\le {(u^{*})}_{*}$ and $r={(r^{*})}_{*}$ for all $u\in L$ and $r\in R(L,[·,·])$.

(2)

If $S\subseteq \mathrm{Cp}(L)/\equiv$ is nonempty and closed with respect to the join and to lower bounds, then $S={(S_{*})}^{*}$. In particular, if $\mathrm{Cp}(L)$ is closed with respect to the commutator and $I\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$, then $I={(I_{*})}^{*}$.

Proof. 

(1) Let $u\in L$ and $r\in R(L,[·,·])$.

For any $a\in \mathrm{Cp}(L)\cap {(u]}_L$, $a/\equiv \in u^{*}$, thus $a\le {(u^{*})}_{*}$ by Lemma 11. Since L is algebraic, it follows that $u=\bigvee (\mathrm{Cp}(L)\cap {(u]}_L)\le {(u^{*})}_{*}$.

Since $r\in R(L,[·,·])$, we have ${(r]}_L/\equiv ={(r/\equiv ]}_{L/\equiv }$, thus $r^{*}=(\mathrm{Cp}(L)\cap {(r]}_L)/\equiv =\mathrm{Cp}(L)/\equiv \cap {(r]}_L/\equiv =\mathrm{Cp}(L)/\equiv \cap {(r/\equiv ]}_{L/\equiv }$, hence ${(r^{*})}_{*}=\bigvee \{a\in \mathrm{Cp}(L)|a/\equiv \in r^{*}\}=\bigvee \{a\in \mathrm{Cp}(L)|a/\equiv \in \mathrm{Cp}(L)/\equiv \cap {(r/\equiv ]}_{L/\equiv }\}=\bigvee \{a\in \mathrm{Cp}(L)|a/\equiv \in {(r/\equiv ]}_{L/\equiv }\}=\bigvee \{a\in \mathrm{Cp}(L)|a\in {(r]}_L\}=\bigvee (\mathrm{Cp}(L)\cap {(r]}_L)=r$ since L is algebraic.

(2) By Lemma 11, for any $x\in \mathrm{Cp}(L)/\equiv$, we have: $x\in {(S_{*})}^{*}=\mathrm{Cp}(L)/\equiv \cap {(S_{*}]}_L/\equiv$ if and only if $x\in {(S_{*}]}_L/\equiv$ if and only if $x=c/\equiv$ for some $c\in \mathrm{Cp}(L)\cap {(S_{*}]}_L$ if and only if $x=c/\equiv$ for some $c\in \mathrm{Cp}(L)$ with $c\le S_{*}$ if and only if $x=c/\equiv$ for some $c\in \mathrm{Cp}(L)$ with $c/\equiv \in S$ if and only if $x\in S$. Therefore ${(S_{*})}^{*}=S$. □

Proposition 5.

If Cp(L) is closed with respect to the commutator, then:

(1) 

the map ${·}_{*}:\mathrm{Id}(\mathrm{Cp}(L)/\equiv )\to L$ is injective;

(2) 

the map ${·}^{*}:L\to \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$ is surjective.

Proof. 

Assume that Cp(L) is closed with respect to the commutator.

(1) Let $I,J\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$ such that $I_{*}=J_{*}$. By Lemma 12. (2), it follows that $I={(I_{*})}^{*}={(J_{*})}^{*}=J$.

(2) Let $I\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$ and denote $u:=I_{*}\in L$. Again by Lemma 12. (2), it follows that $u^{*}={(I_{*})}^{*}=I$. □

Recall that, if $1\in \mathrm{Cp}(L)$ and $[1,1]=1$, then $1/\equiv =\left\{1\right\}$.

Remark 15.

If $1/\equiv =\left\{1\right\}$, then, clearly:

• for any $u\in L\setminus \left\{1\right\}$, we have $1/\equiv \notin u^{*}$;
• $1/\equiv \in 1^{*}$ if and only if $1\in \mathrm{Cp}(L)$.

Now assume that $\mathrm{Cp}(L)$ is closed with respect to the commutator and $1\in \mathrm{Cp}(L)$. If $1/\equiv =\left\{1\right\}$, in particular if $[1,1]=1$, then, for any $I\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$, we have: I is a proper ideal of $\mathrm{Cp}(L)/\equiv$ if and only if $I_{*}\ne 1$.

Indeed, by Lemma 11: $I=\mathrm{Cp}(L)/\equiv$ if and only if $1/\equiv \in I$ if and only if $1\le I_{*}$ if and only if $I_{*}=1$.

Lemma 13.

Assume that L is algebraic, $\mathrm{Cp}(L)$ is closed with respect to the commutator and $1/\equiv =\left\{1\right\}$. Then:

(1) 

for any $p\in {\mathrm{Spec}}_{(L,[·,·])}$, we have $p^{*}\in {\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$;

(2) 

if $1\in \mathrm{Cp}(L)$, that is if L satisfies Condition 1, then, for any $P\in {\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$, we have $P_{*}\in {\mathrm{Spec}}_{(L,[·,·])}$.

Proof. 

(1) Let $p\in {\mathrm{Spec}}_{(L,[·,·])}\subseteq L\setminus \left\{1\right\}$, so that $p^{*}$ is a proper ideal of $\mathrm{Cp}(L)/\equiv$ by Lemma 11 and Remark 15.

Now let $x,y\in \mathrm{Cp}(L)/\equiv$ such that $x\wedge y\in p^{*}$. Then $x=a/\equiv$ and $y=b/\equiv$ for some $a,b\in \mathrm{Cp}(L)$, so that $[a,b]\in \mathrm{Cp}(L)$ by the assumption in the enunciation, and $[a,b]/\equiv =(a\wedge b)/\equiv =a/\equiv \wedge b/\equiv =x\wedge y\in p^{*}$, thus $[a,b]\le {(p^{*})}_{*}=p$ by Lemma 11 and Lemma 12. (1). Since p is a prime element of $(L,[·,·])$, it follows that $a\le p={(p^{*})}_{*}$ or $b\le p={(p^{*})}_{*}$, thus $x=a/\equiv \in p^{*}$ or $y=b/\equiv \in p^{*}$, again by Lemma 11 and Lemma 12. (1). Hence $p^{*}$ is a prime ideal of $\mathrm{Cp}(L)/\equiv$.

(2) Assume that $1\in \mathrm{Cp}(L)$ and let $P\in \mathrm{Spec}(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )$, so that P is a proper ideal of $\mathrm{Cp}(L)/\equiv$ and thus $P_{*}\ne 1$ by Remark 15.

Let $a,b\in \mathrm{Cp}(L)$ such that $[a,b]\le P_{*}$, so that $[a,b]\in \mathrm{Cp}(L)$ and, by Lemma 11 and Lemma 12. (2), $a/\equiv \wedge b/\equiv =(a\wedge b)/\equiv =[a,b]/\equiv \in {(P_{*})}^{*}=P$. Since P is a prime ideal of the lattice $\mathrm{Cp}(L)/\equiv$, it follows that $a/\equiv \in P$ or $b/\equiv \in P$, hence $a\le P_{*}$ or $b\le P_{*}$ by Lemma 11. By Lemma 1, it follows that $P_{*}$ is a prime element of the commutator lattice $(L,[·,·])$. □

Proposition 6.

If L satisfies Condition 1, then the restrictions ${·}^{*}:{\mathrm{Spec}}_{(L,[·,·])}\to {\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$ and ${·}_{*}:{\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}\to {\mathrm{Spec}}_{(L,[·,·])}$ are mutually inverse order isomorphisms.

Proof. 

By Lemma 13, these maps are well defined.

By (1) and (2) from Lemma 12, we have ${(p^{*})}_{*}=p$ for any $p\in {\mathrm{Spec}}_{(L,[·,·])}$ and ${(P_{*})}^{*}=P$ for any $P\in {\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$, respectively. Hence these maps are mutually inverse bijections and thus order isomorphisms by Remark 13. □

Lemma 14.

Assume that $\mathrm{Cp}(L)$ is closed with respect to the commutator and let $a\in L$ and $S\subseteq \mathrm{Cp}(L)/\equiv$.

(1) 

If ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(a^{*})\subseteq S$, then $a^{\perp }\le S_{*}$.

(2) 

If $\rho (0)=0$, $a\in \mathrm{Cp}(L)$ and $S\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$, then: ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(a^{*})\subseteq S$ if and only if $a^{\perp }\le S_{*}$.

Proof. 

Recall that, for any $u,v\in L$, $u/\equiv \wedge v/\equiv =(u\wedge v)/\equiv =[u,v]/\equiv$ and, if $\rho (0)=0$, then $0/\equiv =\left\{0\right\}$, so $u\wedge v=0$ if and only if $[u,v]=0$. Thus:

$a^{\perp }=max{\mathrm{Ann}}_{(L,[·,·])}(a)=\bigvee (\mathrm{Cp}(L)\cap {\mathrm{Ann}}_{(L,[·,·])}(a))=\bigvee \{b\in \mathrm{Cp}(L)|[a,b]=0\}$, so, if $\rho (0)=0$, then $a^{\perp }=\bigvee \{b\in \mathrm{Cp}(L)|a\wedge b=0\}$;

${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(a^{*})={\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}((\mathrm{Cp}(L)\cap {(a]}_L)/\equiv )=\{b/\equiv |b\in \mathrm{Cp}(L),(\forall c\in \mathrm{Cp}(L)\cap {(a]}_L)(b/\equiv \wedge c/\equiv =0/\equiv )\}=\{b/\equiv |b\in \mathrm{Cp}(L),(\forall c\in \mathrm{Cp}(L)\cap {(a]}_L)([b,c]/\equiv =0/\equiv )\}$, so, if $\rho (0)=0$, then ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(a^{*})=\{b/\equiv |b\in \mathrm{Cp}(L),(\forall c\in \mathrm{Cp}(L)\cap {(a]}_L)([b,c]=0)\}=\{b/\equiv |b\in \mathrm{Cp}(L)\cap {\mathrm{Ann}}_{(L,[·,·])}(\mathrm{Cp}(L)\cap {(a]}_L)\}$.

(1) Assume that ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(a^{*})\subseteq S$, that is $b/\equiv \in S$ for any $b\in \mathrm{Cp}(L)$ which satisfies $[b,c]/\equiv =0/\equiv$ for all $c\in \mathrm{Cp}(L)\cap {(a]}_L$.

Now let $b\in \mathrm{Cp}(L)$ such that $[a,b]=0$. Then, for all $c\in \mathrm{Cp}(L)\cap {(a]}_L$, $[b,c]=0$, thus $[b,c]/\equiv =0/\equiv$. Hence $b/\equiv \in S$ and thus $b\le S_{*}$ by Lemma 11.

Therefore $a^{\perp }\le S_{*}$.

(2) Assume that $\rho (0)=0$, $a\in \mathrm{Cp}(L)$ and $S\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$. By (1), we only have to prove the converse implication, so assume that $a^{\perp }\le S_{*}$. Then $a^{\perp }/\equiv \in S$ by Lemma 11, so ${\mathrm{Ann}}_{(L,[·,·])}(a)/\equiv ={(a^{\perp }]}_L/\equiv \subseteq S$.

Since $a\in \mathrm{Cp}(L)$, $a^{*}={(a/\equiv ]}_{\mathrm{Cp}(L)/\equiv }$ according to Lemma 10, thus ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(a^{*})$$={\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}({(a/\equiv ]}_{\mathrm{Cp}(L)/\equiv })={\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(a/\equiv )=\{b/\equiv |b\in \mathrm{Cp}(L),a/\equiv \wedge b/\equiv =0/\equiv \}=\{b/\equiv |b\in \mathrm{Cp}(L),[a,b]/\equiv =0/\equiv \}=\{b/\equiv |b\in \mathrm{Cp}(L),[a,b]=0\}=(\mathrm{Cp}(L)\cap {\mathrm{Ann}}_{(L,[·,·])}(a))/\equiv \subseteq {(a^{\perp }]}_L/\equiv \subseteq S$. □

4.4. Characterizations for Minimal Primes

Remark 16.

For any $a\in L$ and any $p\in {\mathrm{Spec}}_{(L,[·,·])}$, if $a^{\perp }\nleqq p$, then $a\le p$.

Indeed, since $[a,a^{\perp }]=0\le p$ and p is a prime element of $(L,[·,·])$, we have $a\le p$ or $a^{\perp }\le p$, hence the implication above.

Proposition 7.

Assume that $\rho (0)=0$, let $p\in {\mathrm{Spec}}_{(L,[·,·])}$ and let us consider the following statements:

(1) 

$p\in {\mathrm{Min}}_{(L,[·,·])}$;

(2) 

for any $a\in \mathrm{Cp}(L)$, $a\le p$ implies $a^{\perp }\nleqq p$;

(3) 

for any $a\in \mathrm{Cp}(L)$, $a\le p$ if and only if $a^{\perp }\nleqq p$;

(4) 

for any $a\in L$, $a\le p$ implies $a^{\perp }\nleqq p$;

(5) 

for any $a\in L$, $a\le p$ if and only if $a^{\perp }\nleqq p$.

If L satisfies Condition 1, then statements (1), (2) and (3) are equivalent.

If L satisfies Condition 2, then statements (1), (4) and (5) are equivalent.

Thus, if L satisfies Conditions 1 and 2, in particular if L is compact and $1/\equiv =\left\{1\right\}$, then statements (1), (2), (3), (4) and (5) are equivalent.

Proof. 

By Remark 16, (2) is equivalent to (3), while (4) is equivalent to (5).

• Case 1: Assume that L satisfies Condition 1. We have to prove that (1) is equivalent to (2).

The mutually inverse order isomorphisms between ${\mathrm{Spec}}_{(L,[·,·])}$ and ${\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$ from Proposition 6 restrict to mutually inverse order isomorphisms between ${\mathrm{Min}}_{(L,[·,·])}$ and ${\mathrm{Min}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$. Hence: $p\in {\mathrm{Min}}_{(L,[·,·])}$ if and only if $p^{*}\in {\mathrm{Min}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$.

By Lemmas 8 and 10 and Lemma 14. (2), the latter is equivalent to the fact that:

• for any $x\in p^{*}=(\mathrm{Cp}(L)\cap {(p]}_L)/\equiv$, ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(x)⊈p^{*}$,
• that is, for any $a\in \mathrm{Cp}(L)\cap {(p]}_L$, ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(a/\equiv )⊈p^{*}$,
• which means that, for any $a\in \mathrm{Cp}(L)\cap {(p]}_L$, ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}({(a/\equiv ]}_{\mathrm{Cp}(L)/\equiv })⊈p^{*}$,
• that is, for any $a\in \mathrm{Cp}(L)\cap {(p]}_L$, ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(a^{*})⊈p^{*}$,
• which is equivalent to the fact that, for any $a\in \mathrm{Cp}(L)\cap {(p]}_L$, $a^{\perp }\nleqq {(p^{*})}_{*}=p$,
• that is, for any $a\in \mathrm{Cp}(L)$, if $a\le p$, then $a^{\perp }\nleqq p$.

• Case 2: Now assume that L satisfies Condition 2. We have to prove that (1) and (4) are equivalent. By Lemma 9. (4):

• $p\in {\mathrm{Spec}}_{(L,[·,·])}$ if and only if $p/\equiv \in {\mathrm{Spec}}_{(L/\equiv ,\wedge )}$ if and only if ${(p/\equiv ]}_{L/\equiv }\in {\mathrm{Spec}}_{(\mathrm{Id}(L/\equiv ),\cap )}$;
• $p\in {\mathrm{Min}}_{(L,[·,·])}$ if and only if $p/\equiv \in {\mathrm{Min}}_{(L/\equiv ,\wedge )}$, which is equivalent to ${(p/\equiv ]}_{L/\equiv }\in {\mathrm{Min}}_{(\mathrm{Id}(L/\equiv ),\cap )}$ by Condition 2.

According to Lemma 8, Remark 6, Lemma 9. (1) and Lemma 3, the latter is equivalent to the fact that:

• for any $x\in {(p/\equiv ]}_{L/\equiv }$, ${\mathrm{Ann}}_{(L/\equiv ,\wedge )}(x)⊈{(p/\equiv ]}_{L/\equiv }$,
• that is, for any $a\in L$, if $a/\equiv \le p/\equiv$, then ${\mathrm{Ann}}_{(L/\equiv ,\wedge )}(a/\equiv )⊈{(p/\equiv ]}_{L/\equiv }$,
• which means that, for any $a\in L$, if $a\le p$, then ${\mathrm{Ann}}_{(L,[·,·])}(a)/\equiv ⊈{(p/\equiv ]}_{L/\equiv }$,
• that is, for any $a\in L$, if $a\le p$, then ${(a^{\perp }]}_L/\equiv ⊈{(p/\equiv ]}_{L/\equiv }$,
• which means that, for any $a\in L$, if $a\le p$, then ${(a^{\perp }]}_L/\equiv ⊈{(p/\equiv ]}_{L/\equiv }$,
• that is, for any $a\in L$, if $a\le p$, then ${(a^{\perp }]}_L/\equiv ⊈{(p/\equiv ]}_{L/\equiv }$,
• that is, for any $a\in L$, if $a\le p$, then ${(a^{\perp }/\equiv ]}_{L/\equiv }⊈{(p/\equiv ]}_{L/\equiv }$,
• which means that, for any $a\in L$, if $a\le p$, then $a^{\perp }/\equiv \nleqq p/\equiv$,
• which is equivalent to the fact that, for any $a\in L$, if $a\le p$, then $a^{\perp }\nleqq p$. □

Recall from [6] (Example 1) that the equivalence between (1), (4) and (5) in Proposition 7 does not hold for any commutator lattice that satisfies Condition 1 and $\rho (0)=0$.

Remark 17.

• ${\mathrm{Spec}}_{({[\rho (0))}_L,{[·,·]}_{\rho (0)})}={\mathrm{Spec}}_{(L,[·,·])}$, hence ${\mathrm{Min}}_{({[\rho (0))}_L,{[·,·]}_{\rho (0)})}={\mathrm{Min}}_{(L,[·,·])}$.

• For any $p\in {\mathrm{Spec}}_{(L,[·,·])}$ and any $a\in L$, $a\to \rho (0)\nleqq p$ implies $a\le p$.
• If $p\in {\mathrm{Spec}}_{(L,[·,·])}$ and $S\subseteq {[\rho (0))}_L$ is such that:
   for any $a\in S$, $a\le p$ implies $a\to \rho (0)\nleqq p$,
  then: for any $a\in L$ such that $a\vee \rho (0)\in S$, $a\le p$ implies $a\to \rho (0)\nleqq p$.

Indeed, since $\rho (0)=\bigcap {\mathrm{Spec}}_{(L,[·,·])}$, we have ${\mathrm{Spec}}_{(L,[·,·])}\subset {[\rho (0))}_L$ and thus ${\mathrm{Spec}}_{({[\rho (0))}_L,{[·,·]}_{\rho (0)})}={\mathrm{Spec}}_{(L,[·,·])}$ by Remark 9. Hence the equality of the minimal prime spectra.

Now let $p\in {\mathrm{Spec}}_{(L,[·,·])}$ and $a\in L$, so that $a\vee \rho (0)\in {[\rho (0))}_L$ and $\rho (0)=\bigcap {\mathrm{Spec}}_{(L,[·,·])}\le p\in {\mathrm{Spec}}_{({[\rho (0))}_L,{[·,·]}_{\rho (0)})}$. By Remark 8,

$$a\to \rho (0)=a\to (0\vee \rho (0))=(a\vee \rho (0))\to (0\vee \rho (0))=(a\vee \rho (0))\to \rho (0)={(a\vee \rho (0))}^{\perp \rho (0)}.$$

Hence

$${[a\vee \rho (0),a\to \rho (0)]}_{\rho (0)}={[a\vee \rho (0),{(a\vee \rho (0))}^{\perp \rho (0)}]}_{\rho (0)}=\rho (0)\le p\in {\mathrm{Spec}}_{({[\rho (0))}_L,{[·,·]}_{\rho (0)})},$$

thus, if $a\to \rho (0)\nleqq p$, then $a\vee \rho (0)\le p$, so $a\le p$.

Now let $S\subseteq {[\rho (0))}_L$ such that any $a\in S$ satisfies the implication: $a\le p$ implies $a\to \rho (0)\nleqq p$. Let $a\in L$ such that $a\vee \rho (0)\in S$ and $a\le p$.

Since $\rho (0)\le p$, it follows that $a\vee \rho (0)\le p$ and hence $(a\vee \rho (0))\to \rho (0)\nleqq p$ by the assumption on S. By the above, $a\to \rho (0)=(a\vee \rho (0))\to \rho (0)$, hence $a\to \rho (0)\nleqq p$.

${[\rho (0))}_L$ satisfies Condition 1 if and only if ${[\rho (0))}_L$ is algebraic, $\mathrm{Cp}({[\rho (0))}_L)$ is closed with respect to ${[·,·]}_{\rho (0)}$, and $1/\equiv =\left\{1\right\}$.

Note that:

• if L is algebraic, then ${[\rho (0))}_L$ is algebraic;
• if $1\in \mathrm{Cp}(L)$, then $1\in \mathrm{Cp}({[\rho (0))}_L)$.

Thus, if L satisfies Condition 1, then ${[\rho (0))}_L$ satisfies Condition 1.

Corollary 2.

Let $p\in {\mathrm{Spec}}_{(L,[·,·])}$ and let us consider the following statements:

(1) 

$p\in {\mathrm{Min}}_{(L,[·,·])}$;

(2) 

for any $a\in \mathrm{Cp}({[\rho (0))}_L)$, $a\le p$ implies $a\to \rho (0)\nleqq p$;

(3) 

for any $a\in \mathrm{Cp}({[\rho (0))}_L)$, $a\le p$ if and only if $a\to \rho (0)\nleqq p$;

(4) 

for any $a\in \mathrm{Cp}(L)\cup \mathrm{Cp}({[\rho (0))}_L)$, $a\le p$ implies $a\to \rho (0)\nleqq p$;

(5) 

for any $a\in \mathrm{Cp}(L)\cup \mathrm{Cp}({[\rho (0))}_L)$, $a\le p$ if and only if $a\to \rho (0)\nleqq p$;

(6) 

for any $a\in {[\rho (0))}_L$, $a\le p$ implies $a\to \rho (0)\nleqq p$;

(7) 

for any $a\in {[\rho (0))}_L$, $a\le p$ if and only if $a\to \rho (0)\nleqq p$;

(8) 

for any $a\in L$, $a\le p$ implies $a\to \rho (0)\nleqq p$;

(9) 

for any $a\in L$, $a\le p$ if and only if $a\to \rho (0)\nleqq p$.

If ${[\rho (0))}_L$ satisfies Condition 1, in particular if L satisfies Condition 1, then statements (1), (2), (3), (4) and (5) are equivalent.

If ${[\rho (0))}_L$ satisfies Condition 2, then statements (1), (6), (7), (8) and (9) are equivalent.

Thus, if ${[\rho (0))}_L$ satisfies Conditions 1 and 2, in particular if ${[\rho (0))}_L$ is compact and $1/\equiv =\left\{1\right\}$, in particular if L is compact and $1/\equiv =\left\{1\right\}$, then all nine statements above are equivalent.

Proof. 

By Remark 17, $p\in {\mathrm{Spec}}_{({[\rho (0))}_L,{[·,·]}_{\rho (0)})}$ and we have the equivalence: $p\in {\mathrm{Min}}_{(L,[·,·])}$ if and only if $p\in {\mathrm{Min}}_{({[\rho (0))}_L,{[·,·]}_{\rho (0)})}$.

Recall that, for any $a\in {[\rho (0))}_L$, $a\to \rho (0)=a^{\perp \rho (0)}$.

By Remark 9, ${\rho }_{\rho (0)}(\rho (0))=\rho (\rho (0))=\rho (0)$ and $1/{\equiv }_{\rho (0)}=1/\equiv$.

By Remark 10, every $a\in \mathrm{Cp}(L)$ satisfies $a\vee \rho (0)\in \mathrm{Cp}({[\rho (0))}_L)$, hence, according to Remark 17, properties (2), (3), (4) and (5) are equivalent.

Again by Remark 17, conditions (6), (7), (8) and (9) are equivalent.

From Proposition 7 applied to the quotient commutator lattice $({[\rho (0))}_L,{[·,·]}_{\rho (0)})$ we get the rest of the equivalences in the enunciation. □

Corollary 3.

Assume that L is algebraic, $1\in \mathrm{Cp}(L)$, $\mathrm{Cp}(L)$ is closed with respect to the commmutator and, for all $x\in L$, $[x,1]=x$, and let $p\in {\mathrm{Spec}}_{(L,[·,·])}$. Then the following are equivalent:

• $p\in {\mathrm{Min}}_{(L,[·,·])}$;
• for any $a\in \mathrm{Cp}(L)$, $a\le p$ implies $a\to \rho (0)\nleqq p$;
• for any $a\in \mathrm{Cp}(L)$, $a\le p$ if and only if $a\to \rho (0)\nleqq p$.

Proof. 

Recall from Remark 17 that, for any $a\in L$, $a\to \rho (0)\nleqq p$ implies $a\le p$.

Since $1\in \mathrm{Cp}(L)$ and $[1,1]=1$, we have $1/\equiv =\left\{1\right\}$ and thus L satisfies Condition 1.

By Lemma 11, $\mathrm{Cp}({[\rho (0))}_L)=\{a\vee \rho (0)|a\in \mathrm{Cp}(L)\}$. By Corollary 2 it follows that: $p\in {\mathrm{Min}}_{(L,[·,·])}$ if and only if, for any $a\in \mathrm{Cp}(L)$, $a\vee \rho (0)\le p⟹(a\vee \rho (0))\to \rho (0)\nleqq p$.

But $p\in {\mathrm{Spec}}_{(L,[·,·])}$, so $\rho (0)\le p$, thus any $a\in L$ satisfies: $a\vee \rho (0)\le p⟺a\le p$. By Lemma 4. (2), for any $a\in L$, $a\to \rho (0)=a\to (\rho (0)\vee \rho (0))=(a\vee \rho (0))\to (\rho (0)\vee \rho (0))=(a\vee \rho (0))\to \rho (0)$. Hence the equivalences in the enunciation. □

5. Two Topologies on the Minimal Prime Spectrum

Throughout this section, $(L,[·,·])$ will be an arbitrary commutator lattice. As in the previous sections, we denote by $\rho ={\rho }_{(L,[·,·])}$, $\equiv ={\equiv }_{(L,[·,·])}$, $\to ={\to }_{(L,[·,·])}$ and ^⊥ ${=}^{\perp (L,[·,·])}$.

5.1. The Stone and Flat Topologies on the Minimal Prime Spectrum

We have on ${\mathrm{Min}}_{(L,[·,·])}$ the Stone topology ${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$, described in Remark 4.

Lemma 15.

If $\rho (0)=0$, then $x^{\perp }=\bigcap (V_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])})$ for every $x\in L$.

Proof. 

Let $x\in L$. Clearly, $x^{\perp }\le \bigcap (V_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])})$.

Let us denote by $a=\bigcap (V_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])})$. Assume by absurdum that $a\nleqq x^{\perp }$, so that $[a,x]\ne 0=\rho (0)=\bigcap {\mathrm{Min}}_{(L,[·,·])}$ since A is semiprime. Therefore $[a,x]\nleqq p$ for some $p\in {\mathrm{Min}}_{(L,[·,·])}$, which implies that $x\nleqq p$ and $a\nleqq p$, hence $p\notin V_{(L,[·,·])}(x^{\perp })$, that is $x^{\perp }\nleqq p$. So $x\nleqq p$ and $x^{\perp }\nleqq p$, while $[x,x^{\perp }]=0\le p$, which contradicts the fact that $p\in {\mathrm{Min}}_{(L,[·,·])}\subseteq {\mathrm{Spec}}_{(L,[·,·])}$. Therefore $\bigcap (V_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])})=a\le x^{\perp }$, hence the equality. □

Remark 18.

If $\rho (0)=0$, then, by Lemma 15, for any $x,y\in L$, we have: $x^{\perp }=y^{\perp }$ if and only if $V_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}=V_{(L,[·,·])}(y^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}$ if and only if $D_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(y^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}$.

Proposition 8.

Assume that $\rho (0)=0$. For any $x,y,z\in L$, we consider the following statements:

(1) 

$V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=V_{(L,[·,·])}(x^{\perp \perp })\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}$ and $D_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(x^{\perp \perp })\cap {\mathrm{Min}}_{(L,[·,·])}=V_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}$;

(2) 

$x^{\perp }\wedge y^{\perp }=z^{\perp }$ if and only if $V_{(L,[·,·])}(x)\cap V_{(L,[·,·])}(y)\cap {\mathrm{Min}}_{(L,[·,·])}=V_{(L,[·,·])}(z)\cap {\mathrm{Min}}_{(L,[·,·])}$;

(3) 

$x^{\perp \perp }=y^{\perp }$ if and only if $x^{\perp }=y^{\perp \perp }$ if and only if $V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=V_{(L,[·,·])}(y^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}$ if and only if $V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(y)\cap {\mathrm{Min}}_{(L,[·,·])}$ if and only if $D_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(y)\cap {\mathrm{Min}}_{(L,[·,·])}$.

If L satisfies Condition 1, then the statements above hold for all $x,y,z\in \mathrm{Cp}(L)$.

If L satisfies Condition 2, then the statements above hold for all $x,y,z\in L$.

Proof. 

Analogous to the proof of [6] (Proposition 6), using Proposition 7, Lemma 5. (3), Lemma 5. (4) and Remark 18 instead of [6] (Proposition 5), [6] (Lemma 6. (iii)), [6] (Lemma (not Proposition) 6. (iv)) and [6] (Remark 5), respectively. □

Let us denote by ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$ the topology on ${\mathrm{Min}}_{(L,[·,·])}$ generated by $\{V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}|x\in \mathrm{Cp}(L)\}$. We call ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$ the flat topology or the inverse topology on ${\mathrm{Min}}_{(L,[·,·])}$. We denote by $\mathcal{M}i{n}_{(L,[·,·])}$, respectively $\mathcal{M}i{n}_{(L,[·,·])}^{-1}$ the minimal prime spectrum of $(L,[·,·])$ endowed with the Stone, respectively the flat topology: $\mathcal{M}i{n}_{(L,[·,·])}=({\mathrm{Min}}_{(L,[·,·])},{\mathcal{S}}_{\mathrm{Min},(L,[·,·])})$ and $\mathcal{M}i{n}_{(L,[·,·])}^{-1}=({\mathrm{Min}}_{(L,[·,·])},{\mathcal{F}}_{\mathrm{Min},(L,[·,·])})$.

Remark 19.

${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$ has $\{V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}|x\in \mathrm{Cp}(L)\}$ as a basis, since $V_{(L,[·,·])}(0)$$\cap {\mathrm{Min}}_{(L,[·,·])}={\mathrm{Min}}_{(L,[·,·])}$ and, for any $x,y\in \mathrm{Cp}(L)$, $x\vee y\in \mathrm{Cp}(L)$ and $V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}\cap V_{(L,[·,·])}(y)\cap {\mathrm{Min}}_{(L,[·,·])}=V_{(L,[·,·])}(x\vee y)\cap {\mathrm{Min}}_{(L,[·,·])}$, by Proposition 1. (6).

Of course, when L is compact, ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$ is induced by ${\mathcal{F}}_{\mathrm{Spec},(L,[·,·])}$; see Remark 5.

Remark 20.

If L is compact, then:

• the Stone topology on ${\mathrm{Min}}_{(L,[·,·])}$ coincides with its basis $\{D_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}|x\in \mathrm{Cp}(L)=L\}$;
• the flat topology on ${\mathrm{Min}}_{(L,[·,·])}$ coincides with its basis $\{V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}|x\in \mathrm{Cp}(L)=L\}$.

Remark 21.

The map $S\mapsto S\cap {\mathrm{Min}}_{(L,[·,·])}$ is a complete Boolean morphism from $\mathcal{P}({\mathrm{Spec}}_{(L,[·,·])})$ to $\mathcal{P}({\mathrm{Min}}_{(L,[·,·])})$. Hence its restriction to ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ corestricted to ${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$ is a complete surjective (of course, bounded) lattice morphism and so is its restriction to the set of closed sets of ${\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ corestricted to the set of closed sets of ${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$, which, of course, if L is compact, is precisely its restriction to ${\mathcal{F}}_{\mathrm{Spec},(L,[·,·])}$ corestricted to ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$.

Corollary 4.

${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$ is a bounded sublattice of the Boolean algebra $\mathcal{P}({\mathrm{Min}}_{(L,[·,·])})$ closed with respect to arbitrary unions, thus it is a frame.

Its set $\{V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}|x\in L\}$ of closed sets, that is ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$ in the particular case when L is compact, is a bounded sublattice of $\mathcal{P}({\mathrm{Min}}_{(L,[·,·])})$ closed with respect to arbitrary intersections, dually isomorphic to ${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$.

Proof. 

By Proposition 3 and Remark 21. □

Remark 22.

It is straightforward that the map ${\displaystyle S\mapsto \bigcup _{m\in S}{V}_{(L,[·,·])}(m)=\bigcup _{m\in S}({[m)}_L\cap {\mathrm{Spec}}_{(L,[·,·])})}$${\displaystyle =(\bigcup _{m\in S}{[m)}_L)\cap {\mathrm{Spec}}_{(L,[·,·])}}$ is a complete morphism of join–semilattices from $\{V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}|x\in L\}$ to $\left\{V_{(L,[·,·])}(x)\right|x\in L\}$, thus from ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$ to ${\mathcal{F}}_{\mathrm{Spec},(L,[·,·])}$ if L is compact.

Remark 23.

For any $x\in L$ and any $S\subseteq {\mathrm{Spec}}_{(L,[·,·])}$, respectively $S\subseteq {\mathrm{Min}}_{(L,[·,·])}$, we have:

$x\le p$ for all $p\in S$ if and only if $x\le \bigwedge S$;

$x\nleqq q$ for any $q\in {\mathrm{Spec}}_{(L,[·,·])}\setminus S$, respectively any $q\in {\mathrm{Min}}_{(L,[·,·])}\setminus S$, if and only if ${\displaystyle x\notin \bigcup _{q\in {\mathrm{Spec}}_{(L,[·,·])}\setminus S}{(q]}_L}$, respectively ${\displaystyle x\notin \bigcup _{q\in {\mathrm{Min}}_{(L,[·,·])}\setminus S}{(q]}_L}$.

Thus:

${\displaystyle \{x\in L|V_{(L,[·,·])}(x)=S\}=\{x\in L|D_{(L,[·,·])}(x)={\mathrm{Spec}}_{(L,[·,·])}\setminus S\}={(\bigwedge S]}_L\setminus (\bigcup _{q\in {\mathrm{Spec}}_{(L,[·,·])}\setminus S}{(q]}_L)}$;

${\displaystyle \{x\in L|V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=S\}=\{x\in L|D_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}={\mathrm{Min}}_{(L,[·,·])}\setminus S\}={(\bigwedge S]}_L\setminus (\bigcup _{m\in {\mathrm{Min}}_{(L,[·,·])}\setminus S}{(m]}_L)}$.

When are these sets nonempty?

Lemma 16.

If ${\mathrm{Min}}_{(L,[·,·])}$ is finite, in particular if L is finite, then, for any $S\subset {\mathrm{Min}}_{(L,[·,·])}$:

(1) 

for any $m\in {\mathrm{Min}}_{(L,[·,·])}\setminus S$, $\bigwedge S\nleqq m$;

(2) 

$V_{(L,[·,·])}(\bigwedge S)\cap {\mathrm{Min}}_{(L,[·,·])}=S$ and $D_{(L,[·,·])}(\bigwedge S)\cap {\mathrm{Min}}_{(L,[·,·])}={\mathrm{Min}}_{(L,[·,·])}\setminus S$.

Proof. 

(1) $\bigwedge \varnothing =1\nleqq m$ for any $m\in {\mathrm{Min}}_{(L,[·,·])}$.

Now assume that S is nonempty, let $k\in {\mathbb{N}}^{*}$ be its cardinality, $S=\{m_1,\dots ,m_k\}$ and $m\in {\mathrm{Min}}_{(L,[·,·])}\setminus S$. Assume by absurdum that $\bigwedge S\le m$, that is $m_1\wedge \dots \wedge m_k\le m$. Then $[m_1\wedge \dots \wedge m_{k-1},m_k]\le m_1\wedge \dots \wedge m_k\le m\in {\mathrm{Spec}}_{(L,[·,·])}$, thus $m_1\wedge \dots \wedge m_{k-1}\le m$ or $m_k\le m$. But $m,m_k\in {\mathrm{Min}}_{(L,[·,·])}$ with $m\ne m_k$, so they are incomparable. Thus $m_1\wedge \dots \wedge m_{k-1}\le m$. If $k-1\ne 0$, then, analogously, it follows that $m_1\wedge \dots \wedge m_{k-2}\le m$, and so on, until we obtain $m_1\le m$, a contradiction to $m,m_1\in {\mathrm{Min}}_{(L,[·,·])}$ with $m\ne m_1$.

(2) $V_{(L,[·,·])}(\bigwedge \varnothing )\cap {\mathrm{Min}}_{(L,[·,·])}=V_{(L,[·,·])}(1)\cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing$.

If S is nonempty, then $\bigwedge S\le m$ for each $m\in S$, thus $V_{(L,[·,·])}(\bigwedge S)\cap {\mathrm{Min}}_{(L,[·,·])}=S$ by (1), hence $D_{(L,[·,·])}(\bigwedge S)\cap {\mathrm{Min}}_{(L,[·,·])}={\mathrm{Min}}_{(L,[·,·])}\setminus S$. □

Proposition 9.

• If ${\mathrm{Min}}_{(L,[·,·])}$ is finite, in particular if L is finite, then ${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}=$$\mathcal{P}({\mathrm{Min}}_{(L,[·,·])})$.

• If ${\mathrm{Min}}_{(L,[·,·])}$ is finite and all intersections of minimal primes of $(L,[·,·])$ are compact, in particular if ${\mathrm{Min}}_{(L,[·,·])}$ is finite and L is compact, in particular if L is finite, then ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}=\mathcal{P}({\mathrm{Min}}_{(L,[·,·])})$.

Proof. 

If ${\mathrm{Min}}_{(L,[·,·])}$ is finite, then, by Lemma 16. (2), for any $S\subseteq {\mathrm{Min}}_{(L,[·,·])}$:

$D_{(L,[·,·])}(\bigwedge ({\mathrm{Min}}_{(L,[·,·])}\setminus S))\cap {\mathrm{Min}}_{(L,[·,·])}=S$, so $S\in {\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$;

$V_{(L,[·,·])}(\bigwedge S)\cap {\mathrm{Min}}_{(L,[·,·])}=S$, thus $S\in {\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$ if $\bigwedge S$ is compact, in particular if L is compact. □

Remark 24.

By Remark 23, for any $S\subseteq {\mathrm{Min}}_{(L,[·,·])}$ and any $T\subseteq {\mathrm{Spec}}_{(L,[·,·])}$:

• $S\in {\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$ if and only if ${\displaystyle {(\bigwedge ({\mathrm{Min}}_{(L,[·,·])}\setminus S)]}_L\setminus (\bigcup _{m\in S}{(m]}_L)}$ is nonempty;
• if ${\displaystyle \mathrm{Cp}(L)\cap ({(\bigwedge S]}_L\setminus (\bigcup _{m\in {\mathrm{Min}}_{(L,[·,·])}\setminus S}{(m]}_L))}$ is nonempty, then $S\in {\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$;
• $T\in {\mathcal{S}}_{\mathrm{Spec},(L,[·,·])}$ if and only if ${\displaystyle {(\bigwedge ({\mathrm{Spec}}_{(L,[·,·])}\setminus T)]}_L\setminus (\bigcup _{q\in T}{(q]}_L)}$ is nonempty;
• if L is compact and ${\displaystyle {(\bigwedge T]}_L\setminus (\bigcup _{q\in {\mathrm{Spec}}_{(L,[·,·])}\setminus T}{(q]}_L)}$ is nonempty, then $T\in {\mathcal{F}}_{\mathrm{Spec},(L,[·,·])}$.

Recall that, for any $x\in L$, $x^{\perp }$ generates the annihilator of x with respect to the commutator (as well as the meet if $\rho (0)=0$) as a principal ideal.

Note that, in [6] (Proposition 7. (i)), Condition 1. (iv) had to be enforced on the algebra A.

Proposition 10.

If $\rho (0)=0$ and L satisfies one of the Conditions 1 and 2, then:

(1) 

the flat topology on ${\mathrm{Min}}_{(L,[·,·])}$ is coarser than the Stone topology: ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}\subseteq {\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$;

(2) 

if $\mathrm{Cp}(L)$ is closed with respect to ^⊥, in particular if L is compact, then the two topologies coincide: ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}={\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$, that is $\mathcal{M}i{n}_{(L,[·,·])}=\mathcal{M}i{n}_{(L,[·,·])}^{-1}$.

Proof. 

(1) By Proposition 8. (1), $V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}\in$${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$, for any $x\in \mathrm{Cp}(L)$.

(2) Again by Proposition 8. (1), for any $x\in \mathrm{Cp}(L)$, $D_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}=V_{(L,[·,·])}(x^{\perp })$$\cap {\mathrm{Min}}_{(L,[·,·])}$, which belongs to ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$ if $x^{\perp }\in \mathrm{Cp}(L)$. □

5.2. Homeomorphism, Compactness and Separability Results

Lemma 17.

If L satisfies Condition 1, then the maps ${·}^{*}:{\mathrm{Spec}}_{(L,[·,·])}\to {\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$ and ${·}_{*}:{\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}\to {\mathrm{Spec}}_{(L,[·,·])}$ are homeomorphisms with respect to the Stone topologies, thus $\mathcal{S}pe{c}_{(L,[·,·])}$ and $\mathcal{S}pe{c}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$ are homeomorphic.

Proof. 

Assume that L satisfies Condition 1. Then, by Proposition 6, these maps are mutually inverse order isomorphisms. Since $\mathrm{Cp}(L)$ is closed with respect to the commutator and $1\in \mathrm{Cp}(L)$, $\mathrm{Cp}(L)/\equiv$ is a bounded sublattice of $L/\equiv$ and thus a bounded distributive lattice. Since $\mathrm{Cp}(L)$ is closed with respect to the commutator, we have the map ${·}^{*}:L\to \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$, which is surjective by Proposition 5. (2).

Recall that the set of closed sets of the Stone topology on ${\mathrm{Spec}}_{(L,[·,·])}$ is $\left\{V_{(L,[·,·])}(x)\right|x\in L\}$ and that of those of the Stone topology on ${\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$ is $\left\{V_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}(I)\right|I\in \mathrm{Id}(L)\}$, which equals $\left\{V_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}(x^{*})\right|x\in L\}$ by the surjectivity of the map ${·}^{*}:L\to \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$.

Let $x\in L$ and $p\in V_{(L,[·,·])}(x)$, that is $p\in {\mathrm{Spec}}_{(L,[·,·])}$ with $x\le p$. Then $p^{*}\in {\mathrm{Spec}}_{\mathrm{Id}}(\mathrm{Cp}(L)/\equiv )$ and, since the map ${·}^{*}:L\to \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$ is order–preserving, $x^{*}\subseteq p^{*}$. Hence $p^{*}\in V_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}(x^{*})$ and thus the image of $V_{(L,[·,·])}(x)$ through this map: $V_{(L,[·,·])}{(x)}^{*}\subseteq V_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}(x^{*})$.

Now let $P\in V_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}(I)=V_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}(x^{*})$, that is $P\in {\mathrm{Spec}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$ and $x^{*}\subseteq P$. Then $P_{*}\in {\mathrm{Spec}}_{(L,[·,·])}$ and, by Lemma 12. (1), $x\le {(x^{*})}_{*}\le P_{*}$, thus $P_{*}\in V_{(L,[·,·])}(x)$, so $P={(P_{*})}^{*}\in V_{(L,[·,·])}{(x)}^{*}$.

Therefore $V_{(L,[·,·])}{(x)}^{*}=V_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}(x^{*})$, so the direct image of ${·}^{*}$ preserves closed sets and thus also open sets, hence the bijection ${·}^{*}$ is a homeomorphism with respect to the Stone topologies.

Thus so is its inverse ${·}_{*}$: if $I\in \mathrm{Id}(\mathrm{Cp}(L)/\equiv )$, so that $I=x^{*}$ for some $x\in L$, then, again by the above, along with Proposition 6, $V_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}{(I)}_{*}=V_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}{(x^{*})}_{*}={(V_{(L,[·,·])}{(x)}^{*})}_{*}=V_{(L,[·,·])}(x)$. □

Lemma 18.

If L satisfies Condition 1, then:

(1) 

$\mathcal{M}i{n}_{(L,[·,·])}$ is homeomorphic to $\mathcal{M}i{n}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$;

(2) 

$\mathcal{M}i{n}_{(L,[·,·])}^{-1}$ is homeomorphic to $\mathcal{M}i{n}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}^{-1}$.

Proof. 

(1) By Lemma 17, ${·}^{*}$ and ${·}_{*}$ restrict to homeomorphisms between $\mathcal{M}i{n}_{(L,[·,·])}$ and $\mathcal{M}i{n}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$.

(2) Since ${\mathcal{F}}_{\mathrm{Min},(L,[·,·])}$ has $\{V_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}|x\in \mathrm{Cp}(L)\}$ as a basis, while ${\mathcal{F}}_{\mathrm{Min},(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ,\wedge )}$ has $\{V_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}({(a/\equiv ]}_{\mathrm{Cp}(L)/\equiv })\cap {\mathrm{Min}}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}|a\in \mathrm{Cp}(L)\}$ as a basis, we have, by Lemma 10 and the proof of Lemma 17, for all $a\in \mathrm{Cp}(L)$: $V_{(L,[·,·])}{(a)}^{*}=V_{\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}(a^{*})=V_{\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}({(a/\equiv ]}_{\mathrm{Cp}(L)/\equiv })$ and $V_{\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}{({(a/\equiv ]}_{\mathrm{Cp}(L)/\equiv })}_{*}=V_{\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}{(a^{*})}_{*}=V_{(L,[·,·])}(a)$, hence ${·}^{*}$ and ${·}_{*}$ are open and thus, by (1), mutually inverse homeomorphisms between $\mathcal{M}i{n}_{(L,[·,·])}^{-1}$ and $\mathcal{M}i{n}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}^{-1}$. □

Proposition 11.

If L satisfies Condition 1, then $\mathcal{M}i{n}_{(L,[·,·])}^{-1}$ is a compact ${\mathrm{T}}_1$ topological space.

Proof. 

Since $\mathrm{Cp}(L)$ is closed with respect to the commutator and $1\in \mathrm{Cp}(L)$, $\mathrm{Cp}(L)/\equiv$ is a bounded sublattice of the frame $L/\equiv$ and thus a bounded distributive lattice. Therefore, by Hochster’s Theorem [26] (Proposition $3.13$), $\mathrm{Cp}(L)/\equiv$ is lattice isomorphic to the reticulation $\mathcal{L}(R)$ of some commutative unitary ring R. Remember that the commutator lattices $(\mathrm{Con}(R),{[·,·]}_R)$ and $(\mathrm{Id}(R),•)$ are isomorphic, where $\mathrm{Id}(R)$ is the set of ideals of R and $•$ is the multiplication of ideals. Hence the minimal prime spectrum of congruences of R endowed with the flat topology, $\mathcal{M}i{n}_{(\mathrm{Con}(R),{[·,·]}_R)}^{-1}$, is compact and ${\mathrm{T}}_1$ by [34] (Theorem $3.1$), and homeomorphic to $\mathcal{M}i{n}_{(\mathrm{Id}(\mathcal{L}(R)),\cap )}^{-1}$, thus to $\mathcal{M}i{n}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}^{-1}$, which in turn is homeomorphic to $\mathcal{M}i{n}_{(L,[·,·])}^{-1}$ by Lemma 18. (2). Hence $\mathcal{M}i{n}_{(L,[·,·])}^{-1}$ is compact and ${\mathrm{T}}_1$. □

Theorem 1.

If $\rho (0)=0$ and L satisfies Condition 1, then the following are equivalent:

(1) 

$\mathcal{M}i{n}_{(L,[·,·])}=\mathcal{M}i{n}_{(L,[·,·])}^{-1}$;

(2) 

$\mathcal{M}i{n}_{(L,[·,·])}$ is compact;

(3) 

for any $a\in \mathrm{Cp}(L)$, there exists $b\in \mathrm{Cp}(L)$ such that $b\le a^{\perp }$ and ${(a\vee b)}^{\perp }=0$.

Proof. 

Since $\mathrm{Cp}(L)$ is closed with respect to the commutator and $1\in \mathrm{Cp}(L)$, $\mathrm{Cp}(L)/\equiv$ is a bounded distributive lattice and thus a distributive lattice with zero, hence, according to [35] (Proposition $5.1$), the following are equivalent:

$(a)$$\mathcal{M}i{n}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}=\mathcal{M}i{n}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}^{-1}$;

$(b)$$\mathcal{M}i{n}_{(\mathrm{Id}(\mathrm{Cp}(L)/\equiv ),\cap )}$ is compact;

$(c)$ for any $x\in \mathrm{Cp}(L)/\equiv$, there exists a $y\in \mathrm{Cp}(L)/\equiv$ such that $x\wedge y=0/\equiv$ and ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(x\vee y)=\{0/\equiv \}$.

By Lemma 18, (1) is equivalent to $(a)$. By Lemma 18. (1), (2) is equivalent to $(b)$.

To prove that (3) is equivalent to $(c)$, let $a,b\in \mathrm{Cp}(L)$, arbitrary, so that $a/\equiv$ and $b/\equiv$ are arbitrary elements of $\mathrm{Cp}(L)/\equiv$.

We will use the properties of the radical equivalence ≡ recalled in Section 3.

We have $\rho (0)=0$, which is equivalent to $0/\equiv =\left\{0\right\}$, hence, for any $u\in L$, $u=0$ if and only if $u\in 0/\equiv$ if and only if $u/\equiv =0/\equiv$.

Recall that $b\le a^{\perp }$ is equivalent to $[a,b]=0$ and thus to $[a,b]/\equiv =0/\equiv$ by the above, that is $a/\equiv \wedge b/\equiv =0/\equiv$, which means that $(a\wedge b)/\equiv =0/\equiv$, which is equivalent to $a\wedge b=0$ by the above.

${(a\vee b)}^{\perp }=0$ means that ${\mathrm{Ann}}_{(L,[·,·])}(a\vee b)=\left\{0\right\}$, that is ${\mathrm{Ann}}_L(a\vee b)=\left\{0\right\}$, which is equivalent to ${\mathrm{Ann}}_{(L/\equiv ,\wedge )}(a/\equiv \vee b/\equiv )=\{0/\equiv \}$, which in turn is equivalent to ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(a/\equiv \vee b/\equiv )=\{0/\equiv \}$, because, if we denote by $u=a\vee b$, so that $u\in \mathrm{Cp}(L)$ and $u/\equiv =a/\equiv \vee b/\equiv \in \mathrm{Cp}(L)/\equiv$, then we have:

since $\mathrm{Cp}(L)/\equiv$ is a bounded sublattice of $L/\equiv$, ${\mathrm{Ann}}_{(L/\equiv ,\wedge )}(u/\equiv )=\{0/\equiv \}$ implies ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(u/\equiv )={\mathrm{Ann}}_{(L/\equiv ,\wedge )}(u/\equiv )\cap \mathrm{Cp}(L)/\equiv =\{0/\equiv \}$;

for the converse, recall that:

$$max{\mathrm{Ann}}_L(u)=max{\mathrm{Ann}}_{(L,[·,·])}(u)=\bigvee \{c\in \mathrm{Cp}(L)|[u,c]=0\}=$$

$$\bigvee \{c\in \mathrm{Cp}(L)|[u,c]/\equiv =0/\equiv \}=\bigvee \{c\in \mathrm{Cp}(L)|u/\equiv \wedge c/\equiv =0/\equiv \},$$

thus, since $u\in \mathrm{Cp}(L)$ and thus $u/\equiv \in \mathrm{Cp}(L)/\equiv$,

$$max{\mathrm{Ann}}_L(u)=\bigvee \{c\in \mathrm{Cp}(L)|c/\equiv \in {\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(u/\equiv )\};$$

hence, if ${\mathrm{Ann}}_{(\mathrm{Cp}(L)/\equiv ,\wedge )}(u/\equiv )=\{0/\equiv \}$, then

$$max{\mathrm{Ann}}_L(u)=\bigvee \{c\in \mathrm{Cp}(L)|c/\equiv \in \{0/\equiv \}\}=$$

$$\bigvee \{c\in \mathrm{Cp}(L)|c/\equiv =0/\equiv \}=\bigvee \{c\in \mathrm{Cp}(L)|c=0\}=0,$$

thus ${\mathrm{Ann}}_L(u)=\left\{0\right\}$, which is equivalent to ${\mathrm{Ann}}_{L/\equiv }(0/\equiv )=\{0/\equiv \}$. □

Proposition 12.

If $1\in \mathrm{Cp}(L)$ and ${\mathrm{Spec}}_{(L,[·,·])}$ is unordered, then $\mathcal{M}i{n}_{(L,[·,·])}$ is compact.

Proof. 

Assume that $1\in \mathrm{Cp}(L)$ and ${\mathrm{Spec}}_{(L,[·,·])}$ is unordered, that is ${\mathrm{Spec}}_{(L,[·,·])}={\mathrm{Min}}_{(L,[·,·])}$, and let ${\displaystyle {\mathrm{Min}}_{(L,[·,·])}=\bigcup _{i\in I}(D_{(L,[·,·])}(a_i)\cap {\mathrm{Min}}_{(L,[·,·])})}$ for some nonempty family $\left\{a_i\right|i\in I\}\subseteq L$. Then ${\displaystyle {\mathrm{Min}}_{(L,[·,·])}=(\bigcup _{i\in I}{D}_{(L,[·,·])}(a_i))\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(\underset{i\in I}{\bigvee }{a}_i)\cap {\mathrm{Min}}_{(L,[·,·])}}$, thus $V_{(L,[·,·])}({\bigvee }_{i\in I}{a}_i)\cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing$. By Remark 12, this implies that ${\displaystyle \underset{i\in I}{\bigvee }{a}_i=1\in \mathrm{Cp}(L)}$, so that ${\displaystyle 1=\underset{i\in F}{\bigvee }{a}_i}$ for some finite subset F of I, hence ${\displaystyle {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(\underset{i\in F}{\bigvee }{a}_i)\cap {\mathrm{Min}}_{(L,[·,·])}=(\bigcup _{i\in F}{D}_{(L,[·,·])}(a_i))\cap {\mathrm{Min}}_{(L,[·,·])}=\bigcup _{i\in F}(D_{(L,[·,·])}(a_i)\cap {\mathrm{Min}}_{(L,[·,·])})}$, therefore $\mathcal{M}i{n}_{(L,[·,·])}$ is compact. □

Recall from [6] that the converse of the implication in Proposition 12 does not hold. This can also be seen by the two commutator lattices in Example 1 below, that have equal, thus compact (see Theorem 1), Stone and flat topologies on their minimal prime spectra, but their prime spectra are not antichains.

Theorem 2.

If $\rho (0)=0$, L satisfies one of the Conditions 1 and 2 and $\mathrm{Cp}(L)$ is closed with respect to the polar, in particular if $\rho (0)=0$ and L is compact, then the Stone topology ${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$ is Hausdorff and consists solely of clopen sets, thus it is a complete Boolean sublattice of $\mathcal{P}({\mathrm{Min}}_{(L,[·,·])})$. If, moreover, ${\mathrm{Spec}}_{(L,[·,·])}$ is unordered, then ${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$ is also compact.

Proof. 

By Proposition 8. (1), the Stone topology ${\mathcal{S}}_{\mathrm{Min},(L,[·,·])}$ on ${\mathrm{Min}}_{(L,[·,·])}$ consists entirely of clopen sets.

For any $x\in L$, $D_{(L,[·,·])}(x)\cap {\mathrm{Min}}_{(L,[·,·])}\cap D_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(x)\cap D_{(L,[·,·])}(x^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}([x,x^{\perp }])\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(0)\cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing \cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing$.

Let $m,p$ be distinct minimal prime elements of $(L,[·,·])$. Since $m\ne p$, we have $m\nleqq p$.

If L satisfies Condition 1, so that L is algebraic, then $m=\bigvee \{a\in \mathrm{Cp}(L)|a\le m\}$ and $p=\bigvee \{a\in \mathrm{Cp}(L)|a\le p\}$, Hence there exists an $a\in \mathrm{Cp}(L)$ such that $a\le m$, but $a\nleqq p$, so that $a^{\perp }\nleqq m$ by Proposition 7, so $m\in D_{(L,[·,·])}(a^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}$ and $p\in D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$. By the above, $D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}\cap D_{(L,[·,·])}(a^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing$.

Since $m\nleqq p$, we have $p\in D_{(L,[·,·])}(m)\cap {\mathrm{Min}}_{(L,[·,·])}$. If L satisfies Condition 2, then, since $m\le m$, by Proposition 7 it follows that $m^{\perp }\nleqq m$, thus $m\in D_{(L,[·,·])}(m^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}$. By the above, $D_{(L,[·,·])}(m)\cap {\mathrm{Min}}_{(L,[·,·])}\cap D_{(L,[·,·])}(m^{\perp })\cap {\mathrm{Min}}_{(L,[·,·])}=\varnothing$.

Therefore $\mathcal{M}i{n}_{(L,[·,·])}$ is Hausdorff. By Proposition 12, if ${\mathrm{Spec}}_{(L,[·,·])}$ is an antichain, then $\mathcal{M}i{n}_{(L,[·,·])}$ is also compact. □

Remark 25.

Let $(M_1,{\langle ·,·\rangle }_1)$ and $(M_2,{\langle ·,·\rangle }_2)$ be commutator lattices and let us consider their direct product $(M,\langle ·,·\rangle ):=(M_1,{\langle ·,·\rangle }_1)\times (M_2,{\langle ·,·\rangle }_2)$, so $M=M_1\times M_2$, with the direct product bounded lattice structure, and $\langle (a,b),(x,y)\rangle =({\langle a,x\rangle }_1,{\langle b,y\rangle }_2)$ for all $a,x\in L_1$ and $b,y\in L_2$, which is clearly a commutator lattice. If 1 is a neutral element with respect to the commutator in each of $(M_1,{\langle ·,·\rangle }_1)$ and $(M_2,{\langle ·,·\rangle }_2)$, that is ${\langle a,1\rangle }_1=a$ and ${\langle b,1\rangle }_2=b$ for all $a\in L_1$ and all $b\in L_2$, then it is straighforward that:

• ${\mathrm{Spec}}_{(M,\langle ·,·\rangle )}=\left\{(p,1)\right|p\in {\mathrm{Spec}}_{(M_1,{\langle ·,·\rangle }_1)}\}\cup \left\{(1,q)\right|q\in {\mathrm{Spec}}_{(M_2,{\langle ·,·\rangle }_2)}\}$, thus:
• ${\mathrm{Min}}_{(M,\langle ·,·\rangle )}=\left\{(p,1)\right|p\in {\mathrm{Min}}_{(M_1,{\langle ·,·\rangle }_1)}\}\cup \left\{(1,q)\right|q\in {\mathrm{Min}}_{(M_2,{\langle ·,·\rangle }_2)}\}$.

Example 1.

Let us consider the following commutator lattices: $(M_1,{*}_1)$ and $(M_2,{*}_2)$, where $M_1={\mathcal{C}}_2\oplus {\mathcal{N}}_5\oplus {\mathcal{C}}_2$, $M_2={\mathcal{C}}_3\oplus {\mathcal{N}}_5\oplus {\mathcal{C}}_2$ and the commutators are given by the following tables, with the lattice elements denoted as in the following Hasse diagrams:

For the purpose of getting nonsingleton minimal prime spectra (see the spectra below), let us consider the direct products of these commutator lattices with the two–element chain, endowed with the commutator equalling its meet: $(L_1,{[·,·]}_1):=(M_1,{*}_1)\times ({\mathcal{C}}_2,\wedge )$ and $(L_2,{[·,·]}_2):=(M_2,{*}_2)\times ({\mathcal{C}}_2,\wedge )$, with the elements denoted as follows:

$M_1$ and $M_2$ are, of course, nonmodular, $M_1$ is a bounded sublattice of $M_2$, but the commutator lattice $(M_1,{*}_1)$ is not a subalgebra of $(M_2,{*}_2)$, the commutators ${*}_1$ and ${*}_2$ are nonassociative and have 1 as neutral element, thus the same holds for $L_1$ and $L_2$, $(L_1,{[·,·]}_1)$ and $(L_2,{[·,·]}_2)$, respectively ${[·,·]}_1$ and ${[·,·]}_2$.

Lemma 2. (3) makes it easy to determine their prime spectra:

${\mathrm{Spec}}_{(M_1,{*}_1)}={\mathrm{Spec}}_{(M_2,{*}_2)}=\{0,2,3,5\}$, thus ${\mathrm{Min}}_{(M_1,{*}_1)}={\mathrm{Min}}_{(M_2,{*}_2)}=\left\{0\right\}$ and $R(M_1,{*}_1)=R(M_2,{*}_2)=\{0,2,3,5,6,1\}$, so ${\rho }_{(M_1,{*}_1)}(0)={\rho }_{(M_2,{*}_2)}(0)=0$;

thus, since ${\mathrm{Spec}}_{({\mathcal{C}}_2,\wedge )}=\left\{0\right\}$ and 1 is a neutral element with respect to ∧, we have ${\mathrm{Spec}}_{(L_1,{[·,·]}_1)}={\mathrm{Spec}}_{(L_2,{[·,·]}_2)}=\{g,t,w,y,z\}$, thus ${\mathrm{Min}}_{(L_1,{[·,·]}_1)}={\mathrm{Min}}_{(L_2,{[·,·]}_2)}=\{g,t\}$ and $R(L_1,{[·,·]}_1)=R(L_2,{[·,·]}_2)=\{0,c,e,f,g,t,v,w,y,z,1\}$, so ${\rho }_{(L_1,{[·,·]}_1)}(0)={\rho }_{(L_2,{[·,·]}_2)}(0)=0$.

Note that, in any commutator lattice $(L,[·,·])$, the radical of any element $s\in L$ is the smallest radical element of $(L,[·,·])$ larger than s: ${\rho }_{(L,[·,·])}(s)=min({[s)}_L\cap R(L,[·,·]))$. Thus we have the following definitions for the radical in $(L_1,{[·,·]}_1)$ and $(L_2,{[·,·]}_2)$, hence the quotient lattices above through the radical equivalence, which are (self–dual and thus) lattice isomorphic to ${\mathcal{F}}_{\mathrm{Spec},(L_1,{[·,·]}_1)}$, ${\mathcal{S}}_{\mathrm{Spec},(L_1,{[·,·]}_1)}$, ${\mathcal{F}}_{\mathrm{Spec},(L_2,{[·,·]}_2)}$ and ${\mathcal{S}}_{\mathrm{Spec},(L_2,{[·,·]}_2)}$, lattice ordered by set inclusion, as expected by Proposition 3, while ${\mathcal{F}}_{\mathrm{Min},(L_1,{[·,·]}_1)}={\mathcal{S}}_{\mathrm{Min},(L_1,{[·,·]}_1)}={\mathcal{F}}_{\mathrm{Min},(L_2,{[·,·]}_2)}={\mathcal{S}}_{\mathrm{Min},(L_2,{[·,·]}_2)}=\mathcal{P}(\{g,t\})$ so, ordered by set inclusion, they are isomorphic to the four–element Boolean algebra, as expected by Proposition 9; of course, they are Hausdorff, as expected from Theorem 2, since $1{/}_{{\equiv }_{(L_1,{[·,·]}_1)}}=1{/}_{{\equiv }_{(L_2,{[·,·]}_2)}}=\left\{1\right\}$:

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${\rho }_{(L_2,{[·,·]}_2)}(s)$

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We have: $\varnothing =D_{(L_1,{[·,·]}_1)}(0)\cap {\mathrm{Min}}_{(L_1,{[·,·]}_1)}=V_{(L_1,{[·,·]}_1)}(s)\cap {\mathrm{Min}}_{(L_1,{[·,·]}_1)}$ for every $s\in \{v,w,x,y,z,1\}$, $\left\{g\right\}=D_{(L_1,{[·,·]}_1)}(t)\cap {\mathrm{Min}}_{(L_1,{[·,·]}_1)}=V_{(L_1,{[·,·]}_1)}(s)\cap {\mathrm{Min}}_{(L_1,{[·,·]}_1)}$ for each $s\in \{b,c,d,e,f,g\}$, $\left\{t\right\}=D_{(L_1,{[·,·]}_1)}(s)\cap {\mathrm{Min}}_{(L_1,{[·,·]}_1)}=V_{(L_1,{[·,·]}_1)}(t)\cap {\mathrm{Min}}_{(L_1,{[·,·]}_1)}$ for every $s\in \{b,c,d,e,f,g\}$, $\{g,t\}=D_{(L_1,{[·,·]}_1)}(s)\cap {\mathrm{Min}}_{(L_1,{[·,·]}_1)}=V_{(L_1,{[·,·]}_1)}(0)\cap {\mathrm{Min}}_{(L_1,{[·,·]}_1)}$ for each $s\in \{v,w,x,y,z,1\}$.

$\varnothing =D_{(L_2,{[·,·]}_2)}(0)\cap {\mathrm{Min}}_{(L_2,{[·,·]}_2)}=V_{(L_2,{[·,·]}_2)}(s)\cap {\mathrm{Min}}_{(L_2,{[·,·]}_2)}$ for every $s\in \{u,v,w,x,y,$$z,1\}$, $\left\{g\right\}=D_{(L_2,{[·,·]}_2)}(t)\cap {\mathrm{Min}}_{(L_2,{[·,·]}_2)}=V_{(L_2,{[·,·]}_2)}(s)\cap {\mathrm{Min}}_{(L_2,{[·,·]}_2)}$ for each $s\in \{a,b,c,d,e,$$f,g\}$, $\left\{t\right\}=D_{(L_2,{[·,·]}_2)}(s)\cap {\mathrm{Min}}_{(L_2,{[·,·]}_2)}=V_{(L_2,{[·,·]}_2)}(t)\cap {\mathrm{Min}}_{(L_2,{[·,·]}_2)}$ for every $s\in \{a,b,c,d,$$e,f,g\}$, $\{g,t\}=D_{(L_2,{[·,·]}_2)}(s)\cap {\mathrm{Min}}_{(L_2,{[·,·]}_2)}=V_{(L_2,{[·,·]}_2)}(0)\cap {\mathrm{Min}}_{(L_2,{[·,·]}_2)}$ for each $s\in \{u,v,w,x,y,z,1\}$.

The opens of these topologies are very easy to determine, so we will only point out to their lattice structures in what follows. Let us also consider the commutator lattices $(M_3,{*}_3)$, $(M_4,{*}_4)$, $(M_5,{*}_5)$ and $({\mathcal{C}}_6,*)$, with $M_3\cong {\mathcal{N}}_5\oplus {\mathcal{C}}_2$ (nonmodular), $M_4\cong {\mathcal{C}}_3\oplus {\mathcal{M}}_3\oplus {\mathcal{C}}_2$ (modular, nondistributive) and $M_5\cong {\mathcal{C}}_3\oplus {\mathcal{C}}_2^2$ (distributive, as is ${\mathcal{C}}_6$, of course), elements denoted as follows and the definitions below for the commutator operations; note that ${*}_5$ and *, for instance, are nonassociative:

See the (minimal) prime spectra of $(M_1,{*}_1)$ and $(M_2,{*}_2)$ above. We have ${\mathrm{Spec}}_{(M_3,{*}_3)}=\{2,3,5\}$ and ${\mathrm{Spec}}_{(M_4,{*}_4)}={\mathrm{Spec}}_{(M_5,{*}_5)}={\mathrm{Spec}}_{({\mathcal{C}}_6,*)}=\{0,2,3\}$, thus ${\mathrm{Min}}_{(M_3,{*}_3)}=\{3,5\}$, ${\mathrm{Min}}_{(M_4,{*}_4)}={\mathrm{Min}}_{(M_5,{*}_5)}={\mathrm{Min}}_{({\mathcal{C}}_6,*)}=\left\{0\right\}$, the radicals are defined as follows and we have the following quotients through the radical equivalence:

By Proposition 3, it follows that ${\mathcal{S}}_{\mathrm{Spec},(M_1,{*}_1)}\cong {\mathcal{F}}_{\mathrm{Spec},(M_1,{*}_1)}\cong M_1/{\equiv }_{(M_1,{*}_1)}\cong$${\mathcal{S}}_{\mathrm{Spec},(M_2,{*}_2)}\cong {\mathcal{F}}_{\mathrm{Spec},(M_2,{*}_2)}\cong M_2/{\equiv }_{(M_2,{*}_2)}\cong {\mathcal{C}}_2\oplus {\mathcal{C}}_2^2\oplus {\mathcal{C}}_2$ and ${\mathcal{S}}_{\mathrm{Spec},(M_4{*}_4)}\cong {\mathcal{F}}_{\mathrm{Spec},(M_4,{*}_4)}$$\cong M_4/{\equiv }_{(M_4,{*}_4)}\cong {\mathcal{S}}_{\mathrm{Spec},({\mathcal{C}}_6,*)}\cong {\mathcal{F}}_{\mathrm{Spec},({\mathcal{C}}_6,*)}\cong {\mathcal{C}}_6/{\equiv }_{({\mathcal{C}}_6,*)}\cong {\mathcal{C}}_4$, while $(M_3,{*}_3)$ and $(M_5,{*}_5)$ have nonisomorphic Stone and flat topologies on their minimal prime spectra as lattices ordered by set inclusion: ${\mathcal{S}}_{\mathrm{Spec},(M_3,{*}_3)}\cong {\mathcal{F}}_{\mathrm{Spec},(M_5,{*}_5)}\cong M_3/{\equiv }_{(M_3,{*}_3)}\cong {\mathcal{C}}_2^2\oplus {\mathcal{C}}_2$ and ${\mathcal{F}}_{\mathrm{Spec},(M_3,{*}_3)}\cong {\mathcal{S}}_{\mathrm{Spec},(M_5,{*}_5)}\cong M_5/{\equiv }_{(M_5,{*}_5)}\cong {\mathcal{C}}_2\oplus {\mathcal{C}}_2^2$.

By Proposition 9, ${\mathcal{S}}_{\mathrm{Min},(M_1,{*}_1)}={\mathcal{F}}_{\mathrm{Min},(M_1,{*}_1)}={\mathcal{S}}_{\mathrm{Min},(M_2,{*}_2)}={\mathcal{F}}_{\mathrm{Min},(M_2,{*}_2)}=$${\mathcal{S}}_{\mathrm{Min},(M_4,{*}_4)}={\mathcal{F}}_{\mathrm{Min},(M_4,{*}_4)}={\mathcal{S}}_{\mathrm{Min},(M_5,{*}_5)}={\mathcal{F}}_{\mathrm{Min},(M_5,{*}_5)}={\mathcal{S}}_{\mathrm{Min},({\mathcal{C}}_6,*)}={\mathcal{F}}_{\mathrm{Min},({\mathcal{C}}_6,*)}=\mathcal{P}(\left\{0\right\})\cong {\mathcal{C}}_2$ and ${\mathcal{S}}_{\mathrm{Min},(M_3,{*}_3)}={\mathcal{F}}_{\mathrm{Min},(M_3,{*}_3)}=\mathcal{P}(\{3,5\})\cong {\mathcal{C}}_2^2$.

Let’s also consider $(L_3,{[·,·]}_3):=(M_3,{*}_3)\times ({\mathcal{C}}_2,\wedge )$, with the elements denoted as follows:

We have ${\mathrm{Spec}}_{(L_3,{[·,·]}_3)}=\{g,w,y,z\}$, hence the following radicals and thus the quotient lattice $L_3/{\equiv }_{(L_3,{[·,·]}_3)}$ above, and ${\mathrm{Min}}_{(L_3,{[·,·]}_3)}=\{g,w,y\}$:

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Therefore ${\mathcal{S}}_{\mathrm{Spec},(L_3,{[·,·]}_3)}\cong L_3/{\equiv }_{(L_3,{[·,·]}_3)}\cong ({\mathcal{C}}_2^2\oplus {\mathcal{C}}_2)\times {\mathcal{C}}_2$ and thus ${\mathcal{F}}_{\mathrm{Spec},(L_3,{[·,·]}_3)}\cong ({\mathcal{C}}_2\oplus {\mathcal{C}}_2^2)\times {\mathcal{C}}_2$: again, ${\mathcal{S}}_{\mathrm{Spec},(L_3,{[·,·]}_3)}$ and ${\mathcal{F}}_{\mathrm{Spec},(L_3,{[·,·]}_3)}$ are nonisomorphic. Also, ${\mathcal{S}}_{\mathrm{Min},(L_3,{[·,·]}_3)}$$={\mathcal{F}}_{\mathrm{Min},(L_3,{[·,·]}_3)}=\mathcal{P}({\mathrm{Min}}_{(L_3,{[·,·]}_3)})=\mathcal{P}(\{g,w,y\})\cong {\mathcal{C}}_2^3$.

6. (Min–)Admissible Maps

Throughout this section, $(L,[·,·])$ and $(M,\langle ·,·\rangle )$ will be commutator lattices and $h:L\to M$ will be a map that preserves arbitrary joins.

Then h is order–preserving and, according to [32], h has a unique right adjoint $h_{*}$, that is a map $h_{*}:M\to L$ satisfying:

$(\forall a\in L)(\forall b\in M)(h(a)\le b\iff a\le h_{*}(b))$,

namely: for all $b\in M$, ${\displaystyle h_{*}(b)=\bigvee \{a\in L|h(a)\le b\}}$.

Note that $h_{*}$ preserves arbitrary meets, so it is also order–preserving and, for every $a\in L$ and $b\in M$, we have: $a\le h_{*}(h(a))$ and $h(h_{*}(b))\le b$.

We consider the direct image of $h_{*}$. We call the map h:

• admissible if and only if $h_{*}({\mathrm{Spec}}_{(M,\langle ·,·\rangle )})\subseteq {\mathrm{Spec}}_{(L,[·,·])}$;
• $\mathrm{Min}$–admissible if and only if $h_{*}({\mathrm{Min}}_{(M,\langle ·,·\rangle )})\subseteq {\mathrm{Min}}_{(L,[·,·])}$.

If h is admissible, respectively $\mathrm{Min}$–admissible, then we will consider its restriction and corestriction $h_{*}:{\mathrm{Spec}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Spec}}_{(L,[·,·])}$, respectively $h_{*}:{\mathrm{Min}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Min}}_{(L,[·,·])}$.

We say that h preserves compactness if and only if $h(\mathrm{Cp}(L))\subseteq \mathrm{Cp}(M)$.

We also consider two members A and B of a variety $\mathcal{V}$ of similar algebras whose term–condition commutators are commutative and distributive with respect to arbitrary joins, along with a morphism $f:A\to B$ in $\mathcal{V}$.

We denote by $f^{•}:\mathrm{Con}(A)\to \mathrm{Con}(B)$ the map defined by $f^{•}(\alpha )=C{g}_B(f(\alpha ))$ for all $\alpha \in \mathrm{Con}(A)$, where $C{g}_B(Y)$ denotes the congruence of B generated by Y for any $Y\subseteq B^2$. We denote by $f^{*}:={(f\times f)}^{-1}{\mid }_{\mathrm{Con}(B)}:\mathrm{Con}(B)\to \mathrm{Con}(A)$: the inverse image of $f\times f:A^2\to B^2$ restricted to $\mathrm{Con}(B)$ and corestricted to $\mathrm{Con}(A)$.

Then $(\mathrm{Con}(A),{[·,·]}_A)$ and $(\mathrm{Con}(B),{[·,·]}_B)$ are algebraic commutator lattices, $f^{•}$ preserves arbitrary joins and the right adjoint of $f^{•}$ is $f^{*}$: ${(f^{•})}_{*}=f^{*}$. Also, $f^{•}$ preserves compactness, because $f^{•}(C{g}_A(X))=C{g}_B(f(X))$ for any $X\subseteq A^2$ [6,36,37].

Following [6], we call the morphism f:

• admissible if and only if $f^{•}$ is admissible;
• $\mathrm{Min}$–admissible if and only if $f^{•}$ is $\mathrm{Min}$–admissible.

We use the following notations from [6]: $\mathrm{Min}(A):={\mathrm{Min}}_{(\mathrm{Con}(A),{[·,·]}_A)}$, $\mathcal{M}in(A):=\mathcal{M}i{n}_{(\mathrm{Con}(A),{[·,·]}_A)}$ and $\mathcal{M}i{n}^{-1}(A):=\mathcal{M}i{n}_{(\mathrm{Con}(A),{[·,·]}_A)}^{-1}$.

Remark 26.

For any $a\in L$ and $m\in M$, we have: $h(a)\le m\iff a\le h_{*}(m)$, that is:

$$m\in {[h(a))}_M\iff h_{*}(m)\in {[a)}_L,$$

hence also: $m\notin {[h(a))}_M\iff h_{*}(m)\notin {[a)}_L$.

Therefore:

• if h is admissible and $m\in {\mathrm{Spec}}_{(M,\langle ·,·\rangle )}$, then:
  $m\in V_{(M,\langle ·,·\rangle )}(h(a))\iff h_{*}(m)\in V_{(L,[·,·])}(a)$ and
  $m\in D_{(M,\langle ·,·\rangle )}(h(a))\iff h_{*}(m)\in D_{(L,[·,·])}(a)$;
• if h is $\mathrm{Min}$–admissible and $m\in {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$, then:
  $m\in V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}\iff h_{*}(m)\in V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$ and
  $m\in D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}\iff h_{*}(m)\in D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$.

Thus:

• if h is admissible, then:
  $h_{*}(V_{(M,\langle ·,·\rangle )}(h(a)))\subseteq V_{(L,[·,·])}(a)$, so $V_{(M,\langle ·,·\rangle )}(h(a))\subseteq h_{*}^{-1}(V_{(L,[·,·])}(a))$,
  and $h_{*}(D_{(M,\langle ·,·\rangle )}(h(a)))\subseteq D_{(L,[·,·])}(a)$, so $D_{(M,\langle ·,·\rangle )}(h(a))\subseteq h_{*}^{-1}(D_{(L,[·,·])}(a))$;
• if h is $\mathrm{Min}$–admissible, then:
  $h_{*}(V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})\subseteq V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$,
  so $V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}\subseteq h_{*}^{-1}(V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})$,
  and $h_{*}(D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})\subseteq D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$,
  so $D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}\subseteq h_{*}^{-1}(D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})$.

Let us consider the following condition on h:

Condition 3.

For any $a,b\in \mathrm{Cp}(L)$, $h([a,b])\le \langle h(a),h(b)\rangle$.

Remark 27.

If L and M are algebraic and h preserves compactness and satisfies Condition 3, then: for any $x,y\in L$, $h([x,y])\le \langle h(x),h(y)\rangle$.

Indeed, then, for any $x,y\in L$, since the commutator is distributive with respect to arbitrary joins and h preserves arbitrary joins, along with the fact that h preserves compactness and the lattice order, we have:

${\displaystyle h([x,y])=h([\underset{a\in \mathrm{Cp}(L)\cap {(x]}_L}{\bigvee }a,\underset{b\in \mathrm{Cp}(L)\cap {(y]}_L}{\bigvee }b])=h(\underset{a\in \mathrm{Cp}(L)\cap {(x]}_L}{\bigvee }\underset{b\in \mathrm{Cp}(L)\cap {(y]}_L}{\bigvee }[a,b])=}$

${\displaystyle \underset{a\in \mathrm{Cp}(L)\cap {(x]}_L}{\bigvee }\underset{b\in \mathrm{Cp}(L)\cap {(y]}_L}{\bigvee }h([a,b])\le \underset{a\in \mathrm{Cp}(L)\cap {(x]}_L}{\bigvee }\underset{b\in \mathrm{Cp}(L)\cap {(y]}_L}{\bigvee }\langle h(a),h(b)\rangle =}$

${\displaystyle \langle \underset{a\in \mathrm{Cp}(L)\cap {(x]}_L}{\bigvee }h(a),\underset{b\in \mathrm{Cp}(L)\cap {(y]}_L}{\bigvee }h(b)\rangle \le \langle \underset{c\in \mathrm{Cp}(M)\cap {(h(x)]}_M}{\bigvee }c,\underset{d\in \mathrm{Cp}(M)\cap {(h(y)]}_M]}{\bigvee }d\rangle =\langle h(x),h(y)\rangle }$.

Similarly if we replace in Condition 3 the set $\mathrm{Cp}(L)$ with any join–dense subset S of L such that $h(S)$ is join–dense in M.

By [4] (Lemma $4.17$), if $\mathcal{V}$ is congruence–modular, then $f^{•}$ satisfies Condition 3. Note that Condition 3 on $i_{A,B}^{•}$ or congruence–modularity for $\mathcal{V}$ needs to be enforced in Propositions 10–19, Corollaries 3–8, Remark 3 and Theorems 3–7 in [6].

Remark 28.

By [32] (Theorem 1), if $[x,1]=x$ for all $x\in L$ and $\langle y,1\rangle =y$ for all $y\in M$, then the following are equivalent:

• h is admissible;
• for all $a,b\in L$, $V_{(M,\langle ·,·\rangle )}(h([a,b]))=V_{(M,\langle ·,·\rangle )}(\langle h(a),h(b)\rangle )$.

Thus, if h is admissible, $[x,1]=x$ for all $x\in L$ and $\langle y,1\rangle =y$ for all $y\in M$, then:

• for all $a,b\in L$ such that $\langle h(a),h(b)\rangle \in R(M,\langle ·,·\rangle )$, we have, by Proposition 1. (2): $h([a,b])\le {\rho }_{(M,\langle ·,·\rangle )}(h([a,b]))={\rho }_{(M,\langle ·,·\rangle )}(\langle h(a),h(b)\rangle )=\langle h(a),h(b)\rangle$;
• thus, if $\{\langle h(a),h(b)\rangle |a,b\in L\}\subseteq R(M,\langle ·,·\rangle )$, in particular if $R(M,\langle ·,·\rangle )=M$, in particular if M is algebraic and $\langle ·,·\rangle =\wedge$, then h satisfies Condition 3.

Remark 29.

By [32] (Lemma 7), if L and M are algebraic, $[x,1]=x$ for all $x\in L$, $\langle y,1\rangle =y$ for all $y\in M$ and h preserves compactness, then $h(\mathrm{Cp}(L))=\mathrm{Cp}(M)$.

Remark 30.

Trivially, the commutator lattice automorphism $i{d}_L:L\to L$ has ${(i{d}_L)}_{*}=i{d}_L$, hence $i{d}_L$ is admissible and $\mathrm{Min}$–admissible. Trivially, $i{d}_L$ preserves compactness and satisfies Condition 3.

Now let $(L_1,{[·,·]}_1)$, $(L_2,{[·,·]}_2)$, $(M_1,{\langle ·,·\rangle }_1)$ and $(M_2,{\langle ·,·\rangle }_2)$ be commutator lattices, $h_1:L_1\to M_1$ and $h_2:L_2\to M_2$, and let us consider the direct product map $h_1\times h_2:L_1\times L_2\to M_1\times M_2$. Then, clearly:

• $h_1\times h_2$ preserves arbitrary joins if and only if $h_1$ and $h_2$ preserve arbitrary joins;
• $h_1\times h_2$ preserves compactness if and only if $h_1$ and $h_2$ preserve compactness;
• $h_1\times h_2$ satisfies Condition 3 if and only if $h_1$ and $h_2$ satisfy Condition 3.

For all $b_1\in M_1$ and $b_2\in M_2$, ${(h_1\times h_2)}_{*}(b_1,b_2)=\bigvee \{(a_1,a_2)|a_1\in L_1,a_2\in L_2,(h_1(a_1),$$h_2(a_2))\le (b_1,b_2)\}=\bigvee \left\{(a_1,a_2)\right|a_1\in L_1,a_2\in L_2,h_1(a_1)\le b_1,h_2(a_2)\le b_2\}=(\bigvee \{a_1\in L_1|h_1(a_1)\le b_1\},\bigvee \{a_2\in L_2|h_2(a_2)\le b_2\})=({(h_1)}_{*}(b_1),{(h_2)}_{*}(b_2))$, thus ${(h_1\times h_2)}_{*}={(h_1)}_{*}\times {(h_2)}_{*}$.

Consequently, if ${(h_1)}_{*}(1)=1$, ${(h_2)}_{*}(1)=1$ and 1 is a neutral element with respect to the commutator in each of $(L_1,{[·,·]}_1)$, $(L_2,{[·,·]}_2)$, $(M_1,{\langle ·,·\rangle }_1)$ and $(M_2,{\langle ·,·\rangle }_2)$, then, by Remark 25:

• $h_1\times h_2$ is admissible if and only if $h_1$ and $h_2$ are admissible;
• $h_1\times h_2$ is $\mathrm{Min}$–admissible if and only if $h_1$ and $h_2$ are $\mathrm{Min}$–admissible.

Example 2.

Let us consider the commutator lattices in Example 1. Since they are finite and thus compact, trivially any join–preserving map between two of these lattices preserves arbitrary joins and compactness.

Let $k:M_2\to M_1$ be the map defined by: $k(7)=6$ and $k(s)=s$ for all $s\in M_1$, and let $k^{\prime }:=k\times i{d}_{{\mathcal{C}}_2}:L_2\to L_1$, so that $k^{\prime }(a)=b$, $k^{\prime }(u)=v$ and $k^{\prime }(s)=s$ for all $s\in L_1$. Note that k and $k^{\prime }$ are surjective (bounded) lattice morphisms, while $i_{M_1,M_2}$ and $i_{L_1,L_2}=i_{M_1,M_2}\times i{d}_{{\mathcal{C}}_2}$ are, of course, bounded lattice embeddings.

For any $s\in M_2$, ${(i_{M_1,M_2})}_{*}(s)=\bigvee \{r\in M_1|r\le s\}$, thus: ${(i_{M_1,M_2})}_{*}(7)=0$ and ${(i_{M_1,M_2})}_{*}(s)=s$ for all $s\in M_1$. For any $s\in L_2$, ${(i_{L_1,L_2})}_{*}(s)=\bigvee \{r\in L_1|r\le s\}$, thus: ${(i_{L_1,L_2})}_{*}(a)=0$, ${(i_{L_1,L_2})}_{*}(u)=t$ and ${(i_{L_1,L_2})}_{*}(s)=s$ for all $s\in L_1$, which could also have been obtained from the definition of ${(i_{M_1,M_2})}_{*}$ and Remark 30. ${(i_{M_1,M_2})}_{*}$ and ${(i_{L_1,L_2})}_{*}$ are surjective (bounded) lattice morphisms, as well, so we may calculate their right adjoints.

For any $r\in M_1$, $k_{*}(r)=\bigvee \{s\in M_2|k(s)\le r\}$ and ${(i_{M_1,M_2})}_{**}(r)=\bigvee \{s\in M_2|{(i_{M_1,M_2})}_{*}(s)\le r\}$, thus $k_{*}=i_{M_1,M_2}={(i_{M_1,M_2})}_{**}$. Now we can apply Remark 30 or make the direct verification: for any $r\in L_1$, $k_{*}^{\prime }(r)=\bigvee \{s\in L_2|k^{\prime }(s)\le r\}$ and ${(i_{L_1,L_2})}_{**}(r)=\bigvee \{s\in L_2|{(i_{L_1,L_2})}_{*}(s)\le r\}$, thus $k_{*}^{\prime }=i_{L_1,L_2}={(i_{L_1,L_2})}_{**}$.

Notice that none of the maps $i_{M_1,M_2}$, ${(i_{M_1,M_2})}_{*}$, $i_{L_1,L_2}$, ${(i_{L_1,L_2})}_{*}$, k and $k^{\prime }$ is a commutator lattice morphism. k and ${(i_{M_1,M_2})}_{*}$ satisfy Condition 3, while $i_{M_1,M_2}$ fails Condition 3, so, as expected from Remark 30, $k^{\prime }$ and ${(i_{L_1,L_2})}_{*}$ satisfy Condition 3, while $i_{L_1,L_2}$ fails Condition 3.

$i_{M_1,M_2}({\mathrm{Spec}}_{(M_1,{*}_1)})={\mathrm{Spec}}_{(M_1,{*}_1)}={\mathrm{Spec}}_{(M_2,{*}_2)}$ and $i_{M_1,M_2}({\mathrm{Min}}_{(M_1,{*}_1)})={\mathrm{Min}}_{(M_1,{*}_1)}$$={\mathrm{Min}}_{(M_2,{*}_2)}$, thus $i_{L_1,L_2}({\mathrm{Spec}}_{(L_1,{[·,·]}_1)})={\mathrm{Spec}}_{(L_1,{[·,·]}_1)}={\mathrm{Spec}}_{(L_2,{[·,·]}_2)}$ and $i_{L_1,L_2}({\mathrm{Min}}_{(L_1,{[·,·]}_1)})$$={\mathrm{Min}}_{(L_1,{[·,·]}_1)}={\mathrm{Min}}_{(L_2,{[·,·]}_2)}$, hence k, ${(i_{M_1,M_2})}_{*}$, $k^{\prime }$ and ${(i_{L_1,L_2})}_{*}$ are admissible and $\mathrm{Min}$–admissible. Of course, the ($\mathrm{Min}$–)admissibility of the first two is equivalent to that of the last two, by Remark 30.

${(i_{M_1,M_2})}_{*}({\mathrm{Spec}}_{(M_2,{*}_2)})={\mathrm{Spec}}_{(M_1,{*}_1)}$ and ${(i_{M_1,M_2})}_{*}({\mathrm{Min}}_{(M_2,{*}_2)})={\mathrm{Min}}_{(M_1,{*}_1)}$, thus ${(i_{L_1,L_2})}_{*}({\mathrm{Spec}}_{(L_2,{[·,·]}_2)})={\mathrm{Spec}}_{(L_1,{[·,·]}_1)}$ and ${(i_{L_1,L_2})}_{*}({\mathrm{Min}}_{(L_2,{[·,·]}_2)})={\mathrm{Min}}_{(L_1,{[·,·]}_1)}$, hence $i_{M_1,M_2}$ and $i_{L_1,L_2}$ are admissible and $\mathrm{Min}$–admissible. Again, the ($\mathrm{Min}$–)admissibility of the former is equivalent to that of the latter, by Remark 30.

Note that $i_{M_3,M_1}$ is a join–semilattice morphism, but not a lattice morphism and it does not preserve the commutator, but it satisfies Condition 3, thus the same holds for $i_{L_3,L_1}=i_{M_3,M_1}\times i{d}_{{\mathcal{C}}_2}$. Let us also consider the surjective commutator lattice morphisms $l:M_1\to M_3$, defined by $l(6)=0$ and $l(s)=s$ for all $s\in M_3$, and $l^{\prime }:=l\times i{d}_{{\mathcal{C}}_2}:L_1\to L_3$; since they preserve the commutator, in particular they satisfy Condition 3. We will apply Remark 30.

For all $s\in M_1$, ${(i_{M_3,M_1})}_{*}(s)=\bigvee \{r\in M_1|r\le s\}$, thus ${(i_{M_3,M_1})}_{*}=l$, hence ${(i_{L_3,L_1})}_{*}={(i_{M_3,M_1})}_{*}\times {(i{d}_{{\mathcal{C}}_2})}_{*}=l\times i{d}_{{\mathcal{C}}_2}=l^{\prime }$.

For all $s\in M_3$, $l_{*}(s)=\bigvee \{r\in M_1|l(r)\le s\}$, so $l_{*}(0)=6$ and $l_{*}(s)=s$ for all $s\in M_3\setminus \left\{0\right\}$, hence $l_{*}$ is a lattice morphism, but not a bounded lattice morphism, thus clearly so is $l_{*}^{\prime }=l_{*}\times {(i{d}_{{\mathcal{C}}_2})}_{*}=l_{*}\times i{d}_{{\mathcal{C}}_2}$.

For all $s\in M_1$, $l_{**}(s)=\bigvee \{r\in M_3|l_{*}(r)\le s\}$, so $l_{**}=l$, thus $l_{**}^{\prime }={(l_{*}\times i{d}_{{\mathcal{C}}_2})}_{*}=l_{*}*\times {(i{d}_{{\mathcal{C}}_2})}_{*}=l\times i{d}_{{\mathcal{C}}_2}=l^{\prime }$.

$l_{*}({\mathrm{Spec}}_{(M_3,{*}_3)})=l_{*}(\{2,3,5\})=\{2,3,5\}\subset \{0,2,3,5\}={\mathrm{Spec}}_{(M_1,{*}_1)}$, but $l_{*}({\mathrm{Min}}_{(M_3,{*}_3)})$$=l(\{3,5\})=\{3,5\}⊈\left\{0\right\}={\mathrm{Min}}_{(M_1,{*}_1)}$, thus l is admissible, but not $\mathrm{Min}$–admissible, hence $l_{*}^{\prime }$ is admissible, but not $\mathrm{Min}$–admissible.

$l({\mathrm{Spec}}_{(M_1,{*}_1)})=l(\{0,2,3,5\})=\{0,2,3,5\}⊈\{2,3,5\}={\mathrm{Spec}}_{(M_3,{*}_3)}$ and $l({\mathrm{Min}}_{(M_1,{*}_1)})$$=l(\left\{0\right\})=\left\{0\right\}⊈\{3,5\}={\mathrm{Min}}_{(M_3,{*}_3)}$, thus $i_{M_3,M_1}$ and $l_{*}$ are neither admissible, nor $\mathrm{Min}$–admissible, hence $i_{L_3,L_1}$ and $l_{*}^{\prime }$ are neither admissible, nor $\mathrm{Min}$–admissible.

Finally, let us consider the bounded lattice embeddings $i:M_5\to M_4$ and $j:{\mathcal{C}}_6\to M_4$, along with the neither injective, nor surjective bounded lattice morphism $n:M_1\to {\mathcal{C}}_6$, neither of which satisfy Condition 3, in particular neither preserves the commutators:

s	0	1	2	3	4	5	6
$i(s)$	0	1	5	6	4	2	·
$j(s)$	0	1	2	3	5	4	·
$n(s)$	0	1	3	2	2	3	2

We have $i_{*}:M_4\to M_5$ and $j_{*}:M_4\to {\mathcal{C}}_6$, defined by: $i_{*}(s)=\bigvee \{r\in M_5|i(r)\le s\}$ and $j_{*}(s)=\bigvee \{r\in {\mathcal{C}}_6|j(r)\le s\}$ for any $s\in M_4$. Also, $n_{*}:{\mathcal{C}}_6\to M_1$, $n_{*}(s)=\bigvee \{r\in M_1|n(r)\le s\}$ for all $s\in {\mathcal{C}}_6$. So these maps are given by the following tables, which show that $i_{*}$ and $j_{*}$ are not join–preserving, but $n_{*}$ is a bounded lattice morphism, so we may calculate $n_{*}*:M_1\to {\mathcal{C}}_6$, $n_{**}(s)=\bigvee \{r\in {\mathcal{C}}_6|n_{*}(r)\le s\}$ for all $s\in M_1$. We obtain the following definition, which shows that $n_{**}$ does not preserve the join.

s	0	1	2	3	4	5	6	7
$i_{*}(s)$	0	1	5	3	4	2	3	4
$j_{*}(s)$	0	1	2	3	5	4	5	5
$n_{*}(s)$	0	1	3	2	3	3	·	·
$n_{**}(s)$	0	1	3	4	0	0	0	·

$j_{*}({\mathrm{Spec}}_{(M_4,{*}_4)})=j_{*}(\{0,2,3\})=\{0,2,3\}={\mathrm{Spec}}_{({\mathcal{C}}_6,*)}$ and $j_{*}({\mathrm{Min}}_{(M_4,{*}_4)})=j_{*}(\left\{0\right\})=\left\{0\right\}$, while $n_{*}({\mathrm{Spec}}_{({\mathcal{C}}_6,*)})=n_{*}(\{0,2,3\})=\{0,2,3\}\subset \{0,2,3,5\}={\mathrm{Spec}}_{(M_1,{*}_1)}$ and $n_{*}({\mathrm{Min}}_{({\mathcal{C}}_6,*)})=n_{*}(\left\{0\right\})=\left\{0\right\}={\mathrm{Min}}_{({\mathcal{C}}_6,*)}$, thus j and n are admissible and $\mathrm{Min}$–admissible;

$i_{*}({\mathrm{Spec}}_{(M_4,{*}_4)})=i_{*}(\{0,2,3\})=\{0,3,5\}⊈\{0,2,3\}={\mathrm{Spec}}_{(M_5,{*}_5)}$ and $i_{*}({\mathrm{Min}}_{(M_4,{*}_4)})$$=i_{*}(\left\{0\right\})=\left\{0\right\}={\mathrm{Min}}_{(M_5,{*}_5)}$, while $n_{**}({\mathrm{Spec}}_{(M_1,{*}_1)})=n_{**}(\{0,2,3,5\})=\{0,3,4\}⊈{\mathrm{Spec}}_{({\mathcal{C}}_6,*)}$ and $n_{**}({\mathrm{Min}}_{(M_1,{*}_1)})=n_{**}(\left\{0\right\})=\left\{0\right\}={\mathrm{Min}}_{({\mathcal{C}}_6,*)}$, hence i and $n_{*}$ are $\mathrm{Min}$–admissible, but not admissible.

Lemma 19.

If h preserves compactness and satisfies Condition 3, then, for any m–system S in $(L,[·,·])$, $h(S)$ is an m–system in $(M,\langle ·,·\rangle )$.

Proof. 

Since h preserves compactness, $h(S)\subseteq \mathrm{Cp}(M)$.

Let $u,v\in h(S)$, so that $u=h(a)$ and $v=h(b)$ for some $a,b\in S$. Since S is an m–system in $(L,[·,·])$, there exists $c\in S$ such that $c\le [a,b]$. Then $h(c)\in h(S)$ and, since h is order–preserving and satisfies Condition 3, $h(c)\le h([a,b])\le \langle h(a),h(b)\rangle$. □

Lemma 20.

Assume that L and M are algebraic, $1\in \mathrm{Cp}(M)$ and h preserves compactness and satisfies Condition 3.

• If h is admissible, then $h_{*}({\mathrm{Min}}_{(M,\langle ·,·\rangle )})\supseteq {\mathrm{Min}}_{(L,[·,·])}$.
• If h is $\mathrm{Min}$–admissible, then $h_{*}({\mathrm{Min}}_{(M,\langle ·,·\rangle )})={\mathrm{Min}}_{(L,[·,·])}$.

Proof. 

Assume that h is admissible or $\mathrm{Min}$–admissible, so that $h_{*}({\mathrm{Min}}_{(M,\langle ·,·\rangle )})\subseteq {\mathrm{Spec}}_{(L,[·,·])}$.

Let $p\in {\mathrm{Min}}_{(L,[·,·])}$, thus $p\in {\mathrm{Spec}}_{(L,[·,·])}$. Since L is algebraic, it follows that $\mathrm{Cp}(L)\setminus {(p]}_L$ is an m–system in $(L,[·,·])$, hence $h(\mathrm{Cp}(L)\setminus {(p]}_L)$ is an m–system in $(M,\langle ·,·\rangle )$ by Lemmas 6 and 19.

By Lemma 7, there exists a $q\in Max\{y\in M|h(\mathrm{Cp}(L)\setminus {(p]}_L)\cap {(y]}_M=\varnothing \}$, so $h(\mathrm{Cp}(L)\setminus {(p]}_L)\cap {(q]}_M=\varnothing$ and $q\in {\mathrm{Spec}}_{(M,\langle ·,·\rangle )}$, thus there exists an $m\in {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$ with $m\le p$ and hence $h(\mathrm{Cp}(L)\setminus {(p]}_L)\cap {(m]}_M=\varnothing$, that is $h(\mathrm{Cp}(L)\setminus {(p]}_L)\subseteq M\setminus {(m]}_M$, so each $a\in \mathrm{Cp}(L)$ satisfies:

if $a\nleqq p$, then $h(a)\nleqq m$,

that is: if $a\nleqq p$, then $a\nleqq h_{*}(m)$,

otherwise written: if $a\le h_{*}(m)$, then $a\le p$,

therefore, since L is algebraic: $h_{*}(m)=\bigvee (\mathrm{Cp}(L)\cap {(h_{*}(m)]}_L)\le \bigvee (\mathrm{Cp}(L)\cap {(p]}_L)=p$.

Since $h_{*}(m)\in {\mathrm{Spec}}_{(L,[·,·])}$ and $p\in {\mathrm{Min}}_{(L,[·,·])}$, it follows that $h_{*}(m)=p$.

Hence ${\mathrm{Min}}_{(L,[·,·])}\subseteq h_{*}({\mathrm{Min}}_{(M,\langle ·,·\rangle )})$. If h is $\mathrm{Min}$–admissible, then the converse inclusion holds, as well. □

Lemma 21.

If L and M are algebraic, $1\in \mathrm{Cp}(M)$, h is $\mathrm{Min}$–admissible, preserves compactness and satisfies Condition 3, then, for any $a\in L$:

• $h_{*}(V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$;
• $V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}=h_{*}^{-1}(V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})$;
• $h_{*}(D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$;
• $D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}=h_{*}^{-1}(D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})$.

Proof. 

Let $p\in {\mathrm{Min}}_{(L,[·,·])}$. By Lemma 20, there exists an $m\in {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$ such that $h_{*}(m)=p$. By Remark 26:

$p\in V_{(L,[·,·])}(a)$ if and only if $m\in V_{(M,\langle ·,·\rangle )}(h(a))$;

$p\in D_{(L,[·,·])}(a)$ if and only if $m\in D_{(M,\langle ·,·\rangle )}(h(a))$.

Therefore:

$V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}\subseteq h_{*}(V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})$,

$h_{*}^{-1}(V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})\subseteq V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$,

$D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}\subseteq h_{*}(D_{(M,\langle ·,·\rangle )}(h(a)))$,

$h_{*}^{-1}(D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})\subseteq D_{(M,\langle ·,·\rangle )}(h(a))$.

By Remark 26, the converse inclusions hold, as well. □

Lemma 22.

Assume that L and M are algebraic, $1\in \mathrm{Cp}(M)$, h is $\mathrm{Min}$–admissible, preserves compactness and satisfies Condition 3 and let $a\in L$.

(1) 

If ${\rho }_{(L,[·,·])}(0)=0$ and either $(L,[·,·])$ satisfies Condition 2 or $a\in \mathrm{Cp}(L)$ and $(L,[·,·])$ satisfies Condition 1, then:

• $h_{*}(V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}=D_{(L,[·,·])}(a^{\perp (L,[·,·])})$$\cap {\mathrm{Min}}_{(L,[·,·])}=h_{*}(D_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})$;
• $V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}=h_{*}^{-1}(V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=$$h_{*}^{-1}(D_{(L,[·,·])}(a^{\perp (L,[·,·])})\cap {\mathrm{Min}}_{(L,[·,·])})=D_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$;
• $h_{*}(D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}=V_{(L,[·,·])}(a^{\perp (L,[·,·])})$$\cap {\mathrm{Min}}_{(L,[·,·])}=h_{*}(V_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})$;
• $D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}=h_{*}^{-1}(D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=$$h_{*}^{-1}(V_{(L,[·,·])}(a^{\perp (L,[·,·])})\cap {\mathrm{Min}}_{(L,[·,·])})=V_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$.

(2) 

If ${\rho }_{(M,\langle ·,·\rangle )}(0)=0$ and either $(M,\langle ·,·\rangle )$ satisfies Condition 2 or $h(a)\in \mathrm{Cp}(M)$ (in particular if $a\in \mathrm{Cp}(L)$) and $(M,\langle ·,·\rangle )$ satisfies Condition 1, then:

• $h_{*}(D_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}=h_{*}(V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})$;
• $D_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}=h_{*}^{-1}(V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$;
• $h_{*}(V_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}=h_{*}(D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})$;
• $V_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}=h_{*}^{-1}(D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$.

(3) 

If ${\rho }_{(L,[·,·])}(0)=0$, ${\rho }_{(M,\langle ·,·\rangle )}(0)=0$ and either $(L,[·,·])$ and $(M,\langle ·,·\rangle )$ satisfy Condition 2 or $a\in \mathrm{Cp}(L)$ and $(L,[·,·])$ and $(M,\langle ·,·\rangle )$ satisfy Condition 1, then:

• $h_{*}(D_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}=h_{*}(V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=D_{(L,[·,·])}(a^{\perp (L,[·,·])})\cap {\mathrm{Min}}_{(L,[·,·])}=h_{*}(D_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})$;
• $D_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}=h_{*}^{-1}(V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}=h_{*}^{-1}(D_{(L,[·,·])}(a^{\perp (L,[·,·])})\cap {\mathrm{Min}}_{(L,[·,·])})=D_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$;
• $h_{*}(V_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}=h_{*}(D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=V_{(L,[·,·])}(a^{\perp (L,[·,·])})\cap {\mathrm{Min}}_{(L,[·,·])}=h_{*}(V_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})$;
• $V_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}=h_{*}^{-1}(D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}=h_{*}^{-1}(V_{(L,[·,·])}(a^{\perp (L,[·,·])})\cap {\mathrm{Min}}_{(L,[·,·])})=V_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$.

Proof. 

By Lemma 21 and Proposition 8. □

Theorem 3.

If L and M are algebraic, $1\in \mathrm{Cp}(M)$, h is $\mathrm{Min}$–admissible, preserves compactness and satisfies Condition 3, then:

(1) 

• $h_{*}:{\mathrm{Min}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Min}}_{(L,[·,·])}$ is continuous with respect to the Stone topologies and the flat topologies;
• if one of the following holds:
  $\circ$ ${\rho }_{(L,[·,·])}(0)=0$ and $(L,[·,·])$ satisfies one of the Conditions 1 and 2,
  $\circ$ ${\rho }_{(M,\langle ·,·\rangle )}(0)=0$ and $(M,\langle ·,·\rangle )$ satisfies one of the Conditions 1 and 2,
  then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}\to \mathcal{M}i{n}_{(L,[·,·])}^{-1}$ is continuous;
• if ${\rho }_{(L,[·,·])}(0)=0$ and $(L,[·,·])$ satisfies one of the Conditions 1 and 2, then:
  if $\left\{h(a^{\perp (L,[·,·])})\right|a\in \mathrm{Cp}(L)\}\subseteq \mathrm{Cp}(M)$, in particular if $\mathrm{Cp}(L)$ is closed with respect to the polar, then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}^{-1}\to \mathcal{M}i{n}_{(L,[·,·])}$ is continuous;
• if ${\rho }_{(M,\langle ·,·\rangle ))}(0)=0$ and $(M,\langle ·,·\rangle )$ satisfies one of the Conditions 1 and 2, then:
  if $\left\{h{(a)}^{\perp (M,\langle ·,·\rangle )}\right|a\in \mathrm{Cp}(L)\}\subseteq \mathrm{Cp}(M)$, in particular if $\mathrm{Cp}(M)$ is closed with respect to the polar, then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}^{-1}\to \mathcal{M}i{n}_{(L,[·,·])}$ is continuous;

(2) 

if $h_{*}:{\mathrm{Min}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Min}}_{(L,[·,·])}$ is injective, then:

• if $h(L)\supseteq \mathrm{Cp}(M)$, then $h_{*}:{\mathrm{Min}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Min}}_{(L,[·,·])}$ is a homeomorphism with respect to the Stone topologies;
• if $h(\mathrm{Cp}(L))=\mathrm{Cp}(M)$, then $h_{*}:{\mathrm{Min}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Min}}_{(L,[·,·])}$ is a homeomorphism with respect to the flat topologies;
• if one of the following holds:
  $\circ$ ${\rho }_{(L,[·,·])}(0)=0$, $(L,[·,·])$ satisfies one of the Conditions 1 and 2, $h(\mathrm{Cp}(L))=\mathrm{Cp}(M)$ and $\mathrm{Cp}(L)$ is closed with respect to the polar,
  $\circ$ ${\rho }_{(L,[·,·])}(0)=0$, $(L,[·,·])$ satisfies one of the Conditions 1 and 2 and $\left\{h(a^{\perp (L,[·,·])})\right|a\in \mathrm{Cp}(L)\}\supseteq \mathrm{Cp}(M)$,
  $\circ$ ${\rho }_{(M,\langle ·,·\rangle )}(0)=0$, $(M,\langle ·,·\rangle )$ satisfies one of the Conditions 1 and 2 and $\left\{h{(a)}^{\perp (M,\langle ·,·\rangle )}\right|a\in \mathrm{Cp}(L)\}\supseteq \mathrm{Cp}(M)$,
  then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}\to \mathcal{M}i{n}_{(L,[·,·])}^{-1}$ is a homeomorphism;
• if one of the following holds:
  $\circ$ ${\rho }_{(L,[·,·])}(0)=0$, $(L,[·,·])$ satisfies one of the Conditions 1 and 2 and $h(\mathrm{Cp}(L))=\mathrm{Cp}(M)$,
  $\circ$ ${\rho }_{(L,[·,·])}(0)=0$, $(L,[·,·])$ satisfies one of the Conditions 1 and 2 and $\left\{h(a^{\perp (L,[·,·])})\right|a\in \mathrm{Cp}(L)\}=\mathrm{Cp}(M)$,
  $\circ$ ${\rho }_{(L,[·,·])}(0)=0$, $(L,[·,·])$ satisfies Condition 2 and $\left\{h(a^{\perp (L,[·,·])})\right|a\in L\}\supseteq \mathrm{Cp}(M)\supseteq \left\{h(a^{\perp (L,[·,·])})\right|a\in \mathrm{Cp}(L)\}$,
  $\circ$ ${\rho }_{(M,\langle ·,·\rangle )}(0)=0$, $(M,\langle ·,·\rangle )$ satisfies one of the Conditions 1 and 2 and $\left\{h{(a)}^{\perp (M,\langle ·,·\rangle )}\right|a\in \mathrm{Cp}(L)\}=\mathrm{Cp}(M)$,
  $\circ$ ${\rho }_{(M,\langle ·,·\rangle )}(0)=0$, $(M,\langle ·,·\rangle )$ satisfies Condition 2 and $\left\{h{(a)}^{\perp (M,\langle ·,·\rangle )}\right|a\in L\}\supseteq \mathrm{Cp}(M)\supseteq \left\{h{(a)}^{\perp (M,\langle ·,·\rangle )}\right|a\in \mathrm{Cp}(L)\}$,
  then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}^{-1}\to \mathcal{M}i{n}_{(L,[·,·])}$ is a homeomorphism.

Proof. 

Assume that L and M are algebraic, $1\in \mathrm{Cp}(M)$, h is $\mathrm{Min}$–admissible, preserves compactness and satisfies Condition 3.

(1) By Lemma 21:

for any $a\in L$, $h_{*}^{-1}(D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$, therefore $h_{*}:{\mathrm{Min}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Min}}_{(L,[·,·])}$ is continuous with respect to the Stone topologies;

for any $a\in \mathrm{Cp}(L)$, $h_{*}^{-1}(V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$ and $h(a)\in \mathrm{Cp}(M)$, thus $h_{*}:{\mathrm{Min}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Min}}_{(L,[·,·])}$ is continuous with respect to the flat topologies.

By Lemma 22. (1), if ${\rho }_{(L,[·,·])}(0)=0$ and $(L,[·,·])$ satisfies Condition 1, respectively Condition 2, then, for any $a\in \mathrm{Cp}(L)$, respectively any $a\in L$:

$h_{*}^{-1}(V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=D_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$, thus $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}\to \mathcal{M}i{n}_{(L,[·,·])}^{-1}$ is continuous;

$h_{*}^{-1}(D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=V_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$, thus, if $\left\{h(a^{\perp (L,[·,·])})\right|a\in \mathrm{Cp}(L)\}\subseteq \mathrm{Cp}(M)$, in particular if $\mathrm{Cp}(L)$ is closed with respect to the polar (in view of the fact that h preserves compactness), then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}^{-1}\to \mathcal{M}i{n}_{(L,[·,·])}$ is continuous.

By Lemma 22. (2), if ${\rho }_{(M,\langle ·,·\rangle )}(0)=0$ and $(M,\langle ·,·\rangle )$ satisfies Condition 1, respectively Condition 2, then, for any $a\in \mathrm{Cp}(L)$, respectively any $a\in L$:

$h_{*}^{-1}(V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=D_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$, thus $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}\to \mathcal{M}i{n}_{(L,[·,·])}^{-1}$ is continuous;

$h_{*}^{-1}(D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])})=V_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )}$, thus, if $\left\{h(a^{\perp (L,[·,·])})\right|a\in \mathrm{Cp}(L)\}\subseteq \mathrm{Cp}(M)$, in particular if $\mathrm{Cp}(L)$ is closed with respect to the polar (in view of the fact that h preserves compactness), then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}^{-1}\to \mathcal{M}i{n}_{(L,[·,·])}$ is continuous.

(2) Now assume, furthermore, that $h_{*}:{\mathrm{Min}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Min}}_{(L,[·,·])}$ is injective. By Lemma 20, it is also surjective, thus it is bijective.

By Lemma 21, for any $a\in L$:

$h_{*}(D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$, thus, if $h(L)\supseteq \mathrm{Cp}(M)$, then $h_{*}:{\mathrm{Min}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Min}}_{(L,[·,·])}$ is open with respect to the Stone topologies, thus it is a homeomorphism with respect to the Stone topologies by (2);

$h_{*}(V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$, thus, if $h(\mathrm{Cp}(L))=\mathrm{Cp}(M)$, then $h_{*}:{\mathrm{Min}}_{(M,\langle ·,·\rangle )}\to {\mathrm{Min}}_{(L,[·,·])}$ is open with respect to the flat topologies, thus it is a homeomorphism with respect to the flat topologies by (2).

By Lemma 22. (1), if ${\rho }_{(L,[·,·])}(0)=0$ and $(L,[·,·])$ satisfies Condition 1, respectively Condition 2, then, for any $a\in \mathrm{Cp}(L)$, respectively any $a\in L$:

$h_{*}(D_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=V_{(L,[·,·])}(a^{\perp (L,[·,·])})\cap {\mathrm{Min}}_{(L,[·,·])}$, thus, if $h(\mathrm{Cp}(L))=\mathrm{Cp}(M)$ and $\mathrm{Cp}(L)$ is closed with respect to the polar, then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}\to \mathcal{M}i{n}_{(L,[·,·])}^{-1}$ is open, thus it is a homeomorphism by (1);

$h_{*}(D_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$, thus, if $\left\{h(a^{\perp (L,[·,·])})\right|a\in \mathrm{Cp}(L)\}\supseteq \mathrm{Cp}(M)$, then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}\to \mathcal{M}i{n}_{(L,[·,·])}^{-1}$ is open, thus it is a homeomorphism by (1);

$h_{*}(V_{(M,\langle ·,·\rangle )}(h(a))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=D_{(L,[·,·])}(a^{\perp (L,[·,·])})\cap {\mathrm{Min}}_{(L,[·,·])}$, thus, if $h(\mathrm{Cp}(L))=\mathrm{Cp}(M)$, then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}^{-1}\to \mathcal{M}i{n}_{(L,[·,·])}$ is open, thus it is a homeomorphism by (1);

$h_{*}(V_{(M,\langle ·,·\rangle )}(h(a^{\perp (L,[·,·])}))\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$, thus, if either $\left\{h(a^{\perp (L,[·,·])})\right|a\in \mathrm{Cp}(L)\}\supseteq \mathrm{Cp}(M)$ or $(L,[·,·])$ fulfills Condition 2 and $\left\{h(a^{\perp (L,[·,·])})\right|a\in L\}$$\supseteq \mathrm{Cp}(M)$, then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}^{-1}\to \mathcal{M}i{n}_{(L,[·,·])}$ is open, thus it is a homeomorphism by (1).

By Lemma 22. (2), if ${\rho }_{(M,\langle ·,·\rangle )}(0)=0$ and $(M,\langle ·,·\rangle )$ satisfies Condition 1, respectively Condition 2, then, for any $a\in L$ such that $h(a)\in \mathrm{Cp}(M)$ (in particular any $a\in \mathrm{Cp}(L)$), respectively any $a\in L$:

$h_{*}(D_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=V_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$, thus, if $\left\{h{(a)}^{\perp (M,\langle ·,·\rangle )}\right|a\in \mathrm{Cp}(L)\}\supseteq \mathrm{Cp}(M)$, then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}\to \mathcal{M}i{n}_{(L,[·,·])}^{-1}$ is open, thus it is a homeomorphism by (1);

$h_{*}(V_{(M,\langle ·,·\rangle )}(h{(a)}^{\perp (M,\langle ·,·\rangle )})\cap {\mathrm{Min}}_{(M,\langle ·,·\rangle )})=D_{(L,[·,·])}(a)\cap {\mathrm{Min}}_{(L,[·,·])}$, thus, if either $\left\{h{(a)}^{\perp (M,\langle ·,·\rangle )}\right|a\in \mathrm{Cp}(L)\}\supseteq \mathrm{Cp}(M)$ or $(M,\langle ·,·\rangle )$ satisfies Condition 2 and $\left\{h{(a)}^{\perp (M,\langle ·,·\rangle )}\right|$$a\in L\}\supseteq \mathrm{Cp}(M)$, then $h_{*}:\mathcal{M}i{n}_{(M,\langle ·,·\rangle )}^{-1}\to \mathcal{M}i{n}_{(L,[·,·])}$ is open, thus it is a homeomorphism by (1). □

Corollary. 5

If ${\nabla }_B\in \mathrm{Cp}(\mathrm{Con}(B))$, f is $\mathrm{Min}$–admissible and $f^{•}$ satisfies Condition 3, then:

(1)

• $f^{*}:\mathrm{Min}(B)\to \mathrm{Min}(A)$ is continuous with respect to the Stone and the flat topologies;
• if one of the following holds:
  $\circ$ A is semiprime and $\mathrm{Con}(A)$ satisfies one of the Conditions 1 and 2,
  $\circ$ B is semiprime and $\mathrm{Con}(B)$ satisfies one of the Conditions 1 and 2,
  then $f^{*}:\mathcal{M}in(B)\to \mathcal{M}i{n}^{-1}(A)$ is continuous;
• if A is semiprime and $\mathrm{Con}(A)$ satisfies one of the Conditions 1 and 2, then:
  if $\left\{f^{•}({\alpha }^{\perp (\mathrm{Con}(A),{[·,·]}_A)})\right|\alpha \in \mathrm{Cp}(\mathrm{Con}(A))\}\subseteq \mathrm{Cp}(\mathrm{Con}(B))$, in particular if $\mathrm{Cp}(\mathrm{Con}(A))$ is closed with respect to the polar, then $f^{*}:\mathcal{M}i{n}^{-1}(B)\to \mathcal{M}in(A)$ is continuous;
• if B is semiprime and $\mathrm{Con}(B)$ satisfies one of the Conditions 1 and 2, then:
  if $\left\{f^{•}{(\alpha )}^{\perp (\mathrm{Con}(B),{[·,·]}_B)}\right|\alpha \in \mathrm{Cp}(\mathrm{Con}(A))\}\subseteq \mathrm{Cp}(\mathrm{Con}(B))$, in particular if $\mathrm{Cp}(\mathrm{Con}(B))$ is closed with respect to the polar, then $f^{*}:\mathcal{M}i{n}^{-1}(B)\to \mathcal{M}in(A)$ is continuous;

(2)

if $f^{*}:\mathrm{Min}(B)\to \mathrm{Min}(A)$ is injective, then:

• if $f^{•}(\mathrm{Con}(A))\supseteq \mathrm{Cp}(\mathrm{Con}(B))$, then $f^{*}:\mathrm{Min}(B)\to \mathrm{Min}(A)$ is a homeomorphism with respect to the Stone topologies;
• if $f^{•}(\mathrm{Cp}(\mathrm{Con}(A)))=\mathrm{Cp}(\mathrm{Con}(B))$, then $f^{*}:\mathrm{Min}(B)\to \mathrm{Min}(A)$ is a homeomorphism with respect to the flat topologies;
• if one of the following holds:
  $\circ$ A is semiprime, $\mathrm{Con}(A)$ satisfies one of the Conditions 1 and 2, $f^{•}(\mathrm{Cp}(\mathrm{Con}(A)))$$=\mathrm{Cp}(\mathrm{Con}(B))$ and $\mathrm{Cp}(\mathrm{Con}(A))$ is closed with respect to the polar,
  $\circ$ A is semiprime, $\mathrm{Con}(A)$ satisfies one of the Conditions 1 and 2 and $\left\{f^{•}({\alpha }^{\perp (\mathrm{Con}(A),{[·,·]}_A)})\right|\alpha \in \mathrm{Cp}(\mathrm{Con}(A))\}\supseteq \mathrm{Cp}(\mathrm{Con}(B))$,
  $\circ$ B is semiprime, $\mathrm{Con}(B)$ satisfies one of the Conditions 1 and 2 and $\left\{f^{•}{(\alpha )}^{\perp (\mathrm{Con}(B),{[·,·]}_B)}\right|\alpha \in \mathrm{Cp}(\mathrm{Con}(A))\}\supseteq \mathrm{Cp}(\mathrm{Con}(B))$,
  then $f^{*}:\mathcal{M}in(B)\to \mathcal{M}i{n}^{-1}(A)$ is a homeomorphism;
• if one of the following holds:
  $\circ$ A is semiprime, $\mathrm{Con}(A)$ satisfies one of the Conditions 1 and 2 and $f^{•}(\mathrm{Cp}(\mathrm{Con}(A)))=\mathrm{Cp}(\mathrm{Con}(B))$,
  $\circ$ A is semiprime, $\mathrm{Con}(A)$ satisfies one of the Conditions 1 and 2 and $\left\{f^{•}({\alpha }^{\perp (\mathrm{Con}(A),{[·,·]}_A)})\right|\alpha \in \mathrm{Cp}(\mathrm{Con}(A))\}=\mathrm{Cp}(\mathrm{Con}(B))$,
  $\circ$ A is semiprime, $\mathrm{Con}(A)$ satisfies Condition 2 and $\left\{f^{•}({\alpha }^{\perp (\mathrm{Con}(A),{[·,·]}_A)})\right|\alpha \in \mathrm{Con}(A)\}\supseteq \mathrm{Cp}(\mathrm{Con}(B))\supseteq \left\{f^{•}({\alpha }^{\perp (\mathrm{Con}(A),{[·,·]}_A)})\right|\alpha \in \mathrm{Cp}(\mathrm{Con}(A))\}$,
  $\circ$ B is semiprime, $\mathrm{Con}(B)$ satisfies one of the Conditions 1 and 2 and $\left\{f^{•}{(\alpha )}^{\perp (\mathrm{Con}(B),{[·,·]}_B)}\right|\alpha \in \mathrm{Cp}(\mathrm{Con}(A))\}=\mathrm{Cp}(\mathrm{Con}(B))$,
  $\circ$ B is semiprime, $\mathrm{Con}(B)$ satisfies Condition 2 and $\left\{f^{•}{(\alpha )}^{\perp (\mathrm{Con}(B),{[·,·]}_B)}\right|\alpha \in \mathrm{Con}(A)\}\supseteq \mathrm{Cp}(\mathrm{Con}(B))\supseteq \left\{f^{•}{(\alpha )}^{\perp (\mathrm{Con}(B),{[·,·]}_B)}\right|\alpha \in \mathrm{Cp}(\mathrm{Con}(A))\}$,
  then $f^{*}:\mathcal{M}i{n}^{-1}(B)\to \mathcal{M}in(A)$ is a homeomorphism.

Remark 31.

Note, in Theorem 3, that, whenever $h_{*}$ is continuous with respect to two of these topologies, its direct image is a complete morphism of join–semilattices between them as bounded sublattices of the power sets of the minimal prime spectra. Clearly, when it is a homeomorphism, its direct image is a lattice isomorphism.

In view of the definition of $f^{*}$, in Corollary 5, whenever $f^{*}$ is continuous with respect to two of these topologies, its direct image is a complete lattice morphism between them as bounded sublattices of the power sets of the minimal prime spectra of congruences. Of course, again, when it is a homeomorphism, its direct image is a lattice isomorphism.

7. Conclusions

Besides congruence lattices, our work in this abstract case can be applied to various kinds of commutator lattices, for instance those arising in the theory of orthomodular lattices or that of residuated lattices.

More intermediate results from [5,6] can be obtained in this abstract case. It remains to investigate to what degree other ring extension properties such as the ones we have generalized in [6] can be tackled in this abstract case. It is also worth seaching further lattice–theoretical conditions under which these results hold.

In terms of particularly difficult subjects for future research, we first mention obtaining a characterization of the complete join–semilattice morphisms between commutator lattices which correspond, as described in Section 6, to algebra embeddings in the particular case of congruence lattices. While properties such as congruence–extensibility can simplify this task, achieving high levels of generality is likely to prove extremely difficult.

Another very difficult task in cases with high levels of generality, that would be very useful for the study of the lattice structures of the sets of congruences of different kinds of algebras, is characterizing principal congruences as members of the sets of compact congruences, more precisely finding the kinds of compact elements of a commutator lattice which, in the particular case of a congruence lattice, become principal congruences. For the first further abstractions of our congruence extension results from [6], any join–dense subset of the set of compact elements of a commutator lattice obviously suffices instead of identifying one that becomes the set of the principal congruences when applied to a (certain kind of) congruence lattice.