Congruence Extensions in Congruence–Modular Varieties

Abstract

We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences with respect to the term condition commutator. Then we use the topological structure of the minimal prime spectrum to study extensions of universal algebras that generalize certain types of ring extensions. Our results hold for semiprime members of semidegenerate congruence–modular varieties, as well as semiprime algebras whose term condition commutators are commutative and distributive with respect to arbitrary joins and satisfy certain conditions on compact congruences, even if those algebras do not generate congruence–modular varieties.

1. Introduction

Inspired by group theory and initially developped in [1] for congruence–modular varieties, commutator theory has led to the solving of many deep universal algebra problems; it has subsequently been extended by adopting various definitions for the commutator, all of which collapse to the modular commutator in this congruence–modular case.

The congruence lattices of members of congruence–modular varieties, endowed with the modular commutator, form commutator lattices, in which we can introduce the prime elements with respect to the commutator operation. For the purpose of not restricting to this congruence–modular setting, we have introduced the notion of a prime congruence through the term condition commutator. Under certain conditions for this commutator operation which do not have to be satisfied throughout a whole variety, the thus defined set of the prime congruences of an algebra becomes a topological space when endowed with a generalization of the Zariski topology from commutative rings [2,3]. For members of semidegenerate congruence–modular varieties, this topological space has strong properties [4], some of which extend to more general cases.

The first goal of this paper is to study the topology this generalization induces on the antichain of the minimal prime congruences of an algebra whose term condition commutator satisfies certain conditions, all of which hold in any member of a semidegenerate congruence–modular variety.

The second goal of our present work is the study of certain types of extensions of algebras with “well–behaved” commutators, meaning term condition commutators that behave like the modular commutator, generalizing results on ring extensions from [5,6].

In Section 2 we recall some results on congruence lattices and the term condition commutator, as well as the particular case of the modular commutator, along with the prime and minimal prime spectra of congruences of an algebra with ”well–behaved” commutators, where the prime congruences are defined with respect to the commutator operation, as well as the prime and minimal spectra of ideals of a bounded distributive lattice. The following sections are dedicated to our new results.

Section 3 contains arithmetical properties of commutator lattices of congruences and annihilators with respect to the commutator in such lattices, derived from the residuation operation and its associated negation introduced through these annihilators.

In Section 4 we obtain several algebraic properties of the minimal prime spectrum of congruences, including a characterization of minimal prime congruences through their behavior with respect to the negations of congruences, obtained in two different cases from a corresponding characterization of minimal prime ideals of bounded distributive lattices.

In Section 5 we study the Stone (also called spectral) and the flat (also called inverse) topology on the minimal prime spectrum of congruences of an algebra, establish homeomorphisms between these and the corresponding topologies on the minimal prime spectrum of ideals of the reticulation of that algebra (see [7] for the construction of the reticulation in the universal algebra setting) and obtain necessary and sufficient conditions for these two topologies to coincide.

In Section 6, starting from the study of ring extensions in [5,6], we define certain classes of extensions of universal algebras that generalize corresponding classes of ring extensions: m–extensions, rigid, quasirigid and weak rigid extensions, r–extensions and quasi/weak r–extensions, $r^{*}$–extensions and quasi/weak $r^{*}$–extensions, and, generalizing results from [5,6], we obtain relations between these types of extensions, characterizations for these kinds of extensions and topological properties of the minimal prime spectra of the universal algebras that form such extensions.

2. Preliminaries

We refer the reader to [4,8,9,10] for a further study of the following notions from universal algebra, to [11,12,13,14] for the lattice–theoretical ones, to [1,4,10,15] for the results on commutators and to [4,16,17,18,19,20] for the Stone topologies.

All algebras will be nonempty and they will be designated by their underlying sets. By trivial algebra we mean one–element algebra.

$\mathbb{N}$ denotes the set of the natural numbers, ${\mathbb{N}}^{*}$ $=\mathbb{N}\setminus \left\{0\right\}$, and, for any $a,b\in \mathbb{N}$, we denote by $\overline{a,b}$$=\{n\in \mathbb{N}|a\le n\le b\}$ the interval in the lattice $(\mathbb{N},\le )$ bounded by a and b, where ≤ is the natural order (this is to differentiate from the notation for commutators). Let M, N be sets and $S\subseteq M$. Then we denote by $\mathcal{P}\left(M\right)$ the set of the subsets of M, by ${\Delta }_M$$=\left\{\right(x,x\left)\right|x\in M\}$ and ${\nabla }_M$$=M^2$ the smallest and the largest equivalence on M, respectively, and by $i_{S,M}$$:S\to M$ the inclusion map. For any function $f:M\to N$, we denote by $\mathrm{Ker}\left(f\right)$ the kernel of f, by f the direct image of $f^2=f\times f$ and by $f^{*}$ the inverse image of $f^2$.

For any poset P, $Max\left(P\right)$ and $Min\left(P\right)$ will denote the set of the maximal elements and that of the minimal elements of P, respectively. The order on congruences of an algebra or ideals or filters of a lattice will always be the set inclusion.

Let L be a lattice. Then $\mathrm{Cp}\left(L\right)$ and $\mathrm{Mi}\left(L\right)$ denote the set of the compact elements and that of the meet–irreducible elements of L, respectively. $\mathrm{Filt}\left(L\right)$, $\mathrm{Id}\left(L\right)$, $\mathrm{PId}\left(L\right)$ and ${\mathrm{Spec}}_{\mathrm{Id}}\left(L\right)$ denote the sets of the filters, ideals, principal ideals and prime ideals of L, respectively. We denote by ${\mathrm{Min}}_{\mathrm{Id}}\left(L\right)$$=Min\left({\mathrm{Spec}}_{\mathrm{Id}}\left(L\right)\right)$: the set of the minimal prime ideals of L. Let $U\subseteq L$ and $u\in L$. Then ${\left[U\right)}_L$ and ${\left[u\right)}_L$ denote the filters of L generated by U and by u, respectively, while ${\left(U\right]}_L$ and ${\left(u\right]}_L$ denote the ideals of L generated by U and by u, respectively. If L has a 0, then ${\mathrm{Ann}}_L\left(U\right)$$=\{a\in L|(\forall x\in U)(a\wedge x=0)\}$ is the annihilator of U and we denote by ${\mathrm{Ann}}_L\left(u\right)$$={\mathrm{Ann}}_L\left(\left\{u\right\}\right)$ the annihilator of u. The subscript L will be eliminated from these notations when the lattice L is clear from the context. Note that, if L has a 0 and it is distributive, then all annihilators in L are ideals of L. If L is a bounded lattice, then we denote by $\mathcal{B}\left(L\right)$ the set of the complemented elements of L, which, of course, is a Boolean sublattice of L if L is distributive.

Recall that a frame is a complete lattice with the meet distributive with respect to arbitrary joins.

Ⓗ

Throughout the rest of this paper: $\tau$ will be a universal algebras signature, $\mathcal{V}$ a variety of $\tau$–algebras and A an arbitrary member of $\mathcal{V}$.

Everywhere in this paper, we will mark global assumptions as above, for better visibility.

Unless mentioned otherwise, by morphism we mean $\tau$–morphism.

$\mathrm{Con}\left(A\right)$, $\mathrm{Max}\left(A\right)$, $\mathrm{PCon}\left(A\right)$ and $\mathcal{K}\left(A\right)$ denote the sets of the congruences, maximal (proper) congruences, principal congruences and finitely generated congruences of A, respectively; note that $\mathcal{K}\left(A\right)=\mathrm{Cp}\left(\mathrm{Con}\right(A\left)\right)$. $\mathrm{Max}\left(A\right)$ is called the maximal spectrum of A. For any $X\subseteq A^2$ and any $a,b\in A$, $C{g}_A\left(X\right)$ will be the congruence of A generated by X and we shall denote by $C{g}_A(a,b)$$=C{g}_A\left(\left\{(a,b)\right\}\right)$.

For any $\theta \in \mathrm{Con}\left(A\right)$, $p_{\theta }$$:A\to A/\theta$ will be the canonical surjective morphism; given any $X\in A\cup A^2\cup \mathcal{P}\left(A\right)\cup \mathcal{P}\left(A^2\right)$, we denote by $X/\theta$$=p_{\theta }\left(X\right)$. Note that $\mathrm{Ker}\left(p_{\theta }\right)=\theta$ for any $\theta \in \mathrm{Con}\left(A\right)$, and that $C{g}_A\left(C{g}_S\left(X\right)\right)=C{g}_A\left(X\right)$ for any subalgebra S of A and any $X\subseteq S^2$.

Ⓗ

Throughout the rest of this paper, B will be a member of $\mathcal{V}$ and $f:A\to B$ a morphism.

• Recall that, for any $\alpha \in \mathrm{Con}\left(A\right)$ and any $\beta \in \mathrm{Con}\left(B\right)$, we have $f^{*}\left(\beta \right)\in \left[\mathrm{Ker}\left(f\right)\right)\subseteq \mathrm{Con}\left(A\right)$, $f\left(f^{*}\left(\beta \right)\right)=\beta \cap f\left(A^2\right)\subseteq \beta$ and $\alpha \subseteq f^{*}\left(f\left(\alpha \right)\right)$; if $\alpha \in \left[\mathrm{Ker}\right(f\left)\right)$, then $f\left(\alpha \right)\in \mathrm{Con}\left(f\right(A\left)\right)$ and $f^{*}\left(f\left(\alpha \right)\right)=\alpha$. Hence $\theta \mapsto f\left(\theta \right)$ is a lattice isomorphism from $\left[\mathrm{Ker}\right(f\left)\right)$ to $\mathrm{Con}\left(f\right(A\left)\right)$, having $f^{*}$ as inverse, and thus it sets an order isomorphism from $\mathrm{Max}\left(A\right)\cap \left[\mathrm{Ker}\right(f\left)\right)$ to $\mathrm{Max}\left(f\right(A\left)\right)$. In particular, for any $\theta \in \mathrm{Con}\left(A\right)$, the map $\alpha \mapsto \alpha /\theta$ is an order isomorphism from $\left[\theta \right)$ to $\mathrm{Con}(A/\theta )$.

Lemma 1 

([21] (Lemma 1.11), [22] (Proposition 1.2)). For any $X\subseteq A^2$ and any $\alpha ,\theta \in \mathrm{Con}\left(A\right)$:

• $f(C{g}_A\left(X\right)\vee \mathrm{Ker}\left(f\right))=C{g}_{f\left(A\right)}\left(f\left(X\right)\right)$, so $C{g}_B\left(f\left(C{g}_A\left(X\right)\right)\right)=C{g}_B\left(f\left(X\right)\right)$ and $(C{g}_A\left(X\right)\vee \theta )/\theta =C{g}_{A/\theta }(X/\theta )$;
• in particular, $f(\alpha \vee \mathrm{Ker}\left(f\right))=C{g}_{f\left(A\right)}\left(f\left(\alpha \right)\right)$, so $(\alpha \vee \theta )/\theta =C{g}_{A/\theta }(\alpha /\theta )$.

• For any nonempty family ${\left({\alpha }_i\right)}_{i\in I}\subseteq \left[\mathrm{Ker}\left(f\right)\right)$, we have, in $\mathrm{Con}\left(f\right(A\left)\right)$: ${\displaystyle f(\underset{i\in I}{\bigvee }{\alpha }_i)=\underset{i\in I}{\bigvee }f\left({\alpha }_i\right)}$. Indeed, by Lemma 1,

$${\displaystyle f(\underset{i\in I}{\bigvee }{\alpha }_i)=f(C{g}_A(\bigcup _{i\in I}{\alpha }_i))=C{g}_{f\left(A\right)}(f(\bigcup _{i\in I}{\alpha }_i))=}C{g}_{f\left(A\right)}({\bigcup }_{i\in I}f({\alpha }_i))={\bigvee }_{i\in I}f\left({\alpha }_i\right).$$

We denote by $f^{•}$$:\mathrm{Con}\left(A\right)\to \mathrm{Con}\left(B\right)$ the map defined by $f^{•}\left(\alpha \right)=C{g}_B\left(f\left(\alpha \right)\right)$ for all $\alpha \in \mathrm{Con}\left(A\right)$. By the above, if f is surjective, then $f^{•}{\mid }_{\left[\mathrm{Ker}\right(f\left)\right)}:\left[\mathrm{Ker}\left(f\right)\right)\to \mathrm{Con}\left(B\right)$ is the inverse of the lattice isomorphism $f^{*}{\mid }_{\mathrm{Con}\left(B\right)}:\mathrm{Con}\left(B\right)\to \left[\mathrm{Ker}\left(f\right)\right)$.

We use the following definition from [23] for the term condition commutator: let $\alpha ,\beta \in \mathrm{Con}\left(A\right)$. For any $\mu \in \mathrm{Con}\left(A\right)$, by $C(\alpha ,\beta ;\mu )$ we denote the fact that the following condition holds: for all $n,k\in \mathbb{N}$ and any term t over $\tau$ of arity $n+k$, if $(a_i,b_i)\in \alpha$ for all $i\in \overline{1,n}$ and $(c_j,d_j)\in \beta$ for all $j\in \overline{1,k}$, then $(t^A(a_1,\dots ,a_n,c_1,\dots ,c_k),t^A(a_1,\dots ,a_n,d_1,\dots ,d_k))\in \mu$ if and only if $(t^A(b_1,\dots ,b_n,c_1,\dots ,c_k),t^A(b_1,\dots ,b_n,d_1,\dots ,d_k))\in \mu$. We denote by ${[\alpha ,\beta ]}_A$ the commutator of α and β in A, defined by ${[\alpha ,\beta ]}_A:=\bigcap \{\mu \in \mathrm{Con}\left(A\right)|C(\alpha ,\beta ;\mu )\}$. The operation ${[·,·]}_A$$:\mathrm{Con}\left(A\right)\times \mathrm{Con}\left(A\right)\to \mathrm{Con}\left(A\right)$ is called the commutator of A.

By [1], if $\mathcal{V}$ is congruence–modular, then, for each member M of $\mathcal{V}$, ${[·,·]}_M$ is the unique binary operation on $\mathrm{Con}\left(M\right)$ such that, for all $\alpha ,\beta \in \mathrm{Con}\left(M\right)$, ${[\alpha ,\beta ]}_M=min\{\mu \in \mathrm{Con}\left(M\right)|\mu \subseteq \alpha \cap \beta$ and, for any member N of $\mathcal{V}$ and any surjective morphism $h:M\to N$ in $\mathcal{V}$, $\mu \vee \mathrm{Ker}\left(h\right)=h^{*}\left({[h(\alpha \vee \mathrm{Ker}\left(h\right)),h(\beta \vee \mathrm{Ker}\left(h\right))]}_N\right)\}$. Therefore, if $\mathcal{V}$ is congruence–modular, $\alpha ,\beta ,\theta \in \mathrm{Con}\left(A\right)$ and f is surjective, then

$${[f(\alpha \vee \mathrm{Ker}\left(f\right)),f(\beta \vee \mathrm{Ker}\left(f\right))]}_B=f({[\alpha ,\beta ]}_A\vee \mathrm{Ker}\left(f\right)).$$

In particular, ${[(\alpha \vee \theta )/\theta ,(\beta \vee \theta )/\theta ]}_{A/\theta }=({[\alpha ,\beta ]}_A\vee \theta )/\theta$, thus, if $\theta \subseteq \alpha \cap \beta$, then ${[\alpha /\theta ,\beta /\theta ]}_{A/\theta }=({[\alpha ,\beta ]}_A\vee \theta )/\theta$; if, moreover, $\theta \subseteq {[\alpha ,\beta ]}_A$, then ${[\alpha /\theta ,\beta /\theta ]}_{A/\theta }={[\alpha ,\beta ]}_A/\theta$.

By [23] [Lemma 4.6, Lemma 4.7, Theorem 8.3], the commutator is smaller than the intersection and increasing in both arguments; if $\mathcal{V}$ is congruence–modular, then the commutator is also commutative and distributive in both arguments with respect to arbitrary joins.

Hence, if $\mathcal{V}$ is congruence–modular and the commutator of A coincides to the intersection of congruences, then $\mathrm{Con}\left(A\right)$ is a frame, in particular it is distributive. Therefore, if $\mathcal{V}$ is congruence–modular and the commutator coincides to the intersection in each member of $\mathcal{V}$, then $\mathcal{V}$ is congruence–distributive. By [24], the converse holds, as well: if $\mathcal{V}$ is congruence–distributive, then, in each member of $\mathcal{V}$, the commutator coincides to the intersection of congruences.

For any $\alpha ,\beta \in \mathrm{Con}\left(A\right)$, we denote by ${[\alpha ,\beta ]}_A^1={[\alpha ,\beta ]}_A$ and, for any $n\in {\mathbb{N}}^{*}$, by ${[\alpha ,\beta ]}_A^{n+1}{=[[\alpha ,\beta ]}_A^n,$${[\alpha ,\beta ]}_A^n$${]}_A$.

Recall that $\mathcal{V}$ is said to be semidegenerate if and only if no nontrivial algebra in $\mathcal{V}$ has one–element subalgebras. Recall from [10] that, if $\mathcal{V}$ is congruence–modular, then the following are equivalent:

• $\mathcal{V}$ is semidegenerate;
• for all members M of $\mathcal{V}$, ${\nabla }_M\in \mathcal{K}\left(M\right)$.

If ${[·,·]}_A$ is distributive with respect to the join, in particular if $\mathcal{V}$ is congruence–modular, then, if A has principal commutators, that is its set $\mathrm{PCon}\left(A\right)$ of principal congruences is closed with respect to the commutator, then its set $\mathcal{K}\left(A\right)$ of compact congruences is closed with respect to the commutator.

Recall that a prime congruence of A is a proper congruence $\varphi$ of A such that, for any $\alpha ,\beta \in \mathrm{Con}\left(A\right)$, if ${[\alpha ,\beta ]}_A\subseteq \varphi$, then $\alpha \subseteq \varphi$ or $\beta \subseteq \varphi$ [1]. It actually suffices that we enforce this condition for principal congruences $\alpha$, $\beta$ of A:

Lemma 2 

([7,18]). A proper congruence ϕ of A is prime if and only if for any $\alpha ,\beta \in \mathrm{PCon}\left(A\right)$, if ${[\alpha ,\beta ]}_A\subseteq \varphi$, then $\alpha \subseteq \varphi$ or $\beta \subseteq \varphi$.

We denote by $\mathrm{Spec}\left(A\right)$ the (prime) spectrum of A, that is the set of the prime congruences of A. Recall that $\mathrm{Spec}\left(A\right)$ is not necessarily nonempty. However, by [4] [Theorem 5.3], if the commutator of A is distributive with respect to the join of congruences, ${\nabla }_A\in \mathcal{K}\left(A\right)$ and ${[{\nabla }_A,{\nabla }_A]}_A={\nabla }_A$, in particular if $\mathcal{V}$ is congruence–modular and semidegenerate, then:

• $\mathrm{Max}\left(A\right)\subseteq \mathrm{Spec}\left(A\right)$;
• any proper congruence of A is included in a maximal, thus prime congruence of A;
• hence $\mathrm{Max}\left(A\right)$ and thus $\mathrm{Spec}\left(A\right)$ is nonempty whenever A is nontrivial.

For all $\theta \in \mathrm{Con}\left(A\right)$, we set $V_A\left(\theta \right):=\mathrm{Spec}\left(A\right)\cap \left[\theta \right)$ and $D_A\left(\theta \right):=\mathrm{Spec}\left(A\right)\setminus V_A\left(\theta \right)=\mathrm{Spec}\left(A\right)\setminus \left[\theta \right)$.

For all $X\subseteq A^2$ and $a,b\in A$, we set $V_A\left(X\right):=V_A\left(C{g}_A\left(X\right)\right)$, $D_A\left(X\right):=D_A\left(C{g}_A\left(X\right)\right)$, $V_A(a,b):=V_A\left(C{g}_A(a,b)\right)$ and $D_A(a,b):=D_A\left(C{g}_A(a,b)\right)$.

For any $\theta \in \mathrm{Con}\left(A\right)$, we set ${\rho }_A\left(\theta \right):=\bigcap V_A\left(\theta \right)$ and call this congruence the radical of $\theta$. We denote by $\mathrm{RCon}\left(A\right)$$=\left\{{\rho }_A\left(\theta \right)\right|\theta \in \mathrm{Con}\left(A\right)\}=\{\theta \in \mathrm{Con}\left(A\right)|{\rho }_A\left(\theta \right)=\theta \}$. We call the elements of $\mathrm{RCon}\left(A\right)$ the radical congruences of A. Obviously, any prime congruence of A is radical.

By [4] [Lemma 1.6, Proposition 1.2], if the commutator of A is commutative and distributive with respect to arbitrary joins, in particular if $\mathcal{V}$ is congruence–modular, then:

(i)

a congruence $\theta$ of A is radical if and only if it is semiprime, that is, for any $\alpha \in \mathrm{Con}\left(A\right)$, if ${[\alpha ,\alpha ]}_A\subseteq \theta$, then $\alpha \subseteq \theta$;

(ii)

hence $\mathrm{Spec}\left(A\right)=\mathrm{Mi}\left(\mathrm{Con}\right(A\left)\right)\cap \mathrm{RCon}\left(A\right)$.

A is called a semiprime algebra if and only if ${\rho }_A\left({\Delta }_A\right)={\Delta }_A$. By statement (i) above, if the commutator of A equals the intersection, in particular if $\mathcal{V}$ is congruence–distributive, then $\mathrm{RCon}\left(A\right)=\mathrm{Con}\left(A\right)$, thus A is semiprime.

Let us denote by ${\mathcal{S}}_{\mathrm{Spec}}\left(A\right)$$=\left\{D_A\left(\theta \right)\right|\theta \in \mathrm{Con}\left(A\right)\}$. If the commutator of A is commutative and distributive with respect to arbitrary joins, in particular if $\mathcal{V}$ is congruence–modular, then, by [4,7,18], ${\mathcal{S}}_{\mathrm{Spec}}\left(A\right)$ is a topology on $\mathrm{Spec}\left(A\right)$, called the Stone topology or the spectral topology, which satisfies, for all $\alpha ,\beta \in \mathrm{Con}\left(A\right)$ and any family ${\left({\alpha }_i\right)}_{i\in I}\subseteq \mathrm{Con}\left(A\right)$:

• $D_A\left(\alpha \right)\subseteq D_A\left(\beta \right)$ if and only if $V_A\left(\alpha \right)\supseteq V_A\left(\beta \right)$ if and only if ${\rho }_A\left(\alpha \right)\subseteq {\rho }_A\left(\beta \right)$;
• thus $D_A\left(\alpha \right)=D_A\left(\beta \right)$ if and only if $V_A\left(\alpha \right)=V_A\left(\beta \right)$ if and only if ${\rho }_A\left(\alpha \right)={\rho }_A\left(\beta \right)$;
• clearly, $\alpha \subseteq \beta$ implies ${\rho }_A\left(\alpha \right)\subseteq {\rho }_A\left(\beta \right)$;
• clearly, $\alpha \subseteq {\rho }_A\left(\alpha \right)$, thus ${\rho }_A\left(\alpha \right)={\Delta }_A$ implies $\alpha ={\Delta }_A$;
• $D_A\left({\nabla }_A\right)=\mathrm{Spec}\left(A\right)=V_A\left({\Delta }_A\right)$ and $D_A\left({\Delta }_A\right)=\varnothing =V_A\left({\nabla }_A\right)$;
• if A is semiprime, then: $D_A\left(\alpha \right)=\varnothing$ if and only if $V_A\left(\alpha \right)=\mathrm{Spec}\left(A\right)$ if and only if ${\rho }_A\left(\alpha \right)={\Delta }_A$ if and only if $\alpha ={\Delta }_A$;
• if ${\nabla }_A\in \mathcal{K}\left(A\right)$ and ${[{\nabla }_A,{\nabla }_A]}_A={\nabla }_A$, in particular if $\mathcal{V}$ is congruence–modular and semidegenerate, then: $D_A\left(\alpha \right)=\mathrm{Spec}\left(A\right)$ if and only if $V_A\left(\alpha \right)=\varnothing$ if and only if ${\rho }_A\left(\alpha \right)={\nabla }_A$ if and only if $\alpha ={\nabla }_A$;
• $D_A\left({[\alpha ,\beta ]}_A\right)=D_A(\alpha \cap \beta )=D_A\left(\alpha \right)\cap D_A\left(\beta \right)$ and $D_A(\alpha \vee \beta )=D_A\left(\alpha \right)\cup D_A\left(\beta \right)$; thus $V_A\left({[\alpha ,\beta ]}_A\right)=V_A(\alpha \cap \beta )=V_A\left(\alpha \right)\cup V_A\left(\beta \right)$, $V_A(\alpha \vee \beta )=V_A\left(\alpha \right)\cap V_A\left(\beta \right)$,
  ${\rho }_A\left({[\alpha ,\beta ]}_A\right)={\rho }_A(\alpha \cap \beta )={\rho }_A\left(\alpha \right)\cap {\rho }_A\left(\beta \right)$ and ${\rho }_A(\alpha \vee \beta )={\rho }_A({\rho }_A\left(\alpha \right)\vee {\rho }_A\left(\beta \right))$;
• ${\displaystyle D_A(\underset{i\in I}{\bigvee }{\alpha }_i)=D_A(\bigcup _{i\in I}{\alpha }_i)=\bigcup _{i\in I}{D}_A\left({\alpha }_i\right)}$, thus ${\displaystyle V_A(\underset{i\in I}{\bigvee }{\alpha }_i)=V_A(\bigcup _{i\in I}{\alpha }_i)=\bigcap _{i\in I}{V}_A\left({\alpha }_i\right)}$ and ${\displaystyle {\rho }_A(\underset{i\in I}{\bigvee }{\alpha }_i)={\rho }_A(\bigcup _{i\in I}{\alpha }_i)={\rho }_A(\bigcup _{i\in I}{\rho }_A({\alpha }_i))={\rho }_A(\underset{i\in I}{\bigvee }{\rho }_A({\alpha }_i))}$;
• hence, for any $\theta \in \mathrm{Con}\left(A\right)$, ${\displaystyle V_A\left(\theta \right)=\bigcap _{(a,b)\in \theta }{V}_A(a,b)}$ and ${\displaystyle D_A\left(\theta \right)=\bigcup _{(a,b)\in \theta }{D}_A(a,b)}$, therefore the Stone topology ${\mathcal{S}}_{\mathrm{Spec}}\left(A\right)$ has $\left\{D_A(a,b)\right|a,b\in A\}$ as a basis.

If ${[·,·]}_A$ is commutative and distributive with respect to arbitrary joins and $\mathrm{Max}\left(A\right)\subseteq \mathrm{Spec}\left(A\right)$, in particular if $\mathcal{V}$ is congruence–modular and semidegenerate, then the Stone topology ${\mathcal{S}}_{\mathrm{Spec}}\left(A\right)$ on $\mathrm{Spec}\left(A\right)$ induces the Stone or spectral topology on $\mathrm{Max}\left(A\right)$: ${\mathcal{S}}_{\mathrm{Max}}\left(A\right)=\{D_A\left(\theta \right)\cap \mathrm{Max}\left(A\right)|\theta \in \mathrm{Con}\left(A\right)\}$, having $\{D_A(a,b)\cap \mathrm{Max}\left(A\right)|a,b\in A\}$ as a basis. Note that $\mathrm{Max}\left(A\right)\subseteq \mathrm{Spec}\left(A\right)$ if ${[·,·]}_A$ is commutative and distributive with respect to arbitrary joins, ${\nabla }_A\in \mathcal{K}\left(A\right)$ and ${[{\nabla }_A,{\nabla }_A]}_A={\nabla }_A$.

In the same way, but replacing congruences with ideals, one defines the Stone topology on the set of prime ideals and that of maximal ideals of a bounded distributive lattice.

We call f an admissible morphism if and only if $f^{*}\left(\mathrm{Spec}\left(B\right)\right)\subseteq \mathrm{Spec}\left(A\right)$ [18,19]. Recall from [4] that, if $\mathcal{V}$ is congruence–modular, then the map $\alpha \mapsto f\left(\alpha \right)$ is an order isomorphism from $\mathrm{Spec}\left(A\right)\cap \left[\mathrm{Ker}\right(f\left)\right)$ to $\mathrm{Spec}\left(f\right(A\left)\right)$, thus to $\mathrm{Spec}\left(B\right)$ if f is surjective, case in which this map coincides with $f^{•}$ and $f^{*}$ is its inverse, hence f is admissible.

Remark 1. 

By the above, if $\mathcal{V}$ is congruence–modular and f is surjective, then:

• for all $\alpha \in \mathrm{Con}\left(A\right)$, $f\left(V_A\left(\alpha \right)\right)=V_B\left(f\left(\alpha \right)\right)$ and $f\left(D_A\left(\alpha \right)\right)=D_B\left(f\left(\alpha \right)\right)$; in particular:
• for all $a,b\in A$, $f\left(V_A(a,b)\right)=V_B(f\left(a\right),f\left(b\right))$ and $f\left(D_A(a,b)\right)=D_B(f\left(a\right),f\left(b\right))$;

• thus, since $f=f^{•}={\left(f^{*}\right)}^{-1}$, the map $f^{*}{\mid }_{\mathrm{Spec}\left(B\right)}:\mathrm{Spec}\left(B\right)\to \mathrm{Spec}\left(A\right)$ is continuous with respect to the Stone topologies.

A subset S of $A^2$ is called an m–system for A if and only if, for all $a,b,c,d\in A$, if $(a,b),(c,d)\in S$, then ${[C{g}_A(a,b),C{g}_A(c,d)]}_A\cap S\ne \varnothing$. For instance, any congruence of A is an m–system. Also:

Remark 2 

([7,18]). If $\varphi \in \mathrm{Spec}\left(A\right)$, then ${\nabla }_A\setminus \varphi$ is an m–system in A.

Lemma 3 

([4]). Let S be an m–system in A and $\alpha \in \mathrm{Con}\left(A\right)$ such that $\alpha \cap S=\varnothing$. If the commutator of A is distributive with respect to the join, in particular if $\mathcal{V}$ is congruence–modular, then:

• $\mathrm{Max}\{\theta \in \mathrm{Con}(A\left)\right|\alpha \subseteq \theta ,\theta \cap S=\varnothing \}\subseteq \mathrm{Spec}(A)$, in particular, for the case $\alpha ={\Delta }_A$, $\mathrm{Max}\{\theta \in \mathrm{Con}(A\left)\right|\theta \cap S=\varnothing \}\subseteq \mathrm{Spec}(A)$;
• if ${\nabla }_A\in \mathcal{K}\left(A\right)$, in particular if $\mathcal{V}$ is congruence–modular and semidegenerate, then the set $\mathrm{Max}\{\theta \in \mathrm{Con}(A\left)\right|\alpha \subseteq \theta ,\theta \cap S=\varnothing \}$ is nonempty; thus $\mathrm{Max}\{\theta \in \mathrm{Con}(A\left)\right|\theta \cap S=\varnothing \}$ is nonempty.

We denote by $\mathrm{Min}\left(A\right)=Min\left(\mathrm{Spec}\right(A),\subseteq )$. Recall that $\mathrm{Min}\left(A\right)$ is called the minimal prime spectrum of A and its elements are called minimal prime congruences of A.

Now assume that the commutator of A is commutative and distributive with respect to arbitrary joins, which holds if $\mathcal{V}$ is congruence–modular. Then, by [25] [Proposition 5.9], if we define a binary relation ${\equiv }_A$ on $\mathrm{Con}\left(A\right)$ by: for any $\alpha ,\beta \in \mathrm{Con}\left(A\right)$, $\alpha {\equiv }_A\beta$ if and only if ${\rho }_A\left(\alpha \right)={\rho }_A\left(\beta \right)$, then ${\equiv }_A$ is a lattice congruence of $\mathrm{Con}\left(A\right)$ that preserves arbitrary joins such that $\mathrm{Con}\left(A\right)/{\equiv }_A$ is a frame; see also [7].

Following the notations from [25], if $(L,[·,·\left]\right)$ is a commutator lattice, that is a complete lattice L endowed with a binary operation $[·,·]$ which is commutative, smaller than the meet and distributive with respect to arbitrary joins [16,26], then we denote by ${\mathrm{Spec}}_L$ the set of the prime elements of L with respect to the commutator $[·,·]$, by ${\mathrm{Min}}_L$$=Min\left({\mathrm{Spec}}_L\right)$ the set of the minimal prime elements of L and by $R\left(L\right)$ the set of the radical elements of L, that is the meets of subsets of ${\mathrm{Spec}}_L$.

Let $(L,[·,·\left]\right)$ be a commutator lattice, $u\in L$ and $U\subseteq L$. The annihilators with respect to the commutator are defined by ${\mathrm{Ann}}_{(L,[·,·\left]\right)}\left(U\right):=\{a\in L|(\forall x\in U)([a,x]=0)\}$ and ${\mathrm{Ann}}_{(L,[·,·\left]\right)}\left(u\right):={\mathrm{Ann}}_{(L,[·,·\left]\right)}\left(\left\{u\right\}\right)$. Recall from [25] that ${\equiv }_A$ also preserves the commutator and the quotient algebra of $(\mathrm{Con}\left(A\right),{[·,·]}_A)$ through ${\equiv }_A$ is the commutator lattice $(\mathrm{Con}\left(A\right)/{\equiv }_A,\wedge )$. Note that L is a frame if its commutator $[·,·]$ equals the meet, case in which the annihilators in $(L,[·,·\left]\right)$ coincide with those with respect to the meet and ${\mathrm{Spec}}_L$ is exactly the set of the meet–prime elements of L, thus ${\mathrm{Spec}}_L=\mathrm{Mi}\left(L\right)$ since L is distributive.

3. On the Residuated Structure of the Lattice of Congruences

Condition 1. 

Let M be an arbitrary member of the variety $\mathcal{V}$. We will say that M satisfies condition:

(i) 

iff the commutator of M is commutative and distributive with respect to arbitrary joins;

(ii) 

iff ${[\theta ,{\nabla }_M]}_M=\theta$ for all $\theta \in \mathrm{Con}\left(M\right)$;

(iii) 

iff, for all $\alpha ,\beta ,\theta \in \mathrm{Con}\left(M\right)$, $({[\alpha ,\beta ]}_M\vee \theta )/\theta ={[(\alpha \vee \theta )/\theta ,(\beta \vee \theta )/\theta ]}_{M/\theta }$;

(iv) 

iff ${\nabla }_M\in \mathcal{K}\left(M\right)$ and $\mathcal{K}\left(M\right)$ is closed with respect to the commutator of M;

(v) 

iff all principal ideals of $\mathrm{Con}\left(M\right)/{\equiv }_M$ generated by minimal prime elements are minimal prime ideals, that is: for any $p\in {\mathrm{Min}}_{\mathrm{Con}\left(M\right)/{\equiv }_M}$, we have $\left(p\right]\in {\mathrm{Min}}_{\mathrm{Id}}(\mathrm{Con}\left(M\right)/{\equiv }_M)$;

(vi) 

iff ${\alpha }^{\perp }\in \mathcal{K}\left(M\right)$ for any $\alpha \in \mathcal{K}\left(M\right)$.

Recall from Section 2 and [1,10] that:

• if $\mathcal{V}$ is congruence–modular, then A satisfies (i) and (iii) from Condition 1;
• if $\mathcal{V}$ is congruence–modular and semidegenerate, then ${\nabla }_A\in \mathcal{K}\left(A\right)$ and A satisfies Condition 1.(ii);
• if $\mathcal{V}$ is congruence–distributive, then A satisfies Condition 1.(ii).

• Recall that $\mathcal{K}\left(A\right)=\mathrm{Cp}\left(\mathrm{Con}\right(A\left)\right)$. If the lattice $\mathrm{Con}\left(A\right)$ is compact, i.e., $\mathrm{Con}\left(A\right)=\mathcal{K}\left(A\right)$, then A trivially satisfies (iv) and (vi) from Condition 1. Recall from [7] that, if A satisfies Condition 1.(iv), then the reticulation$\mathcal{L}\left(A\right)$ of A can be constructed as $\mathcal{L}\left(A\right)=\mathcal{K}\left(A\right)/{\equiv }_A$, which is a bounded sublattice of the frame $\mathrm{Con}\left(A\right)/{\equiv }_A$ and thus $\mathcal{L}\left(A\right)$ is a bounded distributive lattice.

Since an element of a lattice is prime if and only if the principal ideal it generates is prime, we have that, whenever a principal ideal of a lattice is a minimal prime ideal, it follows that its generator is a minimal prime element of that lattice. Hence Condition 1.(v) for a member M of $\mathcal{V}$ is equivalent to:

• for any $p\in \mathrm{Con}\left(M\right)/{\equiv }_M$, $p\in {\mathrm{Min}}_{\mathrm{Con}\left(M\right)/{\equiv }_M}$ if and only if $\left(p\right]\in {\mathrm{Min}}_{\mathrm{Id}}(\mathrm{Con}\left(M\right)/{\equiv }_M)$.

Note that A satisfies Condition 1.(v) if all prime ideals of $\mathrm{Con}\left(A\right)/{\equiv }_A$ are principal, in particular if all ideals of $\mathrm{Con}\left(A\right)/{\equiv }_A$ are principal, that is if $\mathrm{Con}\left(A\right)/{\equiv }_A$ is compact, in particular if $\mathrm{Con}\left(A\right)/{\equiv }_A=\mathcal{K}\left(A\right)/{\equiv }_A$, that is if $\mathrm{Con}\left(A\right)/{\equiv }_A=\mathcal{L}\left(A\right)$ in the case when $\mathcal{K}\left(A\right)$ is closed with respect to the commutator of A, in particular if $\mathrm{Con}\left(A\right)$ is compact, that is if $\mathrm{Con}\left(A\right)=\mathcal{K}\left(A\right)$.

Ⓗ

Throughout this section, we will assume that A satisfies Condition 1.(i), which holds in the particular case when $\mathcal{V}$ is congruence–modular.

See [7] for the next results. Until mentioned otherwise, let $\alpha ,\beta ,\gamma ,\theta \in \mathrm{Con}\left(A\right)$ and $n\in {\mathbb{N}}^{*}$, arbitrary. An induction argument shows that:

• ${[\alpha ,\beta ]}_A^{n+1}={[{[\alpha ,\beta ]}_A,{[\alpha ,\beta ]}_A]}_A^n$;
• ${\rho }_A\left({[\alpha ,\beta ]}_A^n\right)={\rho }_A\left({[\alpha ,\beta ]}_A\right)={\rho }_A(\alpha \cap \beta )={\rho }_A\left(\alpha \right)\cap {\rho }_A\left(\beta \right)$.

If A is semiprime, then ${\rho }_A\left(\theta \right)={\Delta }_A$ if and only if $\theta ={\Delta }_A$, hence, by the above: ${[\alpha ,\beta ]}_A^n={\Delta }_A$ if and only if ${[\alpha ,\beta ]}_A={\Delta }_A$ if and only if $\alpha \cap \beta ={\Delta }_A$, so ${\mathrm{Ann}}_{\mathrm{Con}\left(A\right)}\left(\alpha \right)={\mathrm{Ann}}_{(\mathrm{Con}\left(A\right),{[·,·]}_A)}\left(\alpha \right)$ and thus, for any $S\subseteq \mathrm{Con}\left(A\right)$, ${\mathrm{Ann}}_{\mathrm{Con}\left(A\right)}\left(S\right)={\mathrm{Ann}}_{(\mathrm{Con}\left(A\right),{[·,·]}_A)}\left(S\right)$.

If $\mathcal{V}$ is congruence–modular and f is surjective, then, for any $X,Y\in \mathcal{P}\left(A^2\right)$ and any $a,b,c,d\in A$:

• ${[f(\alpha \vee \mathrm{Ker}\left(f\right)),f(\beta \vee \mathrm{Ker}\left(f\right))]}_B^n=f({[\alpha ,\beta ]}_A^n\vee \mathrm{Ker}\left(f\right))$, thus ${[(\alpha \vee \theta )/\theta ,(\beta \vee \theta )/\theta ]}_{A/\theta }^n$$=({[\alpha ,\beta ]}_A^n\vee \theta )/\theta$;
• hence ${[C{g}_{A/\theta }(X/\theta ),C{g}_{A/\theta }(Y/\theta )]}_{A/\theta }^n=({[C{g}_A\left(X\right),C{g}_A\left(Y\right)]}_A^n\vee \theta )/\theta$, in particular ${[C{g}_{A/\theta }(a/\theta ,b/\theta ),C{g}_{A/\theta }(c/\theta ,d/\theta )]}_{A/\theta }^n=({[C{g}_A(a,b),C{g}_A(c,d)]}_A^n\vee \theta )/\theta$;
• $\mathrm{Spec}\left(B\right)=\{\varphi /\mathrm{Ker}\left(f\right)|\varphi \in V_A\left(\mathrm{Ker}\left(f\right)\right)\}$, thus $\mathrm{Spec}(A/\theta )=\{\varphi /\theta |\varphi \in V_A\left(\theta \right)\}$.

We denote by $\beta \to \gamma$$=\bigvee \{\delta \in \mathrm{Con}\left(A\right)|{[\delta ,\beta ]}_A\subseteq \gamma \}$ and ${\beta }^{\perp }$$=\beta \to {\Delta }_A$.

Since ${\displaystyle \theta =\underset{(a,b)\in \theta }{\bigvee }C{g}_A(a,b)=\bigvee \{\zeta \in \mathrm{PCon}\left(A\right)|\zeta \subseteq \theta \}=\bigvee \{\zeta \in \mathcal{K}\left(A\right)|\zeta \subseteq \theta \}}$, it follows that:

$$\beta \to \gamma =\bigvee \{\zeta \in \mathcal{K}\left(A\right)|{[\zeta ,\beta ]}_A\subseteq \gamma \}=\bigvee \{\zeta \in \mathrm{PCon}\left(A\right)|{[\zeta ,\beta ]}_A\subseteq \gamma \},$$

$$\mathrm{in\; particular} {\beta }^{\perp }=\bigvee \{\zeta \in \mathcal{K}\left(A\right)|{[\zeta ,\beta ]}_A={\Delta }_A\}=\bigvee \{\zeta \in \mathrm{PCon}\left(A\right)|{[\zeta ,\beta ]}_A={\Delta }_A\}.$$

• Let us note that, for all $a,b\in A$, we have $(a,b)\in {\beta }^{\perp }$ if and only if $C{g}_A(a,b)\subseteq {\beta }^{\perp }$ if and only if ${[C{g}_A(a,b),\beta ]}_A={\Delta }_A$, hence ${\beta }^{\perp }=\{(a,b)\in A^2|{[C{g}_A(a,b),\beta ]}_A={\Delta }_A\}$.

Note that these operations can be defined for any commutator lattice $(L,[·,·\left]\right)$ by: $b\to c=\bigvee \{a\in L|[a,b]\le c\}$ and $b^{\perp }=b\to 0=\bigvee \{a\in L|[a,b]=0\}$ for any $b,c\in L$ and, if L is algebraic, that is compactly generated, then we also have equalities similar to the above.

Since ${[{\Delta }_A,\beta ]}_A={\Delta }_A\subseteq \gamma$ and, for any non–empty family ${\left({\alpha }_i\right)}_{i\in I}$, ${[{\alpha }_i,\beta ]}_A\subseteq \gamma$ for all $i\in I$ implies ${\displaystyle {[\underset{i\in I}{\bigvee }{\alpha }_i,\beta ]}_A=\underset{i\in I}{\bigvee }{[{\alpha }_i,\beta ]}_A\subseteq \gamma }$, it follows that:

$$\beta \to \gamma =max\{\delta \in \mathrm{Con}\left(A\right)|{[\delta ,\beta ]}_A\subseteq \gamma \},$$

$$\mathrm{in\; particular} {\beta }^{\perp }=max\{\delta \in \mathrm{Con}\left(A\right)|{[\delta ,\beta ]}_A={\Delta }_A\},$$

hence ${\beta }^{\perp }=max\left({\mathrm{Ann}}_{(\mathrm{Con}\left(A\right),{[·,·]}_A)}\left(\beta \right)\right)$ and thus ${\mathrm{Ann}}_{(\mathrm{Con}\left(A\right),{[·,·]}_A)}\left(\beta \right)=\left({\beta }^{\perp }\right]\in \mathrm{PId}\left(\mathrm{Con}\left(A\right)\right)$.

Note that ${[\beta ,\beta \to \gamma ]}_A\subseteq \gamma , \mathrm{in\; particular} {[\beta ,{\beta }^{\perp }]}_A={\Delta }_A$; moreover, for all $\delta \in \mathrm{Con}\left(A\right)$:

$${[\delta ,\beta ]}_A\subseteq \gamma \mathrm{if\; and\; only\; if} \delta \subseteq \beta \to \gamma , \mathrm{in\; particular}:{[\delta ,\beta ]}_A={\Delta }_A \mathrm{if\; and\; only\; if} \delta \subseteq {\beta }^{\perp }.$$

• Therefore, in the particular case when the commutator of A is associative, $(\mathrm{Con}\left(A\right),\cap ,\vee ,\to ,{\Delta }_A,{\nabla }_A)$ is a (bounded commutative integral) residuated lattice, in which ${·}^{\perp }$ is the negation.

Lemma 4. 

If the algebra A is semiprime, then ${\theta }^{\perp }\in \mathrm{RCon}\left(A\right)$ for any $\theta \in \mathrm{Con}\left(A\right)$.

Proof. 

Let $\alpha ,\theta \in \mathrm{Con}\left(A\right)$ such that ${[\alpha ,\alpha ]}_A\subseteq {\theta }^{\perp }$. Then, by the above and the fact that A is semiprime, ${[{[\alpha ,\alpha ]}_A,\theta ]}_A={\Delta }_A$, which is equivalent to ${\rho }_A\left({[{[\alpha ,\alpha ]}_A,\theta ]}_A\right)={\Delta }_A$, that is ${\rho }_A(\alpha \cap \theta )={\Delta }_A$, that is ${\rho }_A\left({[\alpha ,\theta ]}_A\right)={\Delta }_A$, which means that ${[\alpha ,\theta ]}_A={\Delta }_A$, which in turn is equivalent to $\alpha \subseteq {\theta }^{\perp }$. Hence ${\theta }^{\perp }$ is a semiprime and thus a radical congruence of A. □

For any $X\subseteq A^2$, we set

$$X^{\perp }:=\{(a,b)\in A^2|(\forall (x,y)\in X)({[C{g}_A(a,b),C{g}_A(x,y)]}_A={\Delta }_A)\}.$$

Thus:

$$\begin{array}{cc}\hfill X^{\perp }& =\{(a,b)\in A^2|{[C{g}_A(a,b),\underset{(x,y)\in X}{\bigvee }C{g}_A(x,y)]}_A={\Delta }_A\}\hfill \\ \hfill & =\{(a,b)\in A^2|{[C{g}_A(a,b),C{g}_A\left(X\right)]}_A={\Delta }_A\}\hfill \\ \hfill & =\bigvee \left\{C{g}_A(a,b)\right|(a,b)\in A^2,{[C{g}_A(a,b),C{g}_A\left(X\right)]}_A={\Delta }_A\}\hfill \\ \hfill & =\bigvee \{\alpha \in \mathrm{Con}\left(A\right)|{[\alpha ,C{g}_A\left(X\right)]}_A={\Delta }_A\}\hfill \\ \hfill & =max\{\alpha \in \mathrm{Con}\left(A\right)|{[\alpha ,C{g}_A\left(X\right)]}_A={\Delta }_A\}=C{g}_A{\left(X\right)}^{\perp }.\hfill \end{array}$$

So this more general notation is consistent with the notation above for the particular case when $X\in \mathrm{Con}\left(A\right)$.

Lemma 5. 

For any $\alpha ,\beta ,\theta \in \mathrm{Con}\left(A\right)$:

(i) 

$\beta \subseteq \alpha \to \beta$;

(ii) 

$(\alpha \vee \theta )\to (\beta \vee \theta )=\alpha \to (\beta \vee \theta )$.

Proof. 

(i) ${[\beta ,\alpha ]}_A\subseteq \beta \cap \alpha \subseteq \beta$, thus $\beta \subseteq max\{\zeta \in \mathcal{K}\left(A\right)|{[\zeta ,\alpha ]}_A\subseteq \beta \}=\alpha \to \beta$.

• (ii) For all $\gamma \in \mathrm{Con}\left(A\right)$, we have, since ${[\gamma ,\theta ]}_A\subseteq \theta \subseteq \beta \vee \theta$: $\gamma \subseteq (\alpha \vee \theta )\to (\beta \vee \theta )$ if and only if ${[\gamma ,\alpha \vee \theta ]}_A\subseteq \beta \vee \theta$ if and only if ${[\gamma ,\alpha ]}_A\vee {[\gamma ,\theta ]}_A\subseteq \beta \vee \theta$ if and only if ${[\gamma ,\alpha ]}_A\subseteq \beta \vee \theta$ if and only if $\gamma \subseteq \alpha \to (\beta \vee \theta )$. By taking $\gamma =(\alpha \vee \theta )\to (\beta \vee \theta )$ and then $\gamma =\alpha \to (\beta \vee \theta )$ in the previous equivalences, we get: $\alpha \to (\beta \vee \theta )=(\alpha \vee \theta )\to (\beta \vee \theta )$. □

Proposition 1. 

If A satisfies Condition 1.(iii), in particular if $\mathcal{V}$ is congruence–modular, then, for any $\alpha ,\beta ,\theta \in \mathrm{Con}\left(A\right)$:

(i) 

$(\alpha \vee \theta )/\theta \to (\beta \vee \theta )/\theta =\left(\right(\alpha \vee \theta )\to (\beta \vee \theta \left)\right)/\theta =(\alpha \to (\beta \vee \theta \left)\right)/\theta$;

(ii) 

${((\alpha \vee \theta )/\theta )}^{\perp }=(\alpha \to \theta )/\theta$.

Proof. 

By Lemma 5.(i), $\alpha \to (\beta \vee \theta )\supseteq \beta \vee \theta \supseteq \theta$ and $(\alpha \vee \theta )\to (\beta \vee \theta )\supseteq \beta \vee \theta \supseteq \theta$.

• (i) For any $\gamma \in \left[\theta \right)$, by the inclusions above, the definition of the binary operation → on $\mathrm{Con}\left(A\right)$ and the assumption that A satisfies Condition 1.(iii), we have:

  $$\begin{array}{cc}\hfill \gamma /\theta \subseteq (\alpha \vee \theta )/\theta \to (\beta \vee \theta )/\theta & \iff {[\gamma /\theta ,(\alpha \vee \theta )/\theta ]}_{A/\theta }\subseteq (\beta \vee \theta )/\theta \hfill \\ \hfill & \iff ({[\gamma ,\alpha \vee \theta ]}_A\vee \theta )/\theta \subseteq (\beta \vee \theta )/\theta \hfill \\ \hfill & \iff {[\gamma ,\alpha \vee \theta ]}_A\vee \theta \subseteq \beta \vee \theta \hfill \\ \hfill & \iff {[\gamma ,\alpha \vee \theta ]}_A\subseteq \beta \vee \theta \hfill \\ \hfill & \iff \gamma \subseteq (\alpha \vee \theta )\to (\beta \vee \theta )\hfill \\ \hfill & \iff \gamma /\theta \subseteq \left(\right(\alpha \vee \theta )\to (\beta \vee \theta \left)\right)/\theta .\hfill \end{array}$$

  Since $(\alpha \vee \theta )/\theta \to (\beta \vee \theta )/\theta ,\left(\right(\alpha \vee \theta )\to (\beta \vee \theta \left)\right)/\theta \in \mathrm{Con}(A/\theta )=\{\zeta /\theta |\zeta \in \left[\theta \right)\}$, we may take $\gamma /\theta =(\alpha \vee \theta )/\theta \to (\beta \vee \theta )/\theta$ and then $\gamma /\theta =\left(\right(\alpha \vee \theta )\to (\beta \vee \theta \left)\right)/\theta$ in the equivalences above and obtain the first equality in the enunciation through double inclusion. The second equality follows from Lemma 5.(ii).
• (ii) Take $\beta ={\Delta }_A$ in (i). □

Lemma 6. 

Let $\alpha ,\beta \in \mathrm{Con}\left(A\right)$. Then:

(i) 

${\Delta }_A^{\perp }={\nabla }_A$ and, if A satisfies Condition 1.(ii), in particular if $\mathcal{V}$ is either congruence–distributive or both congruence–modular and semidegenerate, then ${\nabla }_A^{\perp }={\Delta }_A$;

(ii) 

$\alpha \subseteq \beta$ implies ${\beta }^{\perp }\subseteq {\alpha }^{\perp }$, and: ${\beta }^{\perp }\subseteq {\alpha }^{\perp }$ if and only if ${\alpha }^{\perp \perp }\subseteq {\beta }^{\perp \perp }$, in particular ${\alpha }^{\perp }={\beta }^{\perp }$ if and only if ${\alpha }^{\perp \perp }={\beta }^{\perp \perp }$;

(iii) 

$\alpha \subseteq {\alpha }^{\perp \perp }$ and ${\alpha }^{\perp \perp \perp }={\alpha }^{\perp }$;

(iv) 

${(\alpha \vee \beta )}^{\perp }={\alpha }^{\perp }\cap {\beta }^{\perp }={({\alpha }^{\perp }\cap {\beta }^{\perp })}^{\perp \perp }$;

(v) 

if A is semiprime, then ${[\alpha ,\beta ]}_A^{\perp }={(\alpha \cap \beta )}^{\perp }$ and ${(\alpha \cap \beta )}^{\perp \perp }={\alpha }^{\perp \perp }\cap {\beta }^{\perp \perp }$;

(vi) 

if A is semiprime, then: ${\alpha }^{\perp }\subseteq {\beta }^{\perp }$ if and only if ${[\alpha ,\beta ]}_A^{\perp }={\beta }^{\perp }$;

(vii) 

if A is semiprime, then: $\alpha \subseteq {\alpha }^{\perp }$ if and only if $\alpha ={\Delta }_A$.

Proof. 

(i) ${\Delta }_A^{\perp }=max\{\theta \in \mathrm{Con}\left(A\right)|[\theta ,{\Delta }_A]={\Delta }_A\}=max\left(\mathrm{Con}\left(A\right)\right)={\nabla }_A$.

If ${[\theta ,{\nabla }_A]}_A=\theta$ for all $\theta \in \mathrm{Con}\left(A\right)$, then:

${\nabla }_A^{\perp }=max\{\theta \in \mathrm{Con}\left(A\right)|[\theta ,{\nabla }_A]={\Delta }_A\}=max\{\theta \in \mathrm{Con}\left(A\right)|\theta ={\Delta }_A\}={\Delta }_A$.

• (ii), (iii) If $\alpha \subseteq \beta$, then $\{\theta \in \mathrm{Con}\left(A\right)|{[\alpha ,\theta ]}_A={\Delta }_A\}\supseteq \{\theta \in \mathrm{Con}\left(A\right)|{[\beta ,\theta ]}_A={\Delta }_A\}$, hence ${\beta }^{\perp }\subseteq {\alpha }^{\perp }$, which thus, in turn, implies ${\alpha }^{\perp \perp }\subseteq {\beta }^{\perp \perp }$.

Since ${[\alpha ,{\alpha }^{\perp }]}_A={\Delta }_A$, it follows that $\alpha \subseteq {\alpha }^{\perp \perp }$, hence ${\alpha }^{\perp }\subseteq {\alpha }^{\perp \perp \perp }$ if we replace $\alpha$ by ${\alpha }^{\perp }$ in this inclusion, but also ${\alpha }^{\perp \perp \perp }\subseteq {\alpha }^{\perp }$ by the above, therefore ${\alpha }^{\perp }={\alpha }^{\perp \perp \perp }$.

Hence ${\alpha }^{\perp \perp }\subseteq {\beta }^{\perp \perp }$ implies ${\beta }^{\perp }={\beta }^{\perp \perp \perp }\subseteq {\alpha }^{\perp \perp \perp }={\alpha }^{\perp }$.

• (iv) For any $\theta \in \mathrm{Con}\left(A\right)$, we have: ${[\theta ,\alpha ]}_A={[\theta ,\beta ]}_A={\Delta }_A$ if and only if ${[\theta ,\alpha \vee \beta ]}_A={\Delta }_A$, hence: $\theta \subseteq {\alpha }^{\perp }\cap {\beta }^{\perp }$ if and only if $\theta \subseteq {(\alpha \vee \beta )}^{\perp }$, thus: ${\alpha }^{\perp }\cap {\beta }^{\perp }={(\alpha \vee \beta )}^{\perp }$. By (iii), ${(\alpha \vee \beta )}^{\perp }={(\alpha \vee \beta )}^{\perp \perp \perp }={({\alpha }^{\perp }\cap {\beta }^{\perp })}^{\perp \perp }$.
• (v) If A is semiprime, then, for any $\theta ,\zeta \in \mathrm{Con}\left(A\right)$, we have: $\theta \subseteq {\zeta }^{\perp }$ if and only if ${[\theta ,\zeta ]}_A={\Delta }_A$ if and only if $\theta \cap \zeta ={\Delta }_A$.

Hence, for $\theta \in \mathrm{Con}\left(A\right)$: $\theta \subseteq {[\alpha ,\beta ]}_A^{\perp }$ if and only if ${[\theta ,{[\alpha ,\beta ]}_A]}_A={\Delta }_A$ if and only if $\theta \cap \alpha \cap \beta ={\Delta }_A$ if and only if ${[\theta ,\alpha \cap \beta ]}_A={\Delta }_A$ if and only if $\theta \subseteq {(\alpha \cap \beta )}^{\perp }$. Taking $\theta ={[\alpha ,\beta ]}_A^{\perp }$ and then $\theta ={(\alpha \cap \beta )}^{\perp }$ in the previous equivalences, we obtain ${[\alpha ,\beta ]}_A^{\perp }={(\alpha \cap \beta )}^{\perp }$.

If we denote by $\gamma ={\alpha }^{\perp \perp }\cap {\beta }^{\perp \perp }$ and $\delta ={(\alpha \cap \beta )}^{\perp }={[\alpha ,\beta ]}_A^{\perp }$, then:

$\gamma \subseteq {\alpha }^{\perp \perp }$ and $\gamma \subseteq {\beta }^{\perp \perp }$, thus ${[\gamma ,{\alpha }^{\perp }]}_A={\Delta }_A$ and ${[\gamma ,{\beta }^{\perp }]}_A={\Delta }_A$;

${[\delta ,\alpha \cap \beta ]}_A={\Delta }_A$, so $\delta \cap \alpha \cap \beta ={\Delta }_A$, thus ${[\alpha \cap \delta ,\beta ]}_A={\Delta }_A$, hence $\alpha \cap \delta \subseteq {\beta }^{\perp }$;

therefore ${[\gamma ,\alpha \cap \delta ]}_A={\Delta }_A$, so $\gamma \cap \alpha \cap \delta ={\Delta }_A$, thus ${[\gamma \cap \delta ,\alpha ]}_A={\Delta }_A$, so $\gamma \cap \delta \subseteq {\alpha }^{\perp }$;

hence ${[\gamma ,\gamma \cap \delta ]}_A={\Delta }_A$, so $\gamma \cap \delta =\gamma \cap \gamma \cap \delta ={\Delta }_A$, thus ${[\gamma ,\delta ]}_A={\Delta }_A$, hence ${\alpha }^{\perp \perp }\cap {\beta }^{\perp \perp }=\gamma \subseteq {\delta }^{\perp }={[\alpha ,\beta ]}_A^{\perp \perp }={(\alpha \cap \beta )}^{\perp \perp }$ by the above.

But ${(\alpha \cap \beta )}^{\perp \perp }\subseteq {\alpha }^{\perp \perp }\cap {\beta }^{\perp \perp }$ by (ii). Therefore ${(\alpha \cap \beta )}^{\perp \perp }={\alpha }^{\perp \perp }\cap {\beta }^{\perp \perp }$.

• (vi) By (v), ${[\alpha ,\beta ]}_A^{\perp \perp }={(\alpha \cap \beta )}^{\perp \perp }={\alpha }^{\perp \perp }\cap {\beta }^{\perp \perp }$, thus, according to (ii) and (iii): ${\alpha }^{\perp }\subseteq {\beta }^{\perp }$ if and only if ${\beta }^{\perp \perp }\subseteq {\alpha }^{\perp \perp }$ if and only if ${\alpha }^{\perp \perp }\cap {\beta }^{\perp \perp }={\beta }^{\perp \perp }$ if and only if ${[\alpha ,\beta ]}_A^{\perp \perp }={\beta }^{\perp \perp }$ if and only if ${[\alpha ,\beta ]}_A^{\perp }={\beta }^{\perp }$.
• (vii) If A is semiprime, then: $\alpha \subseteq {\alpha }^{\perp }$ if and only if ${[\alpha ,\alpha ]}_A={\Delta }_A$ if and only if $\alpha \cap \alpha ={\Delta }_A$ if and only if $\alpha ={\Delta }_A$. □

Lemma 7 

([4] (Proposition 4.$\left(1\right)$), [7] (Proposition 18, Corollary 2)). For any $\theta \in \mathrm{Con}\left(A\right)$:

• $\begin{array}{cc}\hfill {\rho }_A\left(\theta \right)& =max\{\alpha \in \mathrm{Con}\left(A\right)|(\exists n\in {\mathbb{N}}^{*})({[\alpha ,\alpha ]}_A^n\subseteq \theta )\}\hfill \\ & =\bigvee \{\alpha \in \mathrm{Con}\left(A\right)|(\exists n\in {\mathbb{N}}^{*})({[\alpha ,\alpha ]}_A^n\subseteq \theta )\}\hfill \\ & =\bigvee \{\alpha \in \mathcal{K}\left(A\right)|(\exists n\in {\mathbb{N}}^{*})({[\alpha ,\alpha ]}_A^n\subseteq \theta )\}\hfill \\ & =\bigvee \{\alpha \in \mathrm{PCon}\left(A\right)|(\exists n\in {\mathbb{N}}^{*})({[\alpha ,\alpha ]}_A^n\subseteq \theta )\}\hfill \\ & =\{(a,b)\in A^2|(\exists n\in {\mathbb{N}}^{*})({[C{g}_A(a,b),C{g}_A(a,b)]}_A^n\subseteq \theta )\};\hfill \end{array}$
• for any $\alpha \in \mathrm{Con}\left(A\right)$, $\alpha \subseteq {\rho }_A\left(\theta \right)$ if and only if there exists an $n\in {\mathbb{N}}^{*}$ such that ${[\alpha ,\alpha ]}_A^n\subseteq \theta$;
• A is semiprime if and only if, for any $\alpha \in \mathrm{PCon}\left(A\right)$ and any $n\in {\mathbb{N}}^{*}$, ${[\alpha ,\alpha ]}_A^n={\Delta }_A$ implies $\alpha ={\Delta }_A$.

Proposition 2. 

If A satisfies Condition 1.(ii), in particular if $\mathcal{V}$ is either congruence–distributive or both congruence–modular and semidegenerate, then: $A/{\theta }^{\perp }$ is semiprime for all $\theta \in \mathrm{Con}\left(A\right)$ if and only if A is semiprime.

Proof. 

By [27] (Proposition $5.22$) and Lemma 4, if A is semiprime, then $A/\zeta$ is semiprime for all $\zeta \in \mathrm{RCon}\left(A\right)$, in particular $A/{\theta }^{\perp }$ is semiprime for all $\theta \in \mathrm{Con}\left(A\right)$.

Conversely, if $A/{\theta }^{\perp }$ is semiprime for all $\theta \in \mathrm{Con}\left(A\right)$, then $A/{\nabla }_A^{\perp }$ is semiprime; but ${\nabla }_A^{\perp }={\Delta }_A$ by Lemma 6.(i), and $A/{\nabla }_A^{\perp }=A/{\Delta }_A$, which is isomorphic to A, thus A is semiprime. □

See also [7] for the following properties. By [25] [Proposition 6.7], if A satisfies Condition 1.(ii), in particular if $\mathcal{V}$ is either congruence–distributive or both congruence–modular and semidegenerate, then:

• for any $\epsilon \in \mathcal{B}\left(\mathrm{Con}\right(A\left)\right)$ and any $\alpha \in \mathrm{Con}\left(A\right)$, ${[\epsilon ,\alpha ]}_A=\epsilon \cap \alpha$;
• $\mathcal{B}\left(\mathrm{Con}\right(A\left)\right)$ is a Boolean sublattice of $\mathrm{Con}\left(A\right)$ whose complementation is ${·}^{\perp }$ and in which, by the above, the commutator equals the intersection.

By [25] (Proposition $6.11$), if ${\nabla }_A\in \mathcal{K}\left(A\right)$ and A satisfies Condition 1.(ii), in particular if $\mathcal{V}$ is congruence–modular and semidegenerate, then $\mathcal{B}\left(\mathrm{Con}\right(A\left)\right)\subseteq \mathcal{K}\left(A\right)$.

Let us also note that, if the commutator of A equals the intersection, in particular if $\mathcal{V}$ is congruence–distributive, then $\mathrm{Con}\left(A\right)$ is a frame, hence $\mathcal{B}\left(\mathrm{Con}\right(A\left)\right)$ is a complete Boolean sublattice of $\mathrm{Con}\left(A\right)$.

Following [8], we say that an algebra A is hyperarchimedean if and only if, for all $\theta \in \mathrm{PCon}\left(A\right)$, there exists an $n\in {\mathbb{N}}^{*}$ such that ${[\theta ,\theta ]}_A^n\in \mathcal{B}\left(\mathrm{Con}\left(A\right)\right)$.

By the above, if the commutator of A equals the intersection, in particular if $\mathcal{V}$ is congruence–distributive, then A is hyperarchimedean if and only if $\mathrm{PCon}\left(A\right)\subseteq \mathcal{B}\left(\mathrm{Con}\right(A\left)\right)$ if and only if $\mathrm{Con}\left(A\right)\subseteq \mathcal{B}\left(\mathrm{Con}\right(A\left)\right)$ if and only if $\mathcal{B}\left(\mathrm{Con}\right(A\left)\right)=\mathrm{Con}\left(A\right)$; furthermore, if the commutator of A equals the intersection and ${\nabla }_A\in \mathcal{K}\left(A\right)$, in particular if $\mathcal{V}$ is congruence–distributive and semidegenerate, then A is hyperarchimedean if and only if $\mathcal{B}\left(\mathrm{Con}\right(A\left)\right)=\mathcal{K}\left(A\right)=\mathrm{Con}\left(A\right)$. Thus the hyperarchimedean members of a congruence–distributive variety are those with Boolean lattices of congruences and, if the variety is also semidegenerate, then all congruences of its hyperarchimedean members are compact.

Extending the terminology used for rings in [25], we call A a strongly Baer, respectively Baer algebra if and only if, for all $\theta \in \mathrm{Con}\left(A\right)$, respectively all $\theta \in \mathrm{PCon}\left(A\right)$, we have ${\theta }^{\perp }\in \mathcal{B}\left(\mathrm{Con}\left(A\right)\right)$, that is if and only if the commutator lattice $(\mathrm{Con}\left(A\right),{[·,·]}_A)$ is strongly Stone, respectively Stone.

Lemma 8. 

If A satisfies Condition 1.(ii), in particular if $\mathcal{V}$ is either congruence–distributive or both congruence–modular and semidegenerate, then: A is Baer if and only if, for all $\theta \in \mathcal{K}\left(A\right)$, we have ${\theta }^{\perp }\in \mathcal{B}\left(\mathrm{Con}\left(A\right)\right)$.

Proof. 

The converse implication is trivial.

If A is Baer and $\theta \in \mathcal{K}\left(A\right)$, so that ${\displaystyle \theta =\underset{i=1}{\overset{n}{\bigvee }}{\theta }_i}$ for some $n\in {\mathbb{N}}^{*}$ and ${\theta }_1,\dots ,{\theta }_n\in \mathrm{PCon}\left(A\right)$, then ${\theta }_1^{\perp },\dots ,{\theta }_n^{\perp }\in \mathcal{B}\left(\mathrm{Con}\left(A\right)\right)$, hence ${\theta }^{\perp }={({\theta }_1\vee \dots \vee {\theta }_n)}^{\perp }={\theta }_1^{\perp }\cap \dots \cap {\theta }_n^{\perp }\in \mathcal{B}\left(\mathrm{Con}\left(A\right)\right)$ by Lemma 6.(iv). □

Proposition 3. 

If A satisfies Condition 1.(ii), in particular if $\mathcal{V}$ is either congruence–distributive or both congruence–modular and semidegenerate, then:

(i) 

if A is hyperarchimedean, then A is strongly Baer;

(ii) 

if A is strongly Baer, then A is semiprime;

(iii) 

if A is Baer and has principal commutators, then A is semiprime.

Proof. 

(i) By the above, if A is hyperarchimedean, then $\mathcal{B}\left(\mathrm{Con}\right(A\left)\right)=\mathrm{Con}\left(A\right)$, thus A is strongly Baer.

• (ii) Assume that A is strongly Baer and let $\theta \in \mathrm{Con}\left(A\right)$ such that ${[\theta ,\theta ]}_A^n={\Delta }_A$ for some $n\in {\mathbb{N}}^{*}$. If $n\ge 2$, then ${[\theta ,\theta ]}_A^n={[{[\theta ,\theta ]}_A^{n-1},{[\theta ,\theta ]}_A^{n-1}]}_A$, hence ${[\theta ,\theta ]}_A^{n-1}\subseteq {\left({[\theta ,\theta ]}_A^{n-1}\right)}^{\perp }$ by the properties of the implication. But, since A is strongly Baer, ${\left({[\theta ,\theta ]}_A^{n-1}\right)}^{\perp }\in \mathcal{B}\left(\mathrm{Con}\left(A\right)\right)$, thus its commutator with any congruence of A equals the intersection, hence ${[\theta ,\theta ]}_A^{n-1}={[\theta ,\theta ]}_A^{n-1}\cap {\left({[\theta ,\theta ]}_A^{n-1}\right)}^{\perp }={[{[\theta ,\theta ]}_A^{n-1},{\left({[\theta ,\theta ]}_A^{n-1}\right)}^{\perp }]}_A={\Delta }_A$. By turning the above into a recursive argument we get that ${[\theta ,\theta ]}_A={\Delta }_A$ and then that $\theta ={\Delta }_A$. By Lemma 7, it follows that A is semiprime.
• (iii) By an analogous argument to that of (ii), taking $\theta \in \mathrm{PCon}\left(A\right)$, so that ${[\theta ,\theta ]}_A^n\in \mathrm{PCon}\left(A\right)$ for any $n\in {\mathbb{N}}^{*}$ since A has principal commutators. □

4. The Minimal Prime Spectrum

Ⓗ

Throughout this section, we will assume that A satisfies Condition 1.(i), which holds if $\mathcal{V}$ is congruence–modular.

By an argument based on Zorn’s Lemma, it follows that:

• any prime congruence of A includes a minimal prime congruence, hence ${\rho }_A\left({\Delta }_A\right)=\bigcap \mathrm{Spec}\left(A\right)=\bigcap \mathrm{Min}\left(A\right)$;
• moreover, for any $\theta \in \mathrm{Con}\left(A\right)$ and any $\psi \in V_A\left(\theta \right)=\left[\theta \right)\cap \mathrm{Spec}\left(A\right)$, there exists a $\varphi \in Min\left(V_A\left(\theta \right)\right)=Min(\left[\theta \right)\cap \mathrm{Spec}\left(A\right))$ such that $\varphi \subseteq \psi$, hence:

Remark 3. 

For any $\theta \in \mathrm{Con}\left(A\right)$, we have:

• ${\rho }_A\left(\theta \right)=\bigcap Min\left(V_A\left(\theta \right)\right)=\bigcap Min(\left[\theta \right)\cap \mathrm{Spec}\left(A\right))$;
• $D_A\left(\theta \right)\cap \mathrm{Min}\left(A\right)=\varnothing$ if and only if $V_A\left(\theta \right)\cap \mathrm{Min}\left(A\right)=\mathrm{Min}\left(A\right)$ if and only if $\left[\theta \right)\cap \mathrm{Min}\left(A\right)=\mathrm{Min}\left(A\right)$ if and only if $\mathrm{Min}\left(A\right)\subseteq \left[\theta \right)$ if and only if $\theta \subseteq \bigcap \mathrm{Min}\left(A\right)$ if and only if $\theta \subseteq {\rho }_A\left({\Delta }_A\right)$ if and only if ${\rho }_A\left(\theta \right)={\rho }_A\left({\Delta }_A\right)$;
• $D_A\left(\theta \right)\cap \mathrm{Min}\left(A\right)=\mathrm{Min}\left(A\right)$ if and only if $V_A\left(\theta \right)\cap \mathrm{Min}\left(A\right)=\varnothing$. $V_A\left(\theta \right)=\varnothing$ if and only if ${\rho }_A\left(\theta \right)={\nabla }_A$, which holds if $\theta ={\nabla }_A$; recall from [7] that, if ${\nabla }_A\in \mathcal{K}\left(A\right)$ and ${[{\nabla }_A,{\nabla }_A]}_A={\nabla }_A$, then ${\nabla }_A/{\equiv }_A=\left\{{\nabla }_A\right\}$, so: ${\rho }_A\left(\theta \right)={\nabla }_A$ if and only if $\theta ={\nabla }_A$. Clearly, $V_A\left(\theta \right)=\varnothing$ implies $V_A\left(\theta \right)\cap \mathrm{Min}\left(A\right)=\varnothing$; the converse implication holds if and only if $\mathrm{Min}\left(A\right)=\mathrm{Spec}\left(A\right)$ if and only if $\mathrm{Spec}\left(A\right)$ is an antichain.

Indeed, $\mathrm{Spec}\left(A\right)$ is an antichain if and only if $\mathrm{Min}\left(A\right)=\mathrm{Spec}\left(A\right)$, case in which $V_A\left(\theta \right)=V_A\left(\theta \right)\cap \mathrm{Min}\left(A\right)$.

Now, if $V_A\left(\theta \right)\cap \mathrm{Min}\left(A\right)=\varnothing$ implies $V_A\left(\theta \right)=\varnothing$, then let us assume by absurdum that $\mathrm{Min}\left(A\right)\ne \mathrm{Spec}\left(A\right)$, that is $\mathrm{Spec}\left(A\right)⊈\mathrm{Min}\left(A\right)$, so that there exists $\varphi \in \mathrm{Spec}\left(A\right)\setminus \mathrm{Min}\left(A\right)$. But then $V_A\left(\varphi \right)\cap \mathrm{Min}\left(A\right)=\varnothing$, while $V_A\left(\varphi \right)\ne \varnothing$ since $\varphi \in V_A\left(\varphi \right)$; a contradiction.

Proposition 4. 

If ${\nabla }_A\in \mathcal{K}\left(A\right)$, in particular if $\mathcal{V}$ is congruence–modular and semidegenerate, then, for any $\theta \in \mathrm{Con}\left(A\right)$ and any $\varphi \in V_A\left(\theta \right)$, the following are equivalent:

(i) 

$\varphi \in Min\left(V_A\left(\theta \right)\right)$;

(ii) 

${\nabla }_A\setminus \varphi$ is a maximal element of the set of m–systems of A which are disjoint from θ.

Proof. 

By Remark 2, ${\nabla }_A\setminus \varphi$ is an m–system, which is, of course, disjoint from $\theta$ since $({\nabla }_A\setminus \varphi )\cap \theta \subseteq ({\nabla }_A\setminus \varphi )\cap \varphi =\varnothing$.

• (i)⇒(ii): By an application of Zorn’s Lemma, it follows that there exists a maximal element M of the set of m–systems of A which include ${\nabla }_A\setminus \varphi$ and are disjoint from $\theta$, so that ${\nabla }_A\setminus \varphi \subseteq M\subseteq {\nabla }_A\setminus \theta$ and, furthermore, M is a maximal element of the set of m–systems of A which are disjoint from $\theta$.

By Lemma 3, there is $\psi \in Max\{\alpha \in \mathrm{Con}(A\left)\right|\theta \subseteq \alpha ,M\cap \alpha =\varnothing \}\subseteq \mathrm{Spec}(A)$, so that $\psi \in V_A\left(\theta \right)$ and $({\nabla }_A\setminus \varphi )\cap \psi \subseteq M\cap \psi =\varnothing$, thus ${\nabla }_A\setminus \varphi \subseteq M\subseteq {\nabla }_A\setminus \psi$, hence $\psi \subseteq \varphi$.

Since $\varphi \in Min\left(V_A\left(\theta \right)\right)$, it follows that $\varphi =\psi$, thus ${\nabla }_A\setminus \varphi =M$, which is a maximal element of the set of m–systems of A disjoint from $\theta$.

• (ii)⇒(i): Let $\mu$ be a minimal element of $V_A\left(\theta \right)$ with $\mu \subseteq \varphi$.

By Remark 2, ${\nabla }_A\setminus \mu$ is an m–system, disjoint from $\theta$ since $({\nabla }_A\setminus \mu )\cap \theta \subseteq ({\nabla }_A\setminus \mu )\cap \mu =\varnothing$, and ${\nabla }_A\setminus \varphi \subseteq {\nabla }_A\setminus \mu$. Since ${\nabla }_A\setminus \varphi$ is a maximal element of the set of m–systems of A which are disjoint from $\theta$, it follows that ${\nabla }_A\setminus \varphi ={\nabla }_A\setminus \mu$, thus $\varphi =\mu \in Min\left(V_A\left(\theta \right)\right)$. □

Corollary 1. 

If ${\nabla }_A\in \mathcal{K}\left(A\right)$, in particular if $\mathcal{V}$ is congruence–modular and semidegenerate, then, for any $\varphi \in \mathrm{Spec}\left(A\right)$, the following are equivalent:

• $\varphi \in \mathrm{Min}\left(A\right)$;
• ${\nabla }_A\setminus \varphi$ is a maximal element of the set of m–systems of A which are disjoint from ${\Delta }_A$.

Proof. 

By Proposition 4 for $\theta ={\Delta }_A$. □

Lemma 9 

([28]). If L is a bounded distributive lattice and $P\in {\mathrm{Spec}}_{\mathrm{Id}}\left(L\right)$, then the following are equivalent:

• $P\in {\mathrm{Min}}_{\mathrm{Id}}\left(L\right)$;
• for any $x\in P$, ${\mathrm{Ann}}_L\left(x\right)⊈P$.

Recall from Section 2 that $\mathrm{Spec}\left(A\right)=\mathrm{Mi}\left(\mathrm{Con}\right(A\left)\right)\cap \mathrm{RCon}\left(A\right)$. By [25] (Proposition $4.4$), if A is semiprime, then all annihilators in $(\mathrm{Con}\left(A\right),{[·,·]}_A)$ are lattice ideals of $\mathrm{Con}\left(A\right)$.

Remember that, in the commutator lattice $(\mathrm{Con}\left(A\right),{[·,·]}_A)$, $R\left(\mathrm{Con}\right(A\left)\right)=\mathrm{RCon}\left(A\right)$ and ${\mathrm{Spec}}_{\mathrm{Con}\left(A\right)}=\mathrm{Spec}\left(A\right)$, and that, since $\mathrm{Con}\left(A\right)/{\equiv }_A$ is a frame, the elements of ${\mathrm{Spec}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}$ are exactly the meet–prime elements of $\mathrm{Con}\left(A\right)/{\equiv }_A$, thus, by the distributivity of $\mathrm{Con}\left(A\right)/{\equiv }_A$, ${\mathrm{Spec}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}=\mathrm{Mi}(\mathrm{Con}\left(A\right)/{\equiv }_A)$.

Lemma 10. 

If A is semiprime, then:

(i) 

for any $U\subseteq \mathrm{Con}\left(A\right)$, ${\mathrm{Ann}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}(U/{\equiv }_A)={\mathrm{Ann}}_{(\mathrm{Con}\left(A\right),{[·,·]}_A)}\left(U\right)/{\equiv }_A$;

(ii) 

${\mathrm{Spec}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}=\{\varphi /{\equiv }_A|\varphi \in \mathrm{Spec}\left(A\right)\}$;

(iii) 

for all $\theta \in \mathrm{RCon}\left(A\right)$, $\theta /{\equiv }_A\cap \mathrm{RCon}\left(A\right)=\left\{\theta \right\}$ and $\theta =max(\theta /{\equiv }_A)$;

(iv) 

$\varphi \mapsto \varphi /{\equiv }_A$ is an order isomorphism from $\mathrm{Spec}\left(A\right)$ to ${\mathrm{Spec}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}$;

(v) 

$R(\mathrm{Con}\left(A\right)/{\equiv }_A)=\{\varphi /{\equiv }_A|\varphi \in \mathrm{RCon}\left(A\right)\}$; moreover, for any $\varphi \in \mathrm{Con}\left(A\right)$, we have: $\varphi \in \mathrm{RCon}\left(A\right)$ if and only if $\varphi /{\equiv }_A\in R(\mathrm{Con}\left(A\right)/{\equiv }_A)$; thus $\varphi \mapsto \varphi /{\equiv }_A$ is an order isomorphism from $\mathrm{RCon}\left(A\right)$ to $R(\mathrm{Con}\left(A\right)/{\equiv }_A)$.

Proof. 

(i) By [25] (Lemma $4.2$).

• (ii) By [25] (Proposition $6.2$).
• (iii) By [25] (Remark $5.11$).
• (iv) By (ii), (iii) and the fact that $\mathrm{Spec}\left(A\right)\subseteq \mathrm{RCon}\left(A\right)$ and ${\mathrm{Spec}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}\subseteq R(\mathrm{Con}\left(A\right)/{\equiv }_A)$.
• (v) The equality follows from (ii) and the definition of radical elements; by (iii), we also obtain the equivalence and the order isomorphism. □

Remark 4. 

For any $\alpha ,\beta \in \mathrm{Con}\left(A\right)$, we have $\alpha /{\equiv }_A\le \beta /{\equiv }_A$ if and only if ${\rho }_A\left(\alpha \right)\subseteq {\rho }_A\left(\beta \right)$. Indeed, $\alpha /{\equiv }_A\le \beta /{\equiv }_A$ if and only if $\alpha /{\equiv }_A\wedge \beta /{\equiv }_A=\alpha /{\equiv }_A$ if and only if $(\alpha \cap \beta )/{\equiv }_A=\alpha /{\equiv }_A$ if and only if ${\rho }_A(\alpha \cap \beta )={\rho }_A\left(\alpha \right)$ if and only if ${\rho }_A\left(\alpha \right)\cap {\rho }_A\left(\beta \right)={\rho }_A\left(\alpha \right)$ if and only if ${\rho }_A\left(\alpha \right)\subseteq {\rho }_A\left(\beta \right)$.

Proposition 5. 

Assume that A is semiprime and let $\varphi \in \mathrm{Spec}\left(A\right)$. Let us consider the following statements:

(i) 

$\varphi \in \mathrm{Min}\left(A\right)$;

(ii) 

for any $\alpha \in \mathcal{K}\left(A\right)$, $\alpha \subseteq \varphi$ implies ${\alpha }^{\perp }⊈\varphi$;

(iii) 

for any $\alpha \in \mathcal{K}\left(A\right)$, $\alpha \subseteq \varphi$ if and only if ${\alpha }^{\perp }⊈\varphi$;

(iv) 

for any $\alpha \in \mathrm{Con}\left(A\right)$, $\alpha \subseteq \varphi$ implies ${\alpha }^{\perp }⊈\varphi$;

(v) 

for any $\alpha \in \mathrm{Con}\left(A\right)$, $\alpha \subseteq \varphi$ if and only if ${\alpha }^{\perp }⊈\varphi$.

If A satisfies Condition 1.(iv), then statements (i), (ii) and (iii) are equivalent.

If A satisfies Condition 1.(v), then statements (i), (iv) and (v) are equivalent.

Proof. Case 1: 

Assume that A satisfies Condition 1.(iv).

• (i)⇔(ii): Recall from [7] [Lemma 11.(i)] that we have the following order–preserving maps:
• $\theta \mapsto {\theta }^{*}$ from $\mathrm{Con}\left(A\right)$ to $\mathrm{Id}\left(\mathcal{L}\right(A\left)\right)$, defined by: ${\theta }^{*}$$=(\left(\theta \right]\cap \mathcal{K}\left(A\right))/{\equiv }_A$ for all $\theta \in \mathrm{Con}\left(A\right)$;
• $I\mapsto I_{*}$ from $\mathrm{Id}\left(\mathcal{L}\right(A\left)\right)$ to $\mathrm{Con}\left(A\right)$, defined by: $I_{*}$$=\bigvee \{\gamma \in \mathcal{K}\left(A\right)|\gamma /{\equiv }_A\in I\}$ for all $I\in \mathrm{Id}\left(\mathcal{L}\right(A\left)\right)$.

By [7] [Proposition 11], these maps restrict to order isomorphisms between $\mathrm{Spec}\left(A\right)$ and ${\mathrm{Spec}}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)$, inverses of each other, thus they further restrict to mutually inverse order isomorphisms between $\mathrm{Min}\left(A\right)$ and ${\mathrm{Min}}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)$.

Let $\beta \in \mathcal{K}\left(A\right)$ and $\psi \in \mathrm{Spec}\left(A\right)$, arbitrary. By the above, ${\left({\psi }^{*}\right)}_{*}=\psi$. Since $\beta \in \mathcal{K}\left(A\right)$,

$${(\beta /{\equiv }_A]}_{\mathcal{L}\left(A\right)}={(\beta /{\equiv }_A]}_{\mathrm{Con}\left(A\right)/{\equiv }_A}\cap \mathcal{L}\left(A\right)=$$

$${\left(\beta \right]}_{\mathrm{Con}\left(A\right)}/{\equiv }_A\cap \mathcal{K}\left(A\right)/{\equiv }_A=({\left(\beta \right]}_{\mathrm{Con}\left(A\right)}\cap \mathcal{K}\left(A\right))/{\equiv }_A={\beta }^{*},$$

hence ${\mathrm{Ann}}_{\mathcal{L}\left(A\right)}(\beta /{\equiv }_A)={\mathrm{Ann}}_{\mathcal{L}\left(A\right)}\left({(\beta /{\equiv }_A]}_{\mathcal{L}\left(A\right)}\right)={\mathrm{Ann}}_{\mathcal{L}\left(A\right)}\left({\beta }^{*}\right)$. By [7] [Lemma 27], since A is semiprime, we have: ${\mathrm{Ann}}_{\mathcal{L}\left(A\right)}\left({\beta }^{*}\right)\subseteq {\psi }^{*}$ if and only if ${\beta }^{\perp }\subseteq {\left({\psi }^{*}\right)}_{*}$, that is ${\beta }^{\perp }\subseteq \psi$.

Hence: $\varphi \in \mathrm{Min}\left(A\right)$ if and only if ${\varphi }^{*}\in {\mathrm{Min}}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)$. By Lemma 9, the latter is equivalent to $(\forall x\in {\varphi }^{*})({\mathrm{Ann}}_{\mathcal{L}\left(A\right)}\left(x\right)⊈{\varphi }^{*})$, i.e., $(\forall \alpha \in \left(\varphi \right]\cap \mathcal{K}\left(A\right))({\mathrm{Ann}}_{\mathcal{L}\left(A\right)}(\alpha /{\equiv }_A)⊈{\varphi }^{*})$, which means that $(\forall \alpha \in \left(\varphi \right]\cap \mathcal{K}\left(A\right))({\mathrm{Ann}}_{\mathcal{L}\left(A\right)}\left({\alpha }^{*}\right)⊈{\varphi }^{*})$, which is equivalent to $(\forall \alpha \in \left(\varphi \right]\cap \mathcal{K}\left(A\right))({\alpha }^{\perp }⊈\varphi )$, that is $(\forall \alpha \in \mathcal{K}\left(A\right))(\alpha \subseteq \varphi \Rightarrow {\alpha }^{\perp }⊈\varphi )$.

• (iii)⇒(ii): Trivial.
• (ii)⇒(iii): If $\alpha \in \mathcal{K}\left(A\right)$ is such that ${\alpha }^{\perp }⊈\varphi$, then, since ${[\alpha ,{\alpha }^{\perp }]}_A={\Delta }_A\subseteq \varphi \in \mathrm{Spec}\left(A\right)$, it follows that $\alpha \subseteq \varphi$.
• Case 2: Now assume that A satisfies Condition 1.(v).
• (v)⇒(iv): Trivial.
• (iv)⇒(v): Analogous to the proof of (ii)⇒(iii).
• (i)⇔(iv): By Lemma 10.(iv), the condition that $\varphi \in \mathrm{Spec}\left(A\right)$ is equivalent to $\varphi /{\equiv }_A\in {\mathrm{Spec}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}$, which is equivalent to $(\varphi /{\equiv }_A]\in {\mathrm{Spec}}_{\mathrm{Id}}(\mathrm{Con}\left(A\right)/{\equiv }_A)$.

Again by Lemma 10.(iv), $\varphi \in \mathrm{Min}\left(A\right)$ if and only if $\varphi /{\equiv }_A\in {\mathrm{Min}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}$, which is equivalent to $(\varphi /{\equiv }_A]$$\in {\mathrm{Min}}_{\mathrm{Id}}(\mathrm{Con}\left(A\right)/{\equiv }_A)$. By Lemma 9 and Lemma 10.(i), the latter is equivalent to the fact that, for any $\alpha \in \left(\varphi \right]$, $({\alpha }^{\perp }/{\equiv }_A]=\left({\alpha }^{\perp }\right]/{\equiv }_A={\mathrm{Ann}}_{(\mathrm{Con}\left(A\right),{[·,·]}_A)}\left(\alpha \right)/{\equiv }_A$$={\mathrm{Ann}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}(\alpha /{\equiv }_A)⊈(\varphi /{\equiv }_A]=\left(\varphi \right]/{\equiv }_A$, that is ${\alpha }^{\perp }/{\equiv }_A\notin (\varphi /{\equiv }_A]=\left(\varphi \right]/{\equiv }_A$. Since $\varphi \in \mathrm{Spec}\left(A\right)\subseteq \mathrm{RCon}\left(A\right)$ and thus $\varphi =max(\varphi /{\equiv }_A)$ by Lemma 10.(iii), this condition is equivalent to ${\alpha }^{\perp }\notin \left(\varphi \right]$, that is ${\alpha }^{\perp }⊈\varphi$. □

Example 1. 

Note that the equivalence in Proposition 5 for the case when A satisfies Condition 1.(iv) does not hold for $\alpha \in \mathrm{Con}\left(A\right)$, arbitrary. Indeed, if we let A be the Boolean subalgebra of the power set $\mathcal{P}\left(\mathbb{N}\right)$ of the set $\mathbb{N}$ of natural numbers formed of the finite and the cofinite subsets of $\mathbb{N}$: $A=\left\{S\right|S\subseteq \mathbb{N},\left|S\right|<{\aleph }_{0}or|\mathbb{N}\setminus S|<{\aleph }_0\}$, then, since A is a Boolean algebra, its lattice of congruences is isomorphic to its lattice of filters, and obviously this lattice isomorphism $\phi :\mathrm{Filt}\left(A\right)\to \mathrm{Con}\left(A\right)$ takes the set ${\mathrm{Spec}}_{\mathrm{Filt}\left(A\right)}$ of the prime elements of the lattice $\mathrm{Filt}\left(A\right)$ of the filters of A, which equals the set ${\mathrm{Spec}}_{\mathrm{Filt}}\left(A\right)={\mathrm{Max}}_{\mathrm{Filt}}\left(A\right)$ of the prime and thus maximal filters of A by a routine proof, to ${\mathrm{Spec}}_{\mathrm{Con}\left(\mathrm{A}\right)}=\mathrm{Spec}\left(A\right)=\mathrm{Max}\left(A\right)=\mathrm{Min}\left(A\right)$ since A is a Boolean algebra, therefore ${\mathrm{Min}}_{\mathrm{Filt}}\left(A\right):=Min\left({\mathrm{Spec}}_{\mathrm{Filt}}\left(A\right)\right)={\mathrm{Spec}}_{\mathrm{Filt}}\left(A\right)={\mathrm{Max}}_{\mathrm{Filt}}\left(A\right)=Max(\mathrm{Filt}\left(A\right)\setminus \left\{A\right\})$. Now let us consider the filter $P:=\left\{S\right|S\subseteq \mathbb{N},|\mathbb{N}\setminus S|<{\aleph }_0\}$. It is well known that ${\mathrm{Spec}}_{\mathrm{Filt}}\left(A\right)=Max(\mathrm{Filt}\left(A\right)\setminus \left\{A\right\})=\{M\cap A|M\in Max(\mathrm{Filt}\left(\mathcal{P}\left(\mathbb{N}\right)\right)\setminus \left\{\mathcal{P}\left(\mathbb{N}\right)\right\})\}\cup \left\{P\right\}=\{{\left[\left\{a\right\}\right)}_{\mathcal{P}\left(\mathbb{N}\right)}\cap A|a\in \mathbb{N}\}\cup \left\{P\right\}$, in particular P is a prime and thus a minimal prime filter of A. P is clearly not a principal, thus not a compact filter of A. Since Boolean algebras are congruence–distributive, the commutator ${[·,·]}_A$ of A equals the intersection, thus the commutator lattice $(\mathrm{Con}\left(A\right),{[·,·]}_A=\cap )$ is isomorphic to the commutator lattice $\left(\mathrm{Filt}\right(A),\cap )$, also endowed with the commutator operation equalling the intersection, in which $P^{\perp }=max\{F\in \mathrm{Filt}\left(A\right)|P\cap F=\left\{\mathbb{N}\right\}\}=max\left\{\left\{\mathbb{N}\right\}\right\}=\left\{\mathbb{N}\right\}$, since any nontrivial filter F of A contains a proper subset S of $\mathbb{N}$, which must thus be such that an $a\in \mathbb{N}$ does not belong to S, hence S is included in the proper cofinite subset $\mathbb{N}\setminus \left\{a\right\}$ of $\mathbb{N}$, so $\mathbb{N}\setminus \left\{a\right\}\in P\cap F$, which means that no nontrivial filter F of A satisfies $P\cap F=\left\{\mathbb{N}\right\}$. So $P^{\perp }=\left\{\mathbb{N}\right\}\subset P$; of course, $P\subseteq P$. Therefore, $\phi \left(P\right)\in \mathrm{Spec}\left(A\right)=\mathrm{Min}\left(A\right)$, and $\phi \left(P\right)\in \mathrm{Con}\left(A\right)\setminus \mathrm{Cp}\left(\mathrm{Con}\right(A\left)\right)=\mathrm{Con}\left(A\right)\setminus \mathcal{K}\left(A\right)$; thus $\phi {\left(P\right)}^{\perp }=\phi \left(P^{\perp }\right)=\phi \left(\left\{\mathbb{N}\right\}\right)={\Delta }_A\subset \phi \left(P\right)$ and $\phi \left(P\right)\subseteq \phi \left(P\right)$, hence $\phi \left(P\right)\subseteq \phi \left(P\right)$ does not imply $\phi {\left(P\right)}^{\perp }⊈\phi \left(P\right)$.

Corollary 2. 

Assume that A satisfies Condition 1.(iii), let $\varphi \in \mathrm{Spec}\left(A\right)$ and let us consider the following statements:

(i) 

$\varphi \in \mathrm{Min}\left(A\right)$;

(ii) 

for any $\alpha \in \mathcal{K}\left(A\right)$, $\alpha \subseteq \varphi$ implies $\alpha \to {\rho }_A\left({\Delta }_A\right)⊈\varphi$;

(iii) 

for any $\alpha \in \mathrm{Con}\left(A\right)$, $\alpha \subseteq \varphi$ implies $\alpha \to {\rho }_A\left({\Delta }_A\right)⊈\varphi$.

If A satisfies Condition 1.(iv), then (i) is equivalent to (ii).

If A satisfies Condition 1.(v), then (i) is equivalent to (iii).

Proof. 

Case 1: Assume that A satisfies Condition 1.(iv). Then we have the following equivalences: $\varphi \in \mathrm{Min}\left(A\right)$ if and only if $\varphi /{\rho }_A\left({\Delta }_A\right)\in \mathrm{Min}(A/{\rho }_A\left({\Delta }_A\right))$, which, by Proposition 5, since $A/{\rho }_A\left({\Delta }_A\right)$ is semiprime, is equivalent to the fact that, for any $\alpha \in \mathcal{K}\left(A\right)$, $(\alpha \vee {\rho }_A\left({\Delta }_A\right))/{\rho }_A\left({\Delta }_A\right)\subseteq \varphi /{\rho }_A\left({\Delta }_A\right)$ implies ${((\alpha \vee {\rho }_A\left({\Delta }_A\right))/{\rho }_A\left({\Delta }_A\right))}^{\perp }⊈\varphi /{\rho }_A\left({\Delta }_A\right)$, that is $\alpha \vee {\rho }_A\left({\Delta }_A\right)\subseteq \varphi$ implies $(\alpha \to {\rho }_A\left({\Delta }_A\right))/{\rho }_A\left({\Delta }_A\right)⊈\varphi /{\rho }_A\left({\Delta }_A\right)$ according to Proposition 1.(ii), that is $\alpha \subseteq \varphi$ implies $\alpha \to {\rho }_A\left({\Delta }_A\right)⊈\varphi$ since $\varphi$ is prime and thus ${\rho }_A\left({\Delta }_A\right)\subseteq \varphi$.

• Case 2: Assume that A satisfies Condition 1.(v). Then the proof goes the same as above, but for all $\alpha \in \mathrm{Con}\left(A\right)$. □

5. Two Topologies on the Minimal Prime Spectrum

Ⓗ

Throughout this section, we will assume that A satisfies Condition 1.(i), which holds in the particular case when $\mathcal{V}$ is congruence–modular.

Clearly, the Stone topology ${\mathcal{S}}_{\mathrm{Spec}}\left(A\right)$ of $\mathrm{Spec}\left(A\right)$ induces the topology ${\mathcal{S}}_{\mathrm{Min}}\left(A\right)=\{D_A\left(\theta \right)\cap \mathrm{Min}\left(A\right)|\theta \in \mathrm{Con}\left(A\right)\}$ on $\mathrm{Min}\left(A\right)$, which has $\{D_A(a,b)\cap \mathrm{Min}\left(A\right)|a,b\in A\}$ as a basis and $\{V_A\left(\theta \right)\cap \mathrm{Min}\left(A\right)|\theta \in \mathrm{Con}\left(A\right)\}$ as the family of closed sets. ${\mathcal{S}}_{\mathrm{Min}}\left(A\right)$ is called the Stone or spectral topology on $\mathrm{Min}\left(A\right)$.

Ⓗ

Throughout the rest of this section, we will also assume that A is semiprime.

Lemma 11. 

${\theta }^{\perp }=\bigcap (V_A\left({\theta }^{\perp }\right)\cap \mathrm{Min}\left(A\right))$ for every $\theta \in \mathrm{Con}\left(A\right)$.

Proof. 

Let $\theta \in \mathrm{Con}\left(A\right)$. Clearly, ${\theta }^{\perp }\subseteq \bigcap (V_A\left({\theta }^{\perp }\right)\cap \mathrm{Min}\left(A\right))$.

Let us denote by $\alpha =\bigcap (V_A\left({\theta }^{\perp }\right)\cap \mathrm{Min}\left(A\right))$, so that $\alpha \subseteq \mu$ for any $\mu \in V_A\left({\theta }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$. Assume by absurdum that $\alpha ⊈{\theta }^{\perp }$, so that ${[\alpha ,\theta ]}_A\ne {\Delta }_A={\rho }_A\left({\Delta }_A\right)=\bigcap \mathrm{Min}\left(A\right)$ since A is semiprime, therefore ${[\alpha ,\theta ]}_A⊈\varphi$ for some $\varphi \in \mathrm{Min}\left(A\right)$, which implies that $\theta ⊈\varphi$ and $\alpha ⊈\varphi$, hence $\varphi \notin V_A\left({\theta }^{\perp }\right)$, that is ${\theta }^{\perp }⊈\varphi$. So $\theta ⊈\varphi$ and ${\theta }^{\perp }⊈\varphi$, while ${[\theta ,{\theta }^{\perp }]}_A={\Delta }_A\subseteq \varphi$, which contradicts the fact that $\varphi \in \mathrm{Min}\left(A\right)\subseteq \mathrm{Spec}\left(A\right)$. Therefore $\bigcap (V_A\left({\theta }^{\perp }\right)\cap \mathrm{Min}\left(A\right))=\alpha \subseteq {\theta }^{\perp }$, hence the equality. □

Remark 5. 

By Lemma 11, for any $\alpha ,\beta \in \mathrm{Con}\left(A\right)$, we have: ${\alpha }^{\perp }={\beta }^{\perp }$ if and only if $V_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)=V_A\left({\beta }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$ if and only if $D_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)=D_A\left({\beta }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$.

Proposition 6. 

For any $\alpha ,\beta ,\gamma \in \mathrm{Con}\left(A\right)$, we consider the following statements:

(i) 

$V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)=V_A\left({\alpha }^{\perp \perp }\right)\cap \mathrm{Min}\left(A\right)=D_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$ and $D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$$=D_A\left({\alpha }^{\perp \perp }\right)\cap \mathrm{Min}\left(A\right)=V_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$;

(ii) 

${\alpha }^{\perp }\cap {\beta }^{\perp }={\gamma }^{\perp }$ if and only if $V_A\left(\alpha \right)\cap V_A\left(\beta \right)\cap \mathrm{Min}\left(A\right)=V_A\left(\gamma \right)\cap \mathrm{Min}\left(A\right)$;

(iii) 

${\alpha }^{\perp \perp }={\beta }^{\perp }$ if and only if ${\alpha }^{\perp }={\beta }^{\perp \perp }$ if and only if $V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)=$$V_A\left({\beta }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$ if and only if $V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)=D_A\left(\beta \right)\cap \mathrm{Min}\left(A\right)$ if and only if $D_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)=D_A\left(\beta \right)\cap \mathrm{Min}\left(A\right)$.

If A satisfies Condition 1.(iv), then the statements above hold for all $\alpha ,\beta ,\gamma \in \mathcal{K}\left(A\right)$.

If A satisfies Condition 1.(v), then the statements above hold for all $\alpha ,\beta ,\gamma \in \mathrm{Con}\left(A\right)$.

Proof. 

Let $\varphi \in \mathrm{Min}\left(A\right)$.

• Case 1: Assume that A satisfies Condition 1.(iv) and let $\alpha ,\beta ,\gamma \in \mathcal{K}\left(A\right)$.
• (i) By Proposition 5, $\varphi \in V_A\left(\alpha \right)$ if and only if $\varphi \in D_A\left({\alpha }^{\perp }\right)$, hence also $\varphi \notin V_A\left(\alpha \right)$ if and only if $\varphi \notin D_A\left({\alpha }^{\perp }\right)$, that is $\varphi \in D_A\left(\alpha \right)$ if and only if $\varphi \in V_A\left({\alpha }^{\perp }\right)$. Therefore $V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)=D_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$ and $D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)=V_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$, hence also $V_A\left({\alpha }^{\perp \perp }\right)\cap \mathrm{Min}\left(A\right)=D_A\left({\alpha }^{\perp \perp \perp }\right)\cap \mathrm{Min}\left(A\right)=D_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$ and $D_A\left({\alpha }^{\perp \perp }\right)\cap \mathrm{Min}\left(A\right)=V_A\left({\alpha }^{\perp \perp \perp }\right)\cap \mathrm{Min}\left(A\right)=V_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$ by Lemma 6.(iii).
• (ii) By (i), along with Proposition 6.(iv), and Remark 5, ${\alpha }^{\perp }\cap {\beta }^{\perp }={\gamma }^{\perp }$ if and only if ${(\alpha \vee \beta )}^{\perp }={\gamma }^{\perp }$ if and only if $V_A\left({(\alpha \vee \beta )}^{\perp }\right)\cap \mathrm{Min}\left(A\right)=V_A\left({\gamma }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$ if and only if $(D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))\cup (D_A\left(\beta \right)\cap \mathrm{Min}\left(A\right))=(D_A\left(\alpha \right)\cup D_A\left(\beta \right))\cap \mathrm{Min}\left(A\right)=D_A(\alpha \vee \beta )\cap \mathrm{Min}\left(A\right)=D_A\left(\gamma \right)\cap \mathrm{Min}\left(A\right)$ if and only if $\mathrm{Min}\left(A\right)\setminus ((D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))\cup (D_A\left(\beta \right)\cap \mathrm{Min}\left(A\right)))=\mathrm{Min}\left(A\right)\setminus (D_A\left(\gamma \right)\cap \mathrm{Min}\left(A\right))$ if and only if $V_A\left(\alpha \right)\cap V_A\left(\beta \right)\cap \mathrm{Min}\left(A\right)=(V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))\cap (V_A\left(\beta \right)\cap \mathrm{Min}\left(A\right))=V_A\left(\gamma \right)\cap \mathrm{Min}\left(A\right)$.
• (iii) By (i) and Remark 5, ${\alpha }^{\perp \perp }={\beta }^{\perp }$ if and only if $V_A\left({\alpha }^{\perp \perp }\right)\cap \mathrm{Min}\left(A\right)=V_A\left({\beta }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$ if and only if $D_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)=V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)=V_A\left({\beta }^{\perp }\right)\cap \mathrm{Min}\left(A\right)=D_A\left(\beta \right)\cap \mathrm{Min}\left(A\right)$. By Lemma 6.(iii), ${\alpha }^{\perp \perp }={\beta }^{\perp }$ implies ${\alpha }^{\perp }={\alpha }^{\perp \perp \perp }={\beta }^{\perp \perp }$, which also proves the converse.
• Case 2: The proof goes similarly in the case when A satisfies Condition 1.(v), but for all $\alpha ,\beta ,\gamma \in \mathrm{Con}\left(A\right)$. □

Let us denote by ${\mathcal{F}}_{\mathrm{Min}}\left(A\right)$ the topology on $\mathrm{Min}\left(A\right)$ generated by $\{V_A(a,b)\cap \mathrm{Min}\left(A\right)|$$a,b\in A\}$, called the flat topology or the inverse topology on $\mathrm{Min}\left(A\right)$. Also, we denote by $\mathcal{M}in\left(A\right)$, respectively $\mathcal{M}in{\left(A\right)}^{-1}$ the minimal prime spectrum of A endowed with the Stone, respectively the flat topology: $\mathcal{M}in\left(A\right)=(\mathrm{Min}\left(A\right),{\mathcal{S}}_{\mathrm{Min}}\left(A\right))$ and $\mathcal{M}in{\left(A\right)}^{-1}=(\mathrm{Min}\left(A\right),{\mathcal{F}}_{\mathrm{Min}}\left(A\right))$.

Remark 6. 

${\mathcal{F}}_{\mathrm{Min}}\left(A\right)$ has $\{V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)|\alpha \in \mathcal{K}\left(A\right)\}$ as a basis, since $V_A\left({\Delta }_A\right)\cap \mathrm{Min}\left(A\right)=\mathrm{Min}\left(A\right)$ and, for $\alpha ,\beta \in \mathcal{K}\left(A\right)$, $\alpha \vee \beta \in \mathcal{K}\left(A\right)$ and $V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)\cap V_A\left(\beta \right)\cap \mathrm{Min}\left(A\right)=V_A(\alpha \vee \beta )\cap \mathrm{Min}\left(A\right)$.

Recall that, for any $\alpha \in \mathrm{Con}\left(A\right)$, ${\alpha }^{\perp }$ generates the annihilator of $\alpha$ in the commutator lattice $(\mathrm{Con}\left(A\right),{[·,·]}_A)$ as a principal ideal.

Proposition 7. 

(i) 

The flat topology on $\mathrm{Min}\left(A\right)$ is coarser than the Stone topology: ${\mathcal{F}}_{\mathrm{Min}}\left(A\right)\subseteq {\mathcal{S}}_{\mathrm{Min}}\left(A\right)$.

(ii) 

If A satisfies Condition 1.(vi), in particular if $\mathrm{Con}\left(A\right)$ is compact, then the two topologies coincide: ${\mathcal{F}}_{\mathrm{Min}}\left(A\right)={\mathcal{S}}_{\mathrm{Min}}\left(A\right)$, that is $\mathcal{M}in\left(A\right)=\mathcal{M}in{\left(A\right)}^{-1}$.

Proof. 

(i) By Proposition 6.(i), $V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)=D_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)\in {\mathcal{S}}_{\mathrm{Min}}\left(A\right)$, for any $\alpha \in \mathcal{K}\left(A\right)$.

• (ii) Again by Proposition 6.(i), for any $\alpha \in \mathcal{K}\left(A\right)$, $D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)=V_A\left({\alpha }^{\perp }\right)\cap \mathrm{Min}\left(A\right)$, which belongs to ${\mathcal{F}}_{\mathrm{Min}}\left(A\right)$ if ${\alpha }^{\perp }\in \mathcal{K}\left(A\right)$. □

Now let L be a bounded distributive lattice. Following [7], we denote, for any $I\in \mathrm{Id}\left(L\right)$ and $a\in L$, by $V_{\mathrm{Id},L}\left(I\right)$$={\mathrm{Spec}}_{\mathrm{Id}}\left(L\right)\cap {\left[I\right)}_{\mathrm{Id}\left(L\right)}$, $D_{\mathrm{Id},L}\left(I\right)$$={\mathrm{Spec}}_{\mathrm{Id}}\left(L\right)\setminus V_{\mathrm{Id},L}\left(I\right)$, $V_{\mathrm{Id},L}\left(a\right)$$=V_{\mathrm{Id},L}\left({\left(a\right]}_L\right)$ and $D_{\mathrm{Id},L}\left(a\right)$$=D_{\mathrm{Id},L}\left({\left(a\right]}_L\right)$.

Let us denote by ${\mathcal{S}}_{\mathrm{Spec},\mathrm{Id}}\left(L\right)$ the Stone topology on ${\mathrm{Spec}}_{\mathrm{Id}}\left(L\right)$ and by ${\mathcal{S}}_{\mathrm{Min},\mathrm{Id}}\left(L\right)$ the Stone topology on ${\mathrm{Min}}_{\mathrm{Id}}\left(L\right)$: ${\mathcal{S}}_{\mathrm{Spec},\mathrm{Id}}\left(L\right)=\left\{D_{\mathrm{Id},L}\left(I\right)\right|I\in \mathrm{Id}\left(L\right)\}$, with $\left\{D_{\mathrm{Id},L}\left(a\right)\right|a\in L\}$ as a basis; ${\mathcal{S}}_{\mathrm{Min},\mathrm{Id}}\left(L\right)=\{D_{\mathrm{Id},L}\left(I\right)\cap {\mathrm{Min}}_{\mathrm{Id}}\left(L\right)|I\in \mathrm{Id}\left(L\right)\}$, with $\{D_{\mathrm{Id},L}\left(a\right)\cap {\mathrm{Min}}_{\mathrm{Id}}\left(L\right)|$$a\in L\}$ as a basis.

And let ${\mathcal{F}}_{\mathrm{Min},\mathrm{Id}}\left(L\right)$ be the flat topology on ${\mathrm{Min}}_{\mathrm{Id}}\left(L\right)$, which has $\{V_{\mathrm{Id},L}\left(a\right)\cap {\mathrm{Min}}_{\mathrm{Id}}\left(L\right)|$$a\in L\}$ as a basis. Let $\mathcal{M}i{n}_{\mathrm{Id}}\left(L\right)$, respectively $\mathcal{M}i{n}_{\mathrm{Id}}{\left(L\right)}^{-1}$ be the minimal prime spectrum of ideals of L endowed with the Stone, respectively the flat topology: $\mathcal{M}i{n}_{\mathrm{Id}}\left(L\right)=({\mathrm{Min}}_{\mathrm{Id}}\left(L\right),{\mathcal{S}}_{\mathrm{Min},\mathrm{Id}}\left(L\right))$ and $\mathcal{M}i{n}_{\mathrm{Id}}{\left(L\right)}^{-1}=({\mathrm{Min}}_{\mathrm{Id}}\left(L\right),{\mathcal{F}}_{\mathrm{Min},\mathrm{Id}}\left(L\right))$.

Lemma 12. 

If A satisfies Condition 1.(iv), then:

(i) 

$\mathcal{M}in\left(A\right)$ is homeomorphic to $\mathcal{M}i{n}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)$;

(ii) 

$\mathcal{M}in{\left(A\right)}^{-1}$ is homeomorphic to $\mathcal{M}i{n}_{\mathrm{Id}}{\left(\mathcal{L}\left(A\right)\right)}^{-1}$.

Proof. 

Assume that A satisfies Condition 1.(iv), so that its reticulation can be constructed as: $\mathcal{L}\left(A\right)$$=\mathcal{K}\left(A\right)/{\equiv }_A$. As in [7], let us denote by $u:\mathrm{Spec}\left(A\right)\to {\mathrm{Spec}}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)$ and $v:{\mathrm{Spec}}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)\to \mathrm{Spec}\left(A\right)$ the mutually inverse homeomorphisms with respect to the Stone topologies mentioned in the proof of Proposition 5: $u\left(\varphi \right)={\varphi }^{*}$ for all $\varphi \in \mathrm{Spec}\left(A\right)$ and $v\left(P\right)=P_{*}$ for all $P\in {\mathrm{Spec}}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)$.

• (i) u and v obviously restrict to homeomorphisms between $\mathcal{M}in\left(A\right)$ and $\mathcal{M}i{n}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)$.
• (ii) Recall that the flat topology ${\mathcal{F}}_{\mathrm{Min}\left(A\right)}$ has $\{V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)|\alpha \in \mathcal{K}\left(A\right)\}$ as a basis, while the flat topology on ${\mathcal{F}}_{\mathrm{Min},\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)$ has $\{V_{\mathrm{Id},\mathcal{L}\left(A\right)}\left({\left(a\right]}_{\mathcal{L}\left(A\right)}\right)\cap {\mathrm{Min}}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)|a\in \mathcal{L}\left(A\right)\}=\{V_{\mathrm{Id},\mathcal{L}\left(A\right)}\left({(\alpha /{\equiv }_A]}_{\mathcal{L}\left(A\right)}\right)\cap {\mathrm{Min}}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)|\alpha \in \mathcal{K}\left(A\right)\}$ as a basis.

In the proof of [7] [Proposition 11] we have obtained that $u\left(V_A\left(\alpha \right)\right)=V_{\mathrm{Id},\mathcal{L}\left(A\right)}\left({\alpha }^{*}\right)$ for all $\alpha \in \mathrm{Con}\left(A\right)$. Note that, if $\alpha \in \mathcal{K}\left(A\right)$, then ${\alpha }^{*}={(\alpha /{\equiv }_A]}_{\mathcal{L}\left(A\right)}$, thus $u\left(V_A\left(\alpha \right)\right)=V_{\mathrm{Id},\mathcal{L}\left(A\right)}\left({(\alpha /{\equiv }_A]}_{\mathcal{L}\left(A\right)}\right)$, hence u is open with respect to the flat topologies on the minimal prime spectra.

Consequently, for all $\alpha \in \mathcal{K}\left(A\right)$, $v\left(V_{\mathrm{Id},\mathcal{L}\left(A\right)}\left({(\alpha /{\equiv }_A]}_{\mathcal{L}\left(A\right)}\right)\right)=v\left(u\left(V_A\left(\alpha \right)\right)\right)=V_A\left(\alpha \right)$, hence v is open with respect to the flat topologies on the minimal prime spectra.

Therefore u and v are mutually inverse homeomorphisms between $\mathcal{M}in{\left(A\right)}^{-1}$ and $\mathcal{M}i{n}_{\mathrm{Id}}{\left(\mathcal{L}\left(A\right)\right)}^{-1}$. □

Proposition 8. 

If A satisfies Condition 1.(iv), then $\mathcal{M}in{\left(A\right)}^{-1}$ is a compact ${\mathrm{T}}_1$ topological space.

Proof. 

Assume that A satisfies Condition 1.(iv), and let us consider the reticulation of A: $\mathcal{L}\left(A\right)=\mathcal{K}\left(A\right)/{\equiv }_A$.

By Hochster’s theorem [20] [Proposition 3.13], there exists a commutative unitary ring R such that the reticulation $\mathcal{L}\left(R\right)$ of R is lattice isomorphic to $\mathcal{L}\left(A\right)$. Recall that the commutator lattice of the ideals of R endowed with the multiplication of ideals as commutator operation is isomorphic to the commutator lattice of its congruences, $(\mathrm{Con}\left(R\right),{[·,·]}_R)$.

By Lemma 12.(ii), the minimal prime spectrum of R endowed with the flat topology, $\mathcal{M}in{\left(R\right)}^{-1}$, is homeomorphic to $\mathcal{M}i{n}_{\mathrm{Id}}{\left(\mathcal{L}\left(R\right)\right)}^{-1}$ and thus to $\mathcal{M}i{n}_{\mathrm{Id}}{\left(\mathcal{L}\left(A\right)\right)}^{-1}$, which in turn is homeomorphic to $\mathcal{M}in{\left(A\right)}^{-1}$, thus $\mathcal{M}in{\left(R\right)}^{-1}$ is homeomorphic to $\mathcal{M}in{\left(A\right)}^{-1}$.

By [29] [Theorem 3.1], $\mathcal{M}in{\left(R\right)}^{-1}$ is compact and ${\mathrm{T}}_1$. Therefore $\mathcal{M}in{\left(A\right)}^{-1}$ is compact and ${\mathrm{T}}_1$. □

Following [7], whenever A satisfies Condition 1.(iv), we will denote the lattice bounds of $\mathcal{L}\left(A\right)$ by $\mathbf{0}$ and $\mathbf{1}$, so $\mathbf{0}={\Delta }_A/{\equiv }_A$ and $\mathbf{1}={\nabla }_A/{\equiv }_A$.

Theorem 1. 

If A satisfies Condition 1.(iv), then the following are equivalent:

(i) 

$\mathcal{M}in\left(A\right)=\mathcal{M}in{\left(A\right)}^{-1}$;

(ii) 

$\mathcal{M}in\left(A\right)$ is compact;

(iii) 

for any $\alpha \in \mathcal{K}\left(A\right)$, there exists $\beta \in \mathcal{K}\left(A\right)$ such that $\beta \subseteq {\alpha }^{\perp }$ and ${(\alpha \vee \beta )}^{\perp }={\Delta }_A$.

Proof. 

Assume that A satisfies Condition 1.(iv). Then the reticulation $\mathcal{L}\left(A\right)$ of A is a bounded distributive lattice and thus a distributive lattice with zero, hence, according to [30] (Proposition $5.1$), the following are equivalent:

(a) 

$\mathcal{M}i{n}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)=\mathcal{M}i{n}_{\mathrm{Id}}{\left(\mathcal{L}\left(A\right)\right)}^{-1}$;

(b) 

$\mathcal{M}i{n}_{\mathrm{Id}}\left(\mathcal{L}\left(A\right)\right)$ is compact;

(c) 

for any $x\in \mathcal{L}\left(A\right)$, there exists $y\in \mathcal{L}\left(A\right)$ such that $x\wedge y=\mathbf{0}$ and ${\mathrm{Ann}}_{\mathcal{L}\left(A\right)}(x\vee y)=\left\{\mathbf{0}\right\}$.

By Lemma 12, (i) is equivalent to $\left(a\right)$. By Lemma 12.(i), (ii) is equivalent to $\left(b\right)$.

To prove that (iii) is equivalent to $\left(c\right)$, let $\alpha ,\beta \in \mathcal{K}\left(A\right)$, arbitrary, so that $\alpha /{\equiv }_A$ and $\beta /{\equiv }_A$ are arbitrary elements of $\mathcal{L}\left(A\right)$.

A is semiprime, that is ${\rho }_A\left({\Delta }_A\right)={\Delta }_A$, which is equivalent to ${\Delta }_A/{\equiv }_A=\left\{{\Delta }_A\right\}$ according to [25] (Remark $5.10$), hence, for any $\theta \in \mathrm{Con}\left(A\right)$, $\theta ={\Delta }_A$ if and only if $\theta \in {\Delta }_A/{\equiv }_A$ if and only if $\theta /{\equiv }_A={\Delta }_A/{\equiv }_A$, that is $\theta /{\equiv }_A=\mathbf{0}$.

Recall that $\beta \subseteq {\alpha }^{\perp }$ is equivalent to ${[\alpha ,\beta ]}_A={\Delta }_A$ and thus to ${[\alpha ,\beta ]}_A/{\equiv }_A=\mathbf{0}$ by the above, that is $\alpha /{\equiv }_A\wedge \beta /{\equiv }_A=\mathbf{0}$.

Furthermore, since A is semiprime, we have, for all $\theta \in \mathrm{Con}\left(A\right)$: by [25] (Lemma $5.18.\left(ii\right)$), ${\mathrm{Ann}}_{(\mathrm{Con}\left(A\right),{[·,·]}_A)}\left(\theta \right)={\mathrm{Ann}}_{\mathrm{Con}\left(A\right)}\left(\theta \right)$, and, by Lemma 10.(i), ${\mathrm{Ann}}_{\mathrm{Con}\left(A\right)}\left(\theta \right)=\left\{{\Delta }_A\right\}$ if and only if ${\mathrm{Ann}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}(\theta /{\equiv }_A)=\left\{\mathbf{0}\right\}$.

${(\alpha \vee \beta )}^{\perp }={\Delta }_A$ means that ${\mathrm{Ann}}_{(\mathrm{Con}\left(A\right),{[·,·]}_A)}(\alpha \vee \beta )=\left\{{\Delta }_A\right\}$, that is ${\mathrm{Ann}}_{\mathrm{Con}\left(A\right)}(\alpha \vee \beta )=\left\{{\Delta }_A\right\}$, which is equivalent to ${\mathrm{Ann}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}(\alpha /{\equiv }_A\vee \beta /{\equiv }_A)=\left\{\mathbf{0}\right\}$, which in turn is equivalent to ${\mathrm{Ann}}_{\mathcal{L}\left(A\right)}(\alpha /{\equiv }_A\vee \beta /{\equiv }_A)=\left\{\mathbf{0}\right\}$, because, if we denote by $\theta =\alpha \vee \beta$, so that $\theta \in \mathcal{K}\left(A\right)$ and $\theta /{\equiv }_A=\alpha /{\equiv }_A\vee \beta /{\equiv }_A\in \mathcal{L}\left(A\right)$, we have:

since $\mathcal{L}\left(A\right)$ is a bounded sublattice of $\mathrm{Con}\left(A\right)/{\equiv }_A$, ${\mathrm{Ann}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}(\theta /{\equiv }_A)=\left\{\mathbf{0}\right\}$ implies ${\mathrm{Ann}}_{\mathcal{L}\left(A\right)}(\theta /{\equiv }_A)={\mathrm{Ann}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}(\theta /{\equiv }_A)\cap \mathcal{L}\left(A\right)=\left\{\mathbf{0}\right\}$;

for the converse, recall that:

$$max{\mathrm{Ann}}_{\mathrm{Con}\left(A\right)}\left(\theta \right)=max{\mathrm{Ann}}_{(\mathrm{Con}\left(A\right),{[·,·]}_A)}\left(\theta \right)=\bigvee \{\gamma \in \mathcal{K}\left(A\right)|{[\theta ,\gamma ]}_A={\Delta }_A\}=$$

$$\bigvee \{\gamma \in \mathcal{K}\left(A\right)|{[\theta ,\gamma ]}_A/{\equiv }_A=\mathbf{0}\}=\bigvee \{\gamma \in \mathcal{K}\left(A\right)|\theta /{\equiv }_A\wedge \gamma /{\equiv }_A=\mathbf{0}\},$$

thus, if $\theta \in \mathcal{K}\left(A\right)$, so that $\theta /{\equiv }_A\in \mathcal{L}\left(A\right)$, then

$$max{\mathrm{Ann}}_{\mathrm{Con}\left(A\right)}\left(\theta \right)=\bigvee \{\gamma \in \mathcal{K}\left(A\right)|\gamma /{\equiv }_A\in {\mathrm{Ann}}_{\mathcal{L}\left(A\right)}(\theta /{\equiv }_A)\};$$

hence, if ${\mathrm{Ann}}_{\mathcal{L}\left(A\right)}(\theta /{\equiv }_A)=\left\{\mathbf{0}\right\}$, then

$$max{\mathrm{Ann}}_{\mathrm{Con}\left(A\right)}\left(\theta \right)=\bigvee \{\gamma \in \mathcal{K}\left(A\right)|\gamma /{\equiv }_A\in \left\{\mathbf{0}\right\}\}=\bigvee \{\gamma \in \mathcal{K}\left(A\right)|\gamma /{\equiv }_A=\mathbf{0}\}=$$

$$\bigvee \{\gamma \in \mathcal{K}\left(A\right)|\gamma ={\Delta }_A\}={\Delta }_A,$$

thus ${\mathrm{Ann}}_{\mathrm{Con}\left(A\right)}\left(\theta \right)=\left\{{\Delta }_A\right\}$, which is equivalent to ${\mathrm{Ann}}_{\mathrm{Con}\left(A\right)/{\equiv }_A}(\theta /{\equiv }_A)=\left\{\mathbf{0}\right\}$. □

Proposition 9. 

If ${\nabla }_A\in \mathcal{K}\left(A\right)$ and $\mathrm{Spec}\left(A\right)$ is unordered, then $\mathcal{M}in\left(A\right)$ is compact.

Proof. 

Assume that ${\nabla }_A\in \mathcal{K}\left(A\right)$ and $\mathrm{Spec}\left(A\right)$ is unordered, that is $\mathrm{Spec}\left(A\right)=\mathrm{Min}\left(A\right)$, and let ${\displaystyle \mathrm{Min}\left(A\right)=\bigcup _{i\in I}(D_A\left({\alpha }_i\right)\cap \mathrm{Min}\left(A\right))}$ for some nonempty family $\left\{{\alpha }_i\right|i\in I\}$ of congruences of A. Then ${\displaystyle \mathrm{Min}\left(A\right)=(\bigcup _{i\in I}{D}_A({\alpha }_i))\cap \mathrm{Min}\left(A\right)=D_A(\underset{i\in I}{\bigvee }{\alpha }_i)\cap \mathrm{Min}\left(A\right)}$, thus $V_A\left({\bigvee }_{i\in I}{\alpha }_i\right)\cap \mathrm{Min}\left(A\right)=\varnothing$. By Remark 3, this implies that ${\displaystyle \underset{i\in I}{\bigvee }{\alpha }_i={\nabla }_A\in \mathcal{K}\left(A\right)}$, so that ${\displaystyle {\nabla }_A=\underset{i\in F}{\bigvee }{\alpha }_i}$ for some finite subset F of I, hence ${\displaystyle \mathrm{Min}\left(A\right)=D_A(\underset{i\in F}{\bigvee }{\alpha }_i)\cap \mathrm{Min}\left(A\right)=(\bigcup _{i\in F}{D}_A({\alpha }_i))\cap \mathrm{Min}\left(A\right)=\bigcup _{i\in F}(D_A({\alpha }_i)\cap \mathrm{Min}\left(A\right))}$, therefore $\mathcal{M}in\left(A\right)$ is compact. □

Remark 7. 

Clearly, if $\mathrm{Con}\left(A\right)$ is finite, in particular if A is finite, then $\mathcal{M}in\left(A\right)$ is compact.

Of course, if $\mathrm{Con}\left(A\right)$ is finite, then $\mathrm{Con}\left(A\right)=\mathrm{Cp}\left(\mathrm{Con}\right(A\left)\right)=\mathcal{K}\left(A\right)$, thus ${\nabla }_A\in \mathcal{K}\left(A\right)$.

However, even if A is finite, its prime spectrum of congruences is not necessarily unordered. For instance, the five–element non–modular lattice ${\mathcal{N}}_5$ has $\mathrm{Con}\left({\mathcal{N}}_5\right)$ isomorphic to the ordinal sum ${\mathcal{L}}_2\oplus {\mathcal{L}}_2^2$ of the two–element chain with the four–element Boolean algebra, so, if we let $\mathrm{Con}\left({\mathcal{N}}_5\right)=\{{\Delta }_{{\mathcal{N}}_5},\alpha ,\beta ,\gamma ,{\nabla }_{{\mathcal{N}}_5}\}$, where $\mathrm{Max}\left({\mathcal{N}}_5\right)=\{\alpha ,\beta \}$ and $\gamma =\alpha \cap \beta$, then $\mathrm{Spec}\left({\mathcal{N}}_5\right)=\{{\Delta }_{{\mathcal{N}}_5},\alpha ,\beta \}=\left\{{\Delta }_{{\mathcal{N}}_5}\right\}\cup \mathrm{Max}\left({\mathcal{N}}_5\right)$, which is obviously not unordered.

Therefore the converse of the implication in Proposition 9 does not hold.

Theorem 2. 

If A satisfies (iv) and (vi) from Condition 1, in particular if the lattice $\mathrm{Con}\left(A\right)$ is compact, then $\mathcal{M}in\left(A\right)$ is a Hausdorff topological space consisting solely of clopen sets, thus the Stone topology ${\mathcal{S}}_{\mathrm{Min}\left(A\right)}$ is a complete Boolean sublattice of $\mathcal{P}\left(\mathrm{Min}\right(A\left)\right)$. If, moreover, $\mathrm{Spec}\left(A\right)$ is unordered, then $\mathcal{M}in\left(A\right)$ is also compact.

Proof. 

By Proposition 6.(i), the Stone topology ${\mathcal{S}}_{\mathrm{Min}}\left(A\right)$ on $\mathrm{Min}\left(A\right)$ consists entirely of clopen sets.

Let $\mu ,\nu$ be distinct minimal prime congruences of A. Then there exist $a,b\in A$ such that $(a,b)\in \mu \setminus \nu$, so that $C{g}_A(a,b)\subseteq \mu$ and $C{g}_A(a,b)⊈\nu$, so that $C{g}_A{(a,b)}^{\perp }⊈\mu$ by Proposition 5, so $\mu \in D_A\left(C{g}_A{(a,b)}^{\perp }\right)\cap \mathrm{Min}\left(A\right)$ and $\nu \in D_A\left(C{g}_A(a,b)\right)\cap \mathrm{Min}\left(A\right)$. $D_A\left(C{g}_A(a,b)\right)\cap \mathrm{Min}\left(A\right)\cap D_A\left(C{g}_A{(a,b)}^{\perp }\right)\cap \mathrm{Min}\left(A\right)=D_A\left(C{g}_A(a,b)\right)\cap D_A\left(C{g}_A{(a,b)}^{\perp }\right)\cap \mathrm{Min}\left(A\right)=D_A\left({[C{g}_A(a,b),C{g}_A{(a,b)}^{\perp }]}_A\right)\cap \mathrm{Min}\left(A\right)=D_A\left({\Delta }_A\right)\cap \mathrm{Min}\left(A\right)=\varnothing \cap \mathrm{Min}\left(A\right)=\varnothing$, therefore the topological space $(\mathrm{Min}\left(A\right),\{D_A\left(\theta \right)\cap \mathrm{Min}\left(A\right)|\theta \in \mathrm{Con}\left(A\right)\})$ is Hausdorff.

By Proposition 9, if $\mathrm{Spec}\left(A\right)$ is an antichain, then $\mathcal{M}in\left(A\right)$ is also compact. □

6. m–Extensions

Ⓗ

Throughout this section, we will assume that A is a subalgebra of B and that the algebras A and B are semiprime and they both satisfy Condition 1.(i).

In particular, the following results hold for extensions of semiprime algebras in congruence–modular varieties.

To avoid any danger of confusion, we will denote by ${\alpha }^{\perp A}=\alpha \to {\Delta }_A$ and $X^{\perp A}=C{g}_A{\left(X\right)}^{\perp A}$ for any $\alpha \in \mathrm{Con}\left(A\right)$ and any $X\subseteq A^2$ and by ${\beta }^{\perp B}=\beta \to {\Delta }_B$ and $Y^{\perp B}=C{g}_B{\left(Y\right)}^{\perp B}$ for any $\beta \in \mathrm{Con}\left(B\right)$ and any $Y\subseteq B^2$. See this notation for arbitrary subsets in Section 3.

We call the extension $A\subseteq B$:

• admissible if and only if the map $i_{A,B}:A\to B$ is admissible, that is if and only if $i_{A,B}^{*}\left(\varphi \right)=\varphi \cap {\nabla }_A\in \mathrm{Spec}\left(A\right)$ for all $\varphi \in \mathrm{Spec}\left(B\right)$;
• $\mathrm{Min}$–admissible or an m–extension if and only if $i_{A,B}^{*}\left(\mu \right)=\mu \cap {\nabla }_A\in \mathrm{Min}\left(A\right)$ for all $\mu \in \mathrm{Min}\left(B\right)$.

Lemma 13. 

Assume that the extension $A\subseteq B$ is admissible and let us consider the following statements:

(i) 

$A\subseteq B$ is an m–extension;

(ii) 

for any $\alpha \in \mathcal{K}\left(A\right)$ and any $\mu \in \mathrm{Min}\left(B\right)$, if $\alpha \subseteq \mu$, then ${\alpha }^{\perp A}⊈\mu$;

(iii) 

for any $\alpha \in \mathcal{K}\left(A\right)$ and any $\mu \in \mathrm{Min}\left(B\right)$, $\alpha \subseteq \mu$ if and only if ${\alpha }^{\perp A}⊈\mu$;

(iv) 

for any $\alpha \in \mathrm{Con}\left(A\right)$ and any $\mu \in \mathrm{Min}\left(B\right)$, if $\alpha \subseteq \mu$, then ${\alpha }^{\perp A}⊈\mu$;

(v) 

for any $\alpha \in \mathrm{Con}\left(A\right)$ and any $\mu \in \mathrm{Min}\left(B\right)$, $\alpha \subseteq \mu$ if and only if ${\alpha }^{\perp A}⊈\mu$.

If A satisfies Condition 1.(iv), then (i), (ii) and (iii) are equivalent.

If A satisfies Condition 1.(v), then (i), (iv) and (v) are equivalent.

Proof. 

For any $\alpha \in \mathrm{Con}\left(A\right)$ and $\mu \in \mathrm{Con}\left(B\right)$, we obviously have: $\alpha \subseteq \mu$ if and only if $\alpha \subseteq \mu \cap {\nabla }_A$, and ${\alpha }^{\perp A}⊈\mu$ if and only if ${\alpha }^{\perp A}⊈\mu \cap {\nabla }_A$.

Now assume that A satisfies (iv) or (v) from Condition 1, and let: $M=\mathcal{K}\left(A\right)$ if A satisfies (iv), and $M=\mathrm{Con}\left(A\right)$ if A satisfies (v).

Since the extension $A\subseteq B$ is admissible, $\mu \cap {\nabla }_A\in \mathrm{Spec}\left(A\right)$ for any $\mu \in \mathrm{Spec}\left(B\right)$. $A\subseteq B$ is an m–extension if and only if $\mu \cap {\nabla }_A\in \mathrm{Min}\left(A\right)$ for any $\mu \in \mathrm{Min}\left(B\right)$, hence, by Proposition 5: $A\subseteq B$ is an m–extension if and only if, for all $\mu \in \mathrm{Min}\left(B\right)$ and all $\alpha \in M$, the following equivalence holds: $\alpha \subseteq \mu \cap {\nabla }_A$ if and only if ${\alpha }^{\perp A}⊈\mu \cap {\nabla }_A$; by the above, this is equivalent to: $\alpha \subseteq \mu$ if and only if ${\alpha }^{\perp A}⊈\mu$. □

If $A\subseteq B$ is an m–extension, then the function $\Gamma$$=i_{A,B}^{*}{\mid }_{\mathrm{Min}\left(B\right)}:\mathrm{Min}\left(B\right)\to \mathrm{Min}\left(A\right)$, $\Gamma \left(\mu \right)=\mu \cap {\nabla }_A$ for all $\mu \in \mathrm{Min}\left(B\right)$, is well defined.

Proposition 10. 

If the extension $A\subseteq B$ is admissible, then, for every $\psi \in \mathrm{Spec}\left(A\right)$, there exists a $\mu \in \mathrm{Min}\left(B\right)$ such that $\mu \cap {\nabla }_A\subseteq \psi$.

Proof. 

Since $\psi \in \mathrm{Spec}\left(A\right)$, ${\nabla }_A\setminus \psi$ is an m–system in A, thus also in B, according to [18] [Lemma 4.18]. Hence there exists a $\nu \in \mathrm{Max}\{\gamma \in \mathrm{Con}\left(B\right)|\gamma \cap ({\nabla }_A\setminus \psi )=\varnothing \}$, so that $\nu \in \mathrm{Spec}\left(B\right)$ by Lemma 3, and thus there exists a $\mu \in \mathrm{Min}\left(B\right)$ with $\mu \subseteq \nu$, so that $\mu \cap ({\nabla }_A\setminus \psi )\subseteq \nu \cap ({\nabla }_A\setminus \psi )=\varnothing$ and thus $(\mu \cap {\nabla }_A)\setminus \psi =\varnothing$, so $\mu \cap {\nabla }_A\subseteq \psi$. □

Corollary 3. 

• If the extension $A\subseteq B$ is admissible, then, for every $\psi \in \mathrm{Min}\left(A\right)$, there exists a $\mu \in \mathrm{Min}\left(B\right)$ such that $\mu \cap {\nabla }_A=\psi$.

• If $A\subseteq B$ is an admissible m–extension, then $\Gamma :\mathrm{Min}\left(B\right)\to \mathrm{Min}\left(A\right)$ is surjective.

Lemma 14. 

If $A\subseteq B$ is an admissible m–extension, then, for any $\theta ,\zeta \in \mathrm{Con}\left(A\right)$: ${[\theta ,\zeta ]}_A={\Delta }_A$ if and only if ${[C{g}_B\left(\theta \right),C{g}_B\left(\zeta \right)]}_B={\Delta }_B$.

Proof. 

Since A and B are semiprime, we have ${\Delta }_A={\rho }_A\left({\Delta }_A\right)=\bigcap \mathrm{Min}\left(A\right)$ and ${\Delta }_B={\rho }_B\left({\Delta }_B\right)=\bigcap \mathrm{Min}\left(B\right)$, therefore: ${[C{g}_B\left(\theta \right),C{g}_B\left(\zeta \right)]}_B={\Delta }_B$ if and only if ${[C{g}_B\left(\theta \right),C{g}_B\left(\zeta \right)]}_B\subseteq \nu$ for all $\nu \in \mathrm{Min}\left(B\right)$ if and only if, for all $\nu \in \mathrm{Min}\left(B\right)$, $C{g}_B\left(\theta \right)\subseteq \nu$ or $C{g}_B\left(\zeta \right)\subseteq \nu$ if and only if, for all $\nu \in \mathrm{Min}\left(B\right)$, $\theta \subseteq \nu$ or $\zeta \subseteq \nu$ if and only if, for all $\nu \in \mathrm{Min}\left(B\right)$, $\theta \subseteq \nu \cap {\nabla }_A$ or $\zeta \subseteq \nu \cap {\nabla }_A$; by Corollary 3, the latter is equivalent to: for all $\mu \in \mathrm{Min}\left(A\right)$, $\theta \subseteq \mu$ or $\zeta \subseteq \mu$, which in turn is equivalent to the fact that ${[\theta ,\zeta ]}_A\subseteq \mu$ for all $\mu \in \mathrm{Min}\left(A\right)$, that is ${[\theta ,\zeta ]}_A={\Delta }_A$. □

Recall from [1] that, if $A\subseteq B$ is an extension of algebras from a congruence–modular variety, then, for all $\alpha ,\beta \in \mathrm{Con}\left(B\right)$, ${[\alpha \cap {\nabla }_A,\beta \cap {\nabla }_A]}_A\subseteq {[\alpha ,\beta ]}_B\cap {\nabla }_A$. So, in this case, the right-to-left implication in Lemma 14 holds without admissibility or $\mathrm{Min}$–admissibility:

Remark 8. 

If the extension $A\subseteq B$ satisfies ${[\alpha \cap {\nabla }_A,\beta \cap {\nabla }_A]}_A\subseteq {[\alpha ,\beta ]}_B\cap {\nabla }_A$ for all $\alpha ,\beta \in \mathrm{Con}\left(B\right)$, in particular if the variety $\mathcal{V}$ is congruence–modular, then, for any $\theta ,\zeta \in \mathrm{Con}\left(A\right)$: ${[C{g}_B\left(\theta \right),C{g}_B\left(\zeta \right)]}_B={\Delta }_B$ implies ${[\theta ,\zeta ]}_A={\Delta }_A$.

Indeed, since $\theta \subseteq C{g}_B\left(\theta \right)\cap {\nabla }_A$ for all $\theta \in \mathrm{Con}\left(A\right)$ and the commutator is increasing in both arguments, it follows that, for all $\theta ,\zeta \in \mathrm{Con}\left(A\right)$:

${[\theta ,\zeta ]}_A\subseteq {[C{g}_B\left(\theta \right)\cap {\nabla }_A,C{g}_B\left(\zeta \right)\cap {\nabla }_A]}_A\subseteq {[C{g}_B\left(\theta \right),C{g}_B\left(\zeta \right)]}_B\cap {\nabla }_A$.

Thus, if ${[C{g}_B\left(\theta \right),C{g}_B\left(\zeta \right)]}_B={\Delta }_B$, then ${[\theta ,\zeta ]}_A\subseteq {\Delta }_B\cap {\nabla }_A={\Delta }_A$, so ${[\theta ,\zeta ]}_A={\Delta }_A$.

Proposition 11. 

If $A\subseteq B$ is an admissible m–extension, then, for any $\theta \in \mathrm{Con}\left(A\right)$:

(i) 

${\theta }^{\perp A}={\theta }^{\perp B}\cap {\nabla }_A$ and $C{g}_B\left({\theta }^{\perp A}\right)\subseteq {\theta }^{\perp B}$;

(ii) 

if, furthermore, ${\theta }^{\perp B}\in \left\{C{g}_B\left(\alpha \right)\right|\alpha \in \mathrm{Con}\left(A\right)\}$, then ${\theta }^{\perp B}=C{g}_B\left({\theta }^{\perp A}\right)$.

Proof. 

(i) By Lemma 14 we have, for any $u,v\in A$: $(u,v)\in {\theta }^{\perp A}$ if and only if ${[C{g}_A(u,v),\theta ]}_A$$={\Delta }_A$ if and only if ${[C{g}_B\left(C{g}_A(u,v)\right),C{g}_B\left(\theta \right)]}_B={[C{g}_B(u,v),C{g}_B\left(\theta \right)]}_B={\Delta }_B$ if and only if $(u,v)\in {\theta }^{\perp B}$ if and only if $(u,v)\in {\theta }^{\perp B}\cap {\nabla }_A$. Therefore ${\theta }^{\perp A}={\theta }^{\perp B}\cap {\nabla }_A$, hence $C{g}_B\left({\theta }^{\perp A}\right)=C{g}_B({\theta }^{\perp B}\cap {\nabla }_A)\subseteq {\theta }^{\perp B}$.

• (ii) If ${\theta }^{\perp B}$ is generated by a congruence of A, then, by Lemma 14 and the fact that the map $\alpha \mapsto C{g}_B\left(\alpha \right)$ from $\mathrm{Con}\left(A\right)$ to $\mathrm{Con}\left(B\right)$ is order–preserving, we have: ${\theta }^{\perp B}=C{g}_B{\left(\theta \right)}^{\perp B}=max\{\beta \in \mathrm{Con}\left(B\right)|{[\beta ,C{g}_B\left(\theta \right)]}_B={\Delta }_B\}=max\left\{C{g}_B\left(\alpha \right)\right|\alpha \in \mathrm{Con}\left(A\right),{[C{g}_B\left(\alpha \right),C{g}_B\left(\theta \right)]}_B={\Delta }_B\}=max\left\{C{g}_B\left(\alpha \right)\right|\alpha \in \mathrm{Con}\left(A\right),{[\alpha ,\theta ]}_A={\Delta }_A\}=C{g}_B(max\{\alpha \in \mathrm{Con}\left(A\right)|{[\alpha ,\theta ]}_A={\Delta }_A\})=C{g}_B\left({\theta }^{\perp A}\right)$. □

Corollary 4. 

If $A\subseteq B$ is an admissible m–extension, then, for any $\theta ,\zeta \in \mathrm{Con}\left(A\right)$:

(i) 

${\theta }^{\perp B}={\zeta }^{\perp B}$ implies ${\theta }^{\perp A}={\zeta }^{\perp A}$;

(ii) 

if, furthermore, ${\theta }^{\perp B},{\zeta }^{\perp B}\in \left\{C{g}_B\left(\alpha \right)\right|\alpha \in \mathrm{Con}\left(A\right)\}$, then: ${\theta }^{\perp A}={\zeta }^{\perp A}$ if and only if ${\theta }^{\perp B}={\zeta }^{\perp B}$.

Corollary 5. 

If $A\subseteq B$ is an admissible m–extension such that A satisfies (v) and B satisfies (iv) or (v) from Condition 1, then, for any $\psi \in \mathrm{Spec}\left(B\right)$, we have: $\psi \in \mathrm{Min}\left(B\right)$ if and only if $\psi \cap {\nabla }_A\in \mathrm{Min}\left(A\right)$.

Proof. 

We have the direct implication by the definition of an m–extension.

Now assume that $\psi \cap {\nabla }_A\in \mathrm{Min}\left(A\right)$ and let $\beta \in \mathrm{Con}\left(B\right)$, arbitrary. Then, by Proposition 5 and Proposition 11.(i): $\beta \subseteq \psi$ implies $\beta \cap {\nabla }_A\subseteq \psi \cap {\nabla }_A$, which is equivalent to ${(\beta \cap {\nabla }_A)}^{\perp A}⊈\psi \cap {\nabla }_A$, hence ${(\beta \cap {\nabla }_A)}^{\perp A}⊈\psi$, thus ${\beta }^{\perp B}\supseteq C{g}_B\left({(\beta \cap {\nabla }_A)}^{\perp A}\right)\supseteq {(\beta \cap {\nabla }_A)}^{\perp A}⊈\psi$, so ${\beta }^{\perp B}⊈\psi$. Therefore, again by Proposition 5, $\psi \in \mathrm{Min}\left(B\right)$. □

Remark 9. 

Note from the proof of Corollary 5 that, if $A\subseteq B$ is an admissible m–extension such that A satisfies (v) and B satisfies (iv) from Condition 1, then B satisfies the equivalence of all statements (i), (ii), (iii), (iv) and (v) in Proposition 5.

By extending the terminology for ring extensions from [5], we call $A\subseteq B$:

• a rigid, a quasirigid, respectively a weak rigid extension if and only if, for any $\beta \in \mathrm{PCon}\left(B\right)$, there exists an $\alpha \in \mathrm{PCon}\left(A\right)$, an $\alpha \in \mathcal{K}\left(A\right)$, respectively an $\alpha \in \mathrm{Con}\left(A\right)$ such that ${\alpha }^{\perp B}={\beta }^{\perp B}$;
• an r–extension, a quasi r–extension, respectively a weak r–extension if and only if, for any $\mu \in \mathrm{Min}\left(B\right)$ and any $\beta \in \mathrm{PCon}\left(B\right)$ such that $\beta ⊈\mu$, there exists an $\alpha \in \mathrm{PCon}\left(A\right)$, an $\alpha \in \mathcal{K}\left(A\right)$, respectively an $\alpha \in \mathrm{Con}\left(A\right)$ such that $\alpha ⊈\mu$ and ${\beta }^{\perp B}\subseteq {\alpha }^{\perp B}$;
• an $r^{*}$–extension, a quasi $r^{*}$–extension, respectively a weak $r^{*}$–extension if and only if, for any $\mu \in \mathrm{Min}\left(B\right)$ and any $\beta \in \mathrm{PCon}\left(B\right)$ such that $\beta \subseteq \mu$, there exists an $\alpha \in \mathrm{PCon}\left(A\right)$, an $\alpha \in \mathcal{K}\left(A\right)$, respectively an $\alpha \in \mathrm{Con}\left(A\right)$ such that $\alpha \subseteq \mu$ and ${\alpha }^{\perp B}\subseteq {\beta }^{\perp B}$.

Remark 10. 

If $A\subseteq B$ is admissible or an m–extension, then, since any $\alpha \in \mathrm{Con}\left(A\right)$ and $\mu \in \mathrm{Con}\left(B\right)$ satisfy the equivalence $\alpha \subseteq \mu$ if and only if $\alpha \subseteq \mu \cap {\nabla }_A$, thus also the equivalence $\alpha ⊈\mu$ if and only if $\alpha ⊈\mu \cap {\nabla }_A$, it follows that $A\subseteq B$ is:

• an r–extension, a quasi r–extension, respectively a weak r–extension if and only if, for any $\beta \in \mathrm{PCon}\left(B\right)$, $\{\mu \cap {\nabla }_A|\mu \in D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right)\}\subseteq \bigcup \left\{D_A\left(\alpha \right)\right|\alpha \in M,$${\beta }^{\perp B}\subseteq {\alpha }^{\perp B}\}=D_A(\bigvee \{\alpha \in M|{\beta }^{\perp B}\subseteq {\alpha }^{\perp B}\})$, where M is equal to $\mathrm{PCon}\left(A\right)$, $\mathcal{K}\left(A\right)$, respectively $\mathrm{Con}\left(A\right)$;
• an $r^{*}$–extension, a quasi $r^{*}$–extension, respectively a weak $r^{*}$–extension if and only if, for any $\beta \in \mathrm{PCon}\left(B\right)$, $\{\mu \cap {\nabla }_A|\mu \in V_B\left(\beta \right)\cap \mathrm{Min}\left(B\right)\}\subseteq \bigcup \left\{V_A\left(\alpha \right)\right|\alpha \in M,$${\alpha }^{\perp B}\subseteq {\beta }^{\perp B}\}$, where M is equal to $\mathrm{PCon}\left(A\right)$, $\mathcal{K}\left(A\right)$, respectively $\mathrm{Con}\left(A\right)$;

• thus, if $A\subseteq B$ is an m–extension, then $A\subseteq B$ is:

• an r–extension, a quasi r–extension, respectively a weak r–extension if and only if, for any $\beta \in \mathrm{PCon}\left(B\right)$, $\Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))\subseteq \bigcup \left\{D_A\left(\alpha \right)\right|\alpha \in M,{\beta }^{\perp B}\subseteq {\alpha }^{\perp B}\}=D_A(\bigvee \{\alpha \in M|{\beta }^{\perp B}\subseteq {\alpha }^{\perp B}\})$, where M is equal to $\mathrm{PCon}\left(A\right)$, $\mathcal{K}\left(A\right)$, respectively $\mathrm{Con}\left(A\right)$;
• an $r^{*}$–extension, a quasi $r^{*}$–extension, respectively a weak $r^{*}$–extension if and only if, for any $\beta \in \mathrm{PCon}\left(B\right)$, $\Gamma (V_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))\subseteq \bigcup \left\{V_A\left(\alpha \right)\right|\alpha \in M,{\alpha }^{\perp B}\subseteq {\beta }^{\perp B}\}$, where M is equal to $\mathrm{PCon}\left(A\right)$, $\mathcal{K}\left(A\right)$, respectively $\mathrm{Con}\left(A\right)$.

Remark 11. 

Note from Lemma 1 that, for any set I and any $\{a_i,b_i|i\in I\}\subseteq A$, $C{g}_B\left(C{g}_A\left(\left\{(a_i,b_i)\right|i\in I\}\right)\right)=C{g}_B\left(\left\{(a_i,b_i)\right|i\in I\}\right)$, hence, for any $\alpha \in \mathrm{PCon}\left(A\right)$ and any $\beta \in \mathcal{K}\left(A\right)$, it follows that $C{g}_B\left(\alpha \right)\in \mathrm{PCon}\left(B\right)$ and $C{g}_B\left(\beta \right)\in \mathcal{K}\left(B\right)$.

Proposition 12. 

If $A\subseteq B$ is an m–extension, then:

(i) 

if B satisfies Condition 1.(v) and $A\subseteq B$ is a weak rigid extension, then it is both a weak r–extension and a weak $r^{*}$–extension;

(ii) 

if B satisfies (iv) or (v) from Condition 1 and $A\subseteq B$ is a quasirigid extension, then it is both a quasi r–extension and a quasi $r^{*}$–extension;

(iii) 

if B satisfies (iv) or (v) from Condition 1 and $A\subseteq B$ is a rigid extension, then it is both an r–extension and an $r^{*}$–extension.

Proof. 

(ii) Assume that $A\subseteq B$ is a quasirigid extension and let $\mu \in \mathrm{Min}\left(B\right)$ and $\beta \in \mathrm{PCon}\left(B\right)$, so that $C{g}_B{\left(\alpha \right)}^{\perp B}={\beta }^{\perp B}$ for some $\alpha \in \mathcal{K}\left(A\right)$, hence, according to Proposition 5 and Remark 11:

$\beta ⊈\mu$ implies $C{g}_B{\left(\alpha \right)}^{\perp B}={\beta }^{\perp B}\subseteq \mu$, thus $C{g}_B\left(\alpha \right)⊈\mu$, hence $\alpha ⊈\mu$;

$\beta \subseteq \mu$ implies $C{g}_B{\left(\alpha \right)}^{\perp B}={\beta }^{\perp B}⊈\mu$, thus $\alpha \subseteq C{g}_B\left(\alpha \right)\subseteq \mu$.

• (i) and (iii) Analogously. □

Proposition 13. 

If $A\subseteq B$ is an m–extension, then:

(i) 

Γ is continuous with respect to the Stone topologies and the inverse topologies;

(ii) 

if A satisfies (iv) or (v) or B satisfies (iv) or (v) from Condition 1, then $\Gamma :\mathcal{M}in\left(B\right)\to \mathcal{M}in{\left(A\right)}^{-1}$ is continuous;

(iii) 

if A satisfies (vi), along with one of (iv) and (v) from Condition 1, or B satisfies (vi), along with one of (iv) and (v) from Condition 1, then $\Gamma :\mathcal{M}in{\left(B\right)}^{-1}\to \mathcal{M}in\left(A\right)$ is continuous.

Proof. 

Let $\alpha \in \mathrm{Con}\left(A\right)$, so that $D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$ is an arbitrary open set in $\mathcal{M}in\left(A\right)$, $D_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$ is an open set in $\mathcal{M}in\left(B\right)$ and, if $\alpha \in \mathcal{K}\left(A\right)$, so that $C{g}_B\left(\alpha \right)\in \mathcal{K}\left(B\right)$, then $V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$ is an arbitrary basic open set in $\mathcal{M}in{\left(A\right)}^{-1}$ and $V_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$ is a basic open set in $\mathcal{M}in{\left(B\right)}^{-1}$.

Since $A\subseteq B$ is an m–extension, we have, for all $\nu \in \mathrm{Min}\left(B\right)$: $\nu \cap {\nabla }_A\in \mathrm{Min}\left(A\right)$, thus:

$\nu \in {\Gamma }^{-1}(V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))$ if and only if $\nu \cap {\nabla }_A\in V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)=\left[\alpha \right)\cap \mathrm{Min}\left(A\right)$ if and only if $\nu \cap {\nabla }_A\in \left[\alpha \right)$ if and only if $\alpha \subseteq \nu \cap {\nabla }_A$ if and only if $\alpha \subseteq \nu$ if and only if $C{g}_B\left(\alpha \right)\subseteq \nu$ if and only if $\nu \in V_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$; hence ${\Gamma }^{-1}(V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))=V_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$;

similarly, $\nu \in {\Gamma }^{-1}(D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))$ if and only if $\alpha ⊈\nu \cap {\nabla }_A$ if and only if $\alpha ⊈\nu$ if and only if $C{g}_B\left(\alpha \right)⊈\nu$ if and only if $\nu \in D_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$; hence ${\Gamma }^{-1}(D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))=D_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$.

• (i) Hence $\Gamma :\mathcal{M}in\left(B\right)\to \mathcal{M}in\left(A\right)$ and $\Gamma :\mathcal{M}in{\left(B\right)}^{-1}\to \mathcal{M}in{\left(A\right)}^{-1}$ are continuous.
• (ii) Assume that $\alpha \in \mathcal{K}\left(A\right)$.

If A satisfies (iv) or (v) from Condition 1, then, by Proposition 5: $\alpha \subseteq \nu \cap {\nabla }_A$ if and only if ${\alpha }^{\perp A}⊈\nu \cap {\nabla }_A$ if and only if ${\alpha }^{\perp A}⊈\nu$ if and only if $C{g}_B\left({\alpha }^{\perp A}\right)⊈\nu$ if and only if $\nu \in D_B\left(C{g}_B\left({\alpha }^{\perp A}\right)\right)\cap \mathrm{Min}\left(B\right)$; hence ${\Gamma }^{-1}(V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))=D_B\left(C{g}_B\left({\alpha }^{\perp A}\right)\right)\cap \mathrm{Min}\left(B\right)$.

If B satisfies (iv) or (v) from Condition 1, then, by Proposition 5: $C{g}_B\left(\alpha \right)\subseteq \nu$ if and only if $C{g}_B{\left(\alpha \right)}^{\perp B}⊈\nu$ if and only if $\nu \in D_B\left(C{g}_B{\left(\alpha \right)}^{\perp B}\right)\cap \mathrm{Min}\left(B\right)$; hence ${\Gamma }^{-1}(V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))=D_B\left(C{g}_B{\left(\alpha \right)}^{\perp B}\right)\cap \mathrm{Min}\left(B\right)$.

Thus, in either of these cases, $\Gamma :\mathcal{M}in\left(B\right)\to \mathcal{M}in{\left(A\right)}^{-1}$ is continuous.

• (iii) Analogous to the proof of (ii) or simply by applying (i), (ii) and Proposition 7.(ii). □

Proposition 14. 

If $A\subseteq B$ is an admissible quasi r–extension and B satisfies (iv) or (v) from Condition 1, then: $A\subseteq B$ is an m–extension and Γ is a bijection.

Proof. 

Assume that $A\subseteq B$ is an admissible quasi r–extension and B satisfies (iv) or (v) from Condition 1.

Assume by absurdum that there exists a $\nu \in \mathrm{Min}\left(B\right)$ with $\nu \cap {\nabla }_A\notin \mathrm{Min}\left(A\right)$, so that $\nu \cap {\nabla }_A\in \mathrm{Spec}\left(A\right)\setminus \mathrm{Min}\left(A\right)$ since $\mathrm{Min}\left(B\right)\subseteq \mathrm{Spec}\left(B\right)$ and $A\subseteq B$ is admissible, hence there exists $\mu \in \mathrm{Min}\left(A\right)$ such that $\mu ⊊\nu \cap {\nabla }_A$.

Since $A\subseteq B$ is admissible, by Corollary 3 it follows that $\mu =\epsilon \cap {\nabla }_A$ for some $\epsilon \in \mathrm{Min}\left(B\right)$. Thus $\epsilon \cap {\nabla }_A=\mu ⊊\nu \cap {\nabla }_A$, therefore $\epsilon$ and $\nu$ are distinct minimal prime congruences of B, hence they are incomparable, thus $\epsilon \setminus \nu \ne \varnothing$, so that $(x,y)\in \epsilon \setminus \nu$ for some $x,y\in B$.

Then $C{g}_B(x,y)⊈\nu$, so that, since $A\subseteq B$ is a quasi r–extension, there exists an $\alpha \in \mathcal{K}\left(A\right)$ such that $\alpha ⊈\nu$ and $C{g}_B{(x,y)}^{\perp B}\subseteq {\alpha }^{\perp B}=C{g}_B{\left(\alpha \right)}^{\perp B}$. Then $C{g}_B\left(\alpha \right)\in \mathcal{K}\left(B\right)$ and $\alpha ⊈\nu$, thus $C{g}_B\left(\alpha \right)⊈\nu$, hence $C{g}_B{\left(\alpha \right)}^{\perp B}\subseteq \nu$ by Proposition 5 and the fact that B satisfies (iv) or (v) from Condition 1.

Also, $C{g}_B(x,y)\subseteq \epsilon$, thus, again by Proposition 5 and the fact that B satisfies (iv) or (v) from Condition 1, $C{g}_B{\left(\alpha \right)}^{\perp B}\supseteq C{g}_B{(x,y)}^{\perp B}⊈\epsilon$, hence $\alpha \subseteq C{g}_B\left(\alpha \right)\subseteq \epsilon$, thus $\alpha \subseteq \epsilon \cap {\nabla }_A=\mu \subset \nu \cap {\nabla }_A\subseteq \nu$, hence $C{g}_B\left(\alpha \right)\subseteq \nu$, so that $C{g}_B{\left(\alpha \right)}^{\perp B}⊈\nu$, which contradicts the above.

Therefore $A\subseteq B$ is an m–extension, hence $\Gamma$ is surjective by Corollary 3 and the admissibility of $A\subseteq B$.

Now let $\varphi ,\psi \in \mathrm{Min}\left(B\right)$ such that $\Gamma \left(\varphi \right)=\Gamma \left(\psi \right)$, that is $\varphi \cap {\nabla }_A=\psi \cap {\nabla }_A$, and assume by absurdum that $\varphi \ne \psi$, so that $\varphi \setminus \psi \ne \varnothing$, that is $(u,v)\in \varphi \setminus \psi$ for some $u,v\in B$, which thus satisfy $C{g}_B(u,v)\subseteq \varphi$ and $C{g}_B(u,v)⊈\psi$, hence $C{g}_B{(u,v)}^{\perp B}⊈\varphi$ and $C{g}_B{(u,v)}^{\perp B}\subseteq \psi$ by Proposition 5.

As above, it follows that there exists a $\gamma \in \mathcal{K}\left(A\right)$ such that $\gamma ⊈\psi$ and $C{g}_B{(u,v)}^{\perp B}\subseteq {\gamma }^{\perp B}$, so that $\gamma ⊈\psi \cap {\nabla }_A$ and $C{g}_B{\left(\gamma \right)}^{\perp B}={\gamma }^{\perp B}⊈\varphi$, thus $\gamma \subseteq C{g}_B\left(\gamma \right)\subseteq \varphi$ by Proposition 5, so $\gamma \subseteq \varphi \cap {\nabla }_A$. We have obtained $\gamma \subseteq \varphi \cap {\nabla }_A=\psi \cap {\nabla }_A⊉\gamma$; a contradiction. Therefore $\Gamma$ is injective. □

Proposition 15. 

If $A\subseteq B$ is an admissible quasi $r^{*}$–extension and B satisfies (iv) or (v) from Condition 1, then: $A\subseteq B$ is an m–extension and Γ is a bijection.

Proof. 

Similar to the proof of Proposition 14. □

Theorem 3. 

If $A\subseteq B$ is an admissible extension such that B satisfies (iv) or (v) from Condition 1, then the following are equivalent:

(i) 

$A\subseteq B$ is an r–extension;

(ii) 

$A\subseteq B$ is a quasi r–extension;

(iii) 

$\Gamma :\mathcal{M}in\left(B\right)\to \mathcal{M}in\left(A\right)$ is a homeomorphism.

Proof. 

(i)⇒(ii): Trivial.

• (ii)⇒(iii): If $A\subseteq B$ is an admissible quasi r–extension such that B satisfies (iv) or (v) from Condition 1, then, by Propositions 14 and 13, it follows that $A\subseteq B$ is an m–extension and $\Gamma :\mathcal{M}in\left(B\right)\to \mathcal{M}in\left(A\right)$ is a continuous bijection.

Let $\beta \in \mathrm{PCon}\left(B\right)$, so that $\Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))=\{\psi \cap {\nabla }_A|\psi \in D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right)\}=\{\psi \cap {\nabla }_A|\psi \in \mathrm{Min}\left(B\right),\beta ⊈\psi \}$. By Remark 10 and the fact that $A\subseteq B$ is a quasi r–extension and an m–extension, $\Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))\subseteq D_A(\bigvee \{\alpha \in \mathcal{K}\left(A\right)|{\beta }^{\perp B}\subseteq {\alpha }^{\perp B}\})\cap \mathrm{Min}\left(A\right)$.

Now let $\varphi \in D_A(\bigvee \{\alpha \in \mathcal{K}\left(A\right)|{\beta }^{\perp B}\subseteq {\alpha }^{\perp B}\})\cap \mathrm{Min}\left(A\right)$, that is $\varphi \in$$(\{\bigcup \left\{D_A\left(\alpha \right)\right|\alpha \in \mathcal{K}\left(A\right),{\beta }^{\perp B}\subseteq {\alpha }^{\perp B}\})\cap \mathrm{Min}\left(A\right)=\bigcup \{D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)|\alpha \in \mathcal{K}\left(A\right),{\beta }^{\perp B}\subseteq {\alpha }^{\perp B}\}\}$, so $\varphi \in D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$ for some $\alpha \in \mathcal{K}\left(A\right)$ such that ${\beta }^{\perp B}\subseteq {\alpha }^{\perp B}=C{g}_B{\left(\alpha \right)}^{\perp B}$. By Corollary 3, there exists $\psi \in \mathrm{Min}\left(B\right)$ such that $\Gamma \left(\psi \right)=\psi \cap {\nabla }_A=\varphi ⊉\alpha$, thus $\psi ⊉C{g}_B\left(\alpha \right)$, hence $\psi \supseteq C{g}_B{\left(\alpha \right)}^{\perp B}={\beta }^{\perp B}$ and thus $\psi ⊉\beta$ by Proposition 5 and the fact that B satisfies (iv) or (v) from Condition 1, so that $\psi \in D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right)$, thus $\varphi \in \Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))$.

Hence we also have the converse inclusion: $D_A(\bigvee \{\alpha \in \mathcal{K}\left(A\right)|{\beta }^{\perp B}\subseteq {\alpha }^{\perp B}\})\cap \mathrm{Min}\left(A\right)\subseteq \Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))$, so $\Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))=D_A(\bigvee \{\alpha \in \mathcal{K}\left(A\right)|{\beta }^{\perp B}\subseteq {\alpha }^{\perp B}\})\cap \mathrm{Min}\left(A\right)$. Therefore $\Gamma :\mathcal{M}in\left(B\right)\to \mathcal{M}in\left(A\right)$ is also open, thus it is a homeomorphism.

• (iii)⇒(i): Assume that $\Gamma$ is a homeomorphism with respect to the Stone topologies, so $A\subseteq B$ is an m–extension and $\Gamma$ maps basic open sets of $\mathcal{M}in\left(B\right)$ to basic open sets of $\mathcal{M}in\left(A\right)$.

Let $\beta \in \mathrm{PCon}\left(B\right)$, so that $D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right)$ is a basic open set of $\mathcal{M}in\left(B\right)$. By the above, there exists $\alpha \in \mathrm{PCon}\left(A\right)$ such that $\{\mu \cap {\nabla }_A|\mu \in D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right)\}=\Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))=D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$. Hence, for all $\mu \in D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right)$, that is $\mu \in \mathrm{Min}\left(B\right)$ such that $\beta ⊈\mu$, we have $\mu \cap {\nabla }_A\in D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$, so $\alpha ⊈\mu \cap {\nabla }_A$, that is $\alpha ⊈\mu$. Since B satisfies (iv) or (v) from Condition 1, we have, by Proposition 5: ${\beta }^{\perp B}\subseteq \mu$ and ${\alpha }^{\perp B}=C{g}_B{\left(\alpha \right)}^{\perp B}\subseteq \mu$.

For any $\gamma \in \mathcal{K}\left(B\right)$ such that $\gamma \subseteq {\beta }^{\perp B}$, all $\epsilon \in \mathrm{Min}\left(B\right)$ satisfy the following:

• if $C{g}_B\left(\alpha \right)\subseteq \epsilon$, then ${[\gamma ,C{g}_B\left(\alpha \right)]}_B\subseteq \epsilon$;
• if $C{g}_B\left(\alpha \right)⊈\epsilon$, then $\alpha ⊈\epsilon \cap {\nabla }_A$, that is $\Gamma \left(\epsilon \right)=\epsilon \cap {\nabla }_A\in D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)=\Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))$, hence $\epsilon \in D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right)$ since $\Gamma$ is a bijection, thus $\beta ⊈\epsilon$, so ${\beta }^{\perp B}\subseteq \epsilon$, again by Proposition 5, thus $\gamma \subseteq \epsilon$ by the above, hence ${[\gamma ,C{g}_B\left(\alpha \right)]}_B\subseteq \epsilon$;

hence ${[\gamma ,C{g}_B\left(\alpha \right)]}_B\subseteq \bigcap \mathrm{Min}\left(B\right)={\Delta }_B$ since B is semiprime, thus ${[\gamma ,C{g}_B\left(\alpha \right)]}_B={\Delta }_B$, that is $\gamma \subseteq C{g}_B{\left(\alpha \right)}^{\perp B}={\alpha }^{\perp B}$.

Therefore ${\beta }^{\perp B}=\bigvee \{\gamma \in \mathcal{K}\left(B\right)|\gamma \subseteq {\beta }^{\perp B}\}\subseteq {\alpha }^{\perp B}$. Hence $A\subseteq B$ is an r–extension. □

Proposition 16. 

If $A\subseteq B$ is an admissible r–extension such that B satisfies (iv) or (v) from Condition 1, then the following are equivalent:

(i) 

$A\subseteq B$ is a rigid extension;

(ii) 

Γ maps basic open sets of $\mathcal{M}in\left(B\right)$ to basic open sets of $\mathcal{M}in\left(A\right)$.

Proof. 

By Proposition 14, $A\subseteq B$ is an m–extension and $\Gamma$ is bijective.

• (i)⇒(ii): Let $\beta \in \mathrm{PCon}\left(B\right)$, so that there exists $\alpha \in \mathrm{PCon}\left(A\right)$ with ${\beta }^{\perp B}={\alpha }^{\perp B}=C{g}_B{\left(\alpha \right)}^{\perp B}$ since $A\subseteq B$ is rigid. By Proposition 5, for any $\nu \in \mathrm{Min}\left(B\right)$, the following equivalences hold: $\beta ⊈\nu$ if and only if ${\beta }^{\perp B}\subseteq \nu$ if and only if $C{g}_B{\left(\alpha \right)}^{\perp B}\subseteq \nu$ if and only if $C{g}_B\left(\alpha \right)⊈\nu$ if and only if $\alpha ⊈\nu$ if and only if $\alpha ⊈\nu \cap {\nabla }_A$, therefore, since $\Gamma$ is bijective, we have $\Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))=\{\nu \cap {\nabla }_A|\nu \in \mathrm{Min}\left(B\right),\beta ⊈\nu \}=\{\mu \in \mathrm{Min}\left(A\right)|\alpha ⊈\mu \}=D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$.
• (ii)⇒(i): Let $\beta \in \mathrm{PCon}\left(B\right)$. By the hypothesis of this implication, $\Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))=D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$ for some $\alpha \in \mathrm{PCon}\left(A\right)$, thus $\{\nu \cap {\nabla }_A|\nu \in \mathrm{Min}\left(A\right),\beta ⊈\nu \}=$$\left\{\mu \right|\mu \in \mathrm{Min}\left(A\right),\alpha ⊈\mu \}$. By Proposition 5, it follows that any $\nu \in \mathrm{Min}\left(B\right)$ satisfies: ${\beta }^{\perp B}\subseteq \nu$ if and only if $\beta ⊈\nu$ if and only if $\alpha ⊈\nu \cap {\nabla }_A$ if and only if $\alpha ⊈\nu$ if and only if $C{g}_B\left(\alpha \right)⊈\nu$ if and only if $C{g}_B{\left(\alpha \right)}^{\perp B}\subseteq \nu$.

As in the proof of the implication (iii)⇒(i) from Theorem 3, it follows that ${\beta }^{\perp B}=\bigcap V_B\left({\beta }^{\perp B}\right)=\bigcap V_B\left(C{g}_B{\left(\alpha \right)}^{\perp B}\right)=C{g}_B{\left(\alpha \right)}^{\perp B}={\alpha }^{\perp B}$. Hence the extension $A\subseteq B$ is rigid. □

Proposition 17. 

If $A\subseteq B$ is an admissible $r^{*}$–extension such that B satisfies (iv) or (v) from Condition 1, then the following are equivalent:

(i) 

$A\subseteq B$ is a quasirigid extension;

(ii) 

Γ maps basic open sets of $\mathcal{M}in{\left(B\right)}^{-1}$ to basic open sets of $\mathcal{M}in{\left(A\right)}^{-1}$.

Proof. 

By Proposition 15, $A\subseteq B$ is an m–extension and $\Gamma$ is bijective.

• (i)⇒(ii): Let $\beta \in \mathcal{K}\left(B\right)$, so that $\beta ={\bigvee }_{i=1}^n{\beta }_i$ for some $n\in {\mathbb{N}}^{*}$ and some ${\beta }_1,\dots ,{\beta }_n\in \mathrm{PCon}\left(B\right)$. By the hypothesis of this implication, for each $i\in \overline{1,n}$, there exists ${\alpha }_i\in \mathcal{K}\left(A\right)$ such that ${\beta }_i^{\perp B}={\alpha }_i^{\perp B}={\left(C{g}_B\left({\alpha }_i\right)\right)}^{\perp B}$.

Analogously to the proof of (i)⇒(ii) from Proposition 16, it follows that $\Gamma (V_B\left({\beta }_i\right)\cap \mathrm{Min}\left(B\right))=V_A\left({\alpha }_i\right)\cap \mathrm{Min}\left(A\right)$ for all $i\in \overline{1,n}$, hence $\Gamma (V_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))=\Gamma ({\bigcap }_{i=1}^n{V}_B\left({\beta }_i\right)\cap \mathrm{Min}\left(B\right))={\bigcap }_{i=1}^n\Gamma (V_B\left({\beta }_i\right)\cap \mathrm{Min}\left(B\right))={\bigcap }_{i=1}^n{V}_A\left({\alpha }_i\right)\cap \mathrm{Min}\left(A\right)=$$V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$, where $\alpha ={\bigvee }_{i=1}^n{\alpha }_i\in \mathcal{K}\left(A\right)$.

• (ii)⇒(i): Similar to the proof of (ii)⇒(i) in Proposition 16. □

Corollary 6. 

If $A\subseteq B$ is an admissible r–extension and $r^{*}$–extension such that B satisfies (iv) or (v) from Condition 1, then the following are equivalent:

• $A\subseteq B$ is a quasirigid extension;
• $A\subseteq B$ is a rigid extension.

Proof. 

By Proposition 15, $A\subseteq B$ is an m–extension and $\Gamma$ is bijective.

Clearly, if the extension $A\subseteq B$ is rigid, then it is quasirigid.

Now assume that $A\subseteq B$ is quasirigid. Then, by Proposition 17, $\Gamma$ maps basic open sets of $\mathcal{M}in{\left(B\right)}^{-1}$ to basic open sets of $\mathcal{M}in{\left(A\right)}^{-1}$. Since $\Gamma$ is bijective, it follows that $\Gamma$ maps basic open sets of $\mathcal{M}in\left(B\right)$ to basic open sets of $\mathcal{M}in\left(A\right)$, hence $A\subseteq B$ is rigid according to Proposition 16. □

Theorem 4. 

If $A\subseteq B$ is an admissible extension such that B satisfies (iv) or (v) from Condition 1, then the following are equivalent:

(i) 

$A\subseteq B$ is a quasi $r^{*}$–extension;

(ii) 

$\Gamma :\mathcal{M}in{\left(B\right)}^{-1}\to \mathcal{M}in{\left(A\right)}^{-1}$ is a homeomorphism.

Proof. 

By adapting the proof of Theorem 3. □

We say that A satisfies the annihilator condition (AC for short) if for all $\alpha ,\beta \in \mathrm{PCon}\left(A\right)$ there exists $\gamma \in \mathrm{PCon}\left(A\right)$ such that ${\gamma }^{\perp A}={\alpha }^{\perp A}\cap {\beta }^{\perp A}$.

Remark 12. 

By Proposition 6.(ii), A satisfies AC if and only if the family $\{V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)|\alpha \in \mathrm{PCon}\left(A\right)\}$ is closed under finite intersections. Thus, for any semiprime algebra A that satisfies AC, the family $\{V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)|\alpha \in \mathrm{PCon}\left(A\right)\}$ is a basis for the inverse topology ${\mathcal{F}}_{\mathrm{Min}\left(A\right)}$ of $\mathrm{Min}\left(A\right)$.

Proposition 18. 

Let $A\subseteq B$ be an admissible extension such that B satisfies (iv) or (v) from Condition 1.

(i) 

If A satisfies AC, then: $A\subseteq B$ is a quasi $r^{*}$–extension if and only if $A\subseteq B$ is an $r^{*}$–extension.

(ii) 

If $A\subseteq B$ is an r–extension and both A and B satisfy AC, then: $A\subseteq B$ is a quasirigid extension if and only if $A\subseteq B$ is a rigid extension.

Proof. 

(i) Assume that $A\subseteq B$ is a quasi $r^{*}$–extension. Then, by Theorem 4, $\Gamma :\mathcal{M}in{\left(B\right)}^{-1}\to \mathcal{M}in{\left(A\right)}^{-1}$ is a homeomorphism.

Let $\nu \in \mathrm{Min}\left(B\right)$ and $\beta \in \mathrm{PCon}\left(B\right)$ such that $\beta \subseteq \nu$, so that $\nu \in V_B\left(\beta \right)\cap \mathrm{Min}\left(B\right)$, thus, by the above, $\nu \cap {\nabla }_A=\Gamma \left(\nu \right)\in \Gamma (V_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))$, which, according to Remark 12, equals $V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$ for some $\alpha \in \mathrm{PCon}\left(A\right)$. As in the proof of (ii)⇒(i) from Proposition 16, it follows that ${\alpha }^{\perp B}\subseteq {\beta }^{\perp B}$. Therefore $A\subseteq B$ is an $r^{*}$–extension.

The converse implication is trivial.

• (ii) By Propositions 16 and 17 and the clear fact that, in this case, condition (ii) from Proposition 16 is equivalent to (ii) from Proposition 17. □

Let us denote, for any subset $X\subseteq A^2$, by $S\left(X\right)=\{\psi \in \mathrm{Min}\left(B\right)|X⊈\psi \cap {\nabla }_A\}$, thus $S\left(X\right)=\{\psi \in \mathrm{Min}\left(B\right)|\psi \cap {\nabla }_A\in D_A\left(C{g}_A\left(X\right)\right)\}$.

Proposition 19. 

Let $A\subseteq B$ be an admissible extension such that B is hyperarchimedean and satisfies (iv) or (v) from Condition 1. If the extension $A\subseteq B$ has the property that, for any $\theta ,\zeta \in \mathrm{Con}\left(A\right)$, ${\theta }^{\perp A}={\zeta }^{\perp A}$ implies ${\theta }^{\perp B}={\zeta }^{\perp B}$, in particular if $A\subseteq B$ is an m–extension such that $\left\{{\beta }^{\perp B}\right|\beta \in \mathrm{Con}\left(B\right)\}\subseteq \left\{C{g}_B\left(\alpha \right)\right|\alpha \in \mathrm{Con}\left(A\right)\}$, then the following are equivalent:

(i) 

$\mathcal{M}in\left(A\right)$ is a compact space;

(ii) 

$A\subseteq B$ is an m–extension;

(iii) 

for any $\alpha \in \mathrm{PCon}\left(A\right)$ there exists $\beta \in \mathcal{K}\left(A\right)$ such that $S\left(\beta \right)=\mathrm{Spec}\left(B\right)\setminus S\left(\alpha \right)$;

(iv) 

for any $\alpha \in \mathrm{PCon}\left(A\right)$ there exists $\beta \in \mathcal{K}\left(A\right)$ such that $\beta \subseteq {\alpha }^{\perp A}$ and ${(\alpha \vee \beta )}^{\perp A}={\Delta }_A$.

Proof. 

First, note from Corollary 4.(ii) that, if $A\subseteq B$ is an admissible m–extension such that ${\beta }^{\perp B}$ is generated by a congruence of A for every $\beta \in \mathrm{Con}\left(B\right)$, then this extension satisfies the implication: ${\theta }^{\perp A}={\zeta }^{\perp A}$ implies ${\theta }^{\perp B}={\zeta }^{\perp B}$.

Now assume that $A\subseteq B$ is an admissible extension such that, for any $\theta ,\zeta \in \mathrm{Con}\left(A\right)$, ${\theta }^{\perp A}={\zeta }^{\perp A}$ implies ${\theta }^{\perp B}={\zeta }^{\perp B}$, and that B is hyperarchimedean and satisfies (iv) or (v) from Condition 1.

• (i)⇔(iv) By Theorem 1.
• (iv)⇒(iii): Assume that $\alpha \in \mathrm{PCon}\left(A\right)$. By the hypothesis (iv), there exists $\beta \in \mathcal{K}\left(A\right)$ such that $\beta \subseteq {\alpha }^{\perp A}$ and ${(\alpha \vee \beta )}^{\perp A}={\Delta }_A$.

To show that $\mathrm{Spec}\left(B\right)\setminus S\left(\alpha \right)=S\left(\beta \right)$, let $\psi \in \mathrm{Spec}\left(B\right)\setminus S\left(\alpha \right)$, hence $\alpha \subseteq \psi \cap {\nabla }_A$. Since $A\subseteq B$ satisfies the implication in the enunciation, by Proposition 5 it follows that these implications hold: if ${(\alpha \vee \beta )}^{\perp B}={\Delta }_B$, i.e., ${\left(C{g}_B(\alpha \vee \beta )\right)}^{\perp B}={\Delta }_B$, then ${\left(C{g}_B(\alpha \vee \beta )\right)}^{\perp B}\subseteq \psi$, thus $C{g}_B(\alpha \vee \beta )⊈\psi$, so $\alpha \vee \beta ⊈\psi$, thus $\beta ⊈\psi \cap {\nabla }_A$, so $\psi \in S\left(\beta \right)$, which proves the inclusion $\mathrm{Spec}\left(B\right)\setminus S\left(\alpha \right)\subseteq S\left(\beta \right)$.

Conversely, let $\psi \in S\left(\beta \right)$, so that $\beta ⊈\psi \cap {\nabla }_A$. But ${[\alpha ,\beta ]}_A={\Delta }_A\subseteq \psi \cap {\nabla }_A\in \mathrm{Spec}\left(A\right)$, hence $\alpha \subseteq \psi \cap {\nabla }_A$, so $\psi \in \mathrm{Spec}\left(B\right)\setminus S\left(\alpha \right)$. Therefore $S\left(\beta \right)\subseteq \mathrm{Spec}\left(B\right)\setminus S\left(\alpha \right)$.

• (iii)⇒(ii): We have to prove that $\psi \cap {\nabla }_A\in \mathrm{Min}\left(A\right)$ for any $\psi \in \mathrm{Min}\left(B\right)$. Assume by absurdum that there exists $\psi \in \mathrm{Min}\left(B\right)$ such that $\psi \cap {\nabla }_A\notin \mathrm{Min}\left(A\right)$. But $\psi \cap {\nabla }_A\in \mathrm{Spec}\left(A\right)$ since $A\subseteq B$ is admissible, thus $\varphi ⊊\psi \cap {\nabla }_A$ for some $\varphi \in \mathrm{Min}\left(A\right)$. By Corollary 3, there exists $\epsilon \in \mathrm{Min}\left(B\right)$ such that $\varphi =\epsilon \cap {\nabla }_A$. So $\epsilon \cap {\nabla }_A⊊\psi \cap {\nabla }_A$, hence there exists $(a,b)\in (\psi \cap {\nabla }_A)\setminus (\epsilon \cap {\nabla }_A)$, so that, if we denote by $\alpha =C{g}_A(a,b)\in \mathrm{PCon}\left(A\right)$, then $\alpha ⊈\epsilon \cap {\nabla }_A$ and $\alpha \subseteq \psi \cap {\nabla }_A$, therefore $\epsilon \in S\left(\alpha \right)$ and $\psi \notin S\left(\alpha \right)$. Since $S\left(\beta \right)=\mathrm{Spec}\left(B\right)\setminus S\left(\alpha \right)$, it follows that $\epsilon \notin S\left(\beta \right)$ and $\psi \in S\left(\beta \right)$, hence $\beta \subseteq \epsilon \cap {\nabla }_A\subseteq \psi \cap {\nabla }_A$ and $\beta ⊈\psi \cap {\nabla }_A$. We have obtained a contradiction, thus $A\subseteq B$ is an m–extension.
• (ii)⇒(i): Assume that $A\subseteq B$ is an m–extension, so the map $\Gamma$ is surjective and continuous with respect to the Stone topologies by Corollary 3 and Proposition 13.(i).

By [7] [Theorem 8], it follows that the reticulation $\mathcal{L}\left(B\right)$ of the hyperarchimedean algebra B is a Boolean algebra. Since $\mathcal{M}in\left(B\right)$ and $\mathcal{M}i{n}_{Id}\left(\mathcal{L}\left(B\right)\right)$ are homeomorphic, it follows that $\mathcal{M}in\left(B\right)$ is a Boolean space, hence $\mathcal{M}in\left(B\right)$ is a compact space, therefore $\mathcal{M}in\left(A\right)$ is also a compact space. □

Remark 13. 

Let A be a reduced (that is semiprime) commutative ring and $Q\left(A\right)$ the complete ring of A (see [3]). In this case, $Q\left(A\right)$ is a regular ring [3], i.e., a hyperarchimedean ring. In accordance with [6] [Proposition 7.2.(2)], $A\subseteq Q\left(A\right)$ is a Baer extension of rings, so one can apply our Proposition 6.18. Then we obtain [31] [Theorem 4.3] as a particular case. It also results that, if A is a reduced ring, then: $\mathcal{M}in\left(A\right)$ is compact if and only if $A\subseteq Q\left(A\right)$ is an m–extension.

Theorem 5. 

If $A\subseteq B$ is an admissible m–extension such that $\left\{{\beta }^{\perp B}\right|\beta \in \mathrm{Con}\left(B\right)\}\subseteq \left\{C{g}_B\left(\alpha \right)\right|\alpha \in \mathrm{Con}\left(A\right)\}$, both A and B satisfy (v) and (vi) from Condition 1 and Γ is injective, then $\Gamma :\mathcal{M}in\left(B\right)\to \mathcal{M}in\left(A\right)$ is a homeomorphism and $A\subseteq B$ is a weak rigid extension.

Proof. 

We will be using Proposition 5, Proposition 11.(ii) and Lemma 6.(ii).

Let $\alpha \in \mathrm{Con}\left(A\right)$ and $\mu \in \mathrm{Min}\left(A\right)$, so that $\mu =\nu \cap {\nabla }_A$ for some $\nu \in \mathrm{Min}\left(B\right)$ by Corollary 3. Then the fact that $\mu \in D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$, that is $\alpha ⊈\mu$, is equivalent to ${\alpha }^{\perp A}\subseteq \mu$, which implies ${\alpha }^{\perp B}=C{g}_B{\left(\alpha \right)}^{\perp B}=C{g}_B\left({\alpha }^{\perp A}\right)\subseteq C{g}_B\left(\mu \right)\subseteq \nu$, thus $C{g}_B\left(\alpha \right)⊈\nu$, that is $\nu \in D_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$. On the other hand, $\nu \in D_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$ means that $C{g}_B\left(\alpha \right)⊈\nu$, so that ${\alpha }^{\perp B}=C{g}_B{\left(\alpha \right)}^{\perp B}\subseteq \nu$, hence ${\alpha }^{\perp A}={\alpha }^{\perp B}\cap {\nabla }_A\subseteq \nu \cap {\nabla }_A=\mu$, thus $\alpha ⊈\mu$, that is $\mu \in D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$.

Hence, $\nu \in D_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$ if and only if $\Gamma \left(\nu \right)=\mu \in D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$ if and only if $\nu \in {\Gamma }^{-1}(D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))$, therefore ${\Gamma }^{-1}(D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))=D_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$. Thus $\Gamma$ is continuous and, by Proposition 6.(i), for all $\theta \in \mathrm{Con}\left(A\right)$, ${\Gamma }^{-1}(V_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))={\Gamma }^{-1}(D_A\left({\alpha }^{\perp A}\right)\cap \mathrm{Min}\left(A\right))=D_B\left(C{g}_B\left({\alpha }^{\perp A}\right)\right)\cap \mathrm{Min}\left(B\right)=V_B\left(C{g}_B{\left({\alpha }^{\perp A}\right)}^{\perp B}\right)\cap \mathrm{Min}\left(B\right)=V_B\left({\alpha }^{\perp B\perp B}\right)\cap \mathrm{Min}\left(B\right)=V_B\left(C{g}_B{\left(\alpha \right)}^{\perp B\perp B}\right)\cap \mathrm{Min}\left(B\right)=V_B\left(C{g}_B\left(\alpha \right)\right)\cap \mathrm{Min}\left(B\right)$.

Hence, if $\Gamma$ is injective and thus bijective according to Corollary 3, then $\Gamma$ is a homeomorphism, in particular $\Gamma$ is open, thus, for every $\beta \in \mathrm{Con}\left(B\right)$, there is $\alpha \in \mathrm{Con}\left(A\right)$ such that $\Gamma (D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right))=D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right)$, hence, by the above, along with Proposition 6.(i) and Proposition 11.(ii),

$V_B\left({\beta }^{\perp B}\right)\cap \mathrm{Min}\left(B\right)=D_B\left(\beta \right)\cap \mathrm{Min}\left(B\right)={\Gamma }^{-1}(D_A\left(\alpha \right)\cap \mathrm{Min}\left(A\right))$

$\phantom{V_B\left({\beta }^{\perp B}\right)\cap \mathrm{Min}\left(B\right)}={\Gamma }^{-1}(D_A\left({\alpha }^{\perp A\perp A}\right)\cap \mathrm{Min}\left(A\right))=D_B\left(C{g}_B\left({\alpha }^{\perp A\perp A}\right)\right)\cap \mathrm{Min}\left(B\right)$

$\phantom{V_B\left({\beta }^{\perp B}\right)\cap \mathrm{Min}\left(B\right)}=D_B\left(C{g}_B{\left({\alpha }^{\perp A}\right)}^{\perp B}\right)\cap \mathrm{Min}\left(B\right)=V_B\left(C{g}_B\left({\alpha }^{\perp A}\right)\right)\cap \mathrm{Min}\left(B\right)$

$\phantom{V_B\left({\beta }^{\perp B}\right)\cap \mathrm{Min}\left(B\right)}=V_B\left(C{g}_B{\left(\alpha \right)}^{\perp B}\right)\cap \mathrm{Min}\left(B\right)$.

• By Lemma 11, ${\beta }^{\perp B}=\bigcap (V_B\left({\beta }^{\perp B}\right)\cap \mathrm{Min}\left(B\right))=\bigcap (V_B\left(C{g}_B{\left(\alpha \right)}^{\perp B}\right)\cap \mathrm{Min}\left(B\right))=C{g}_B{\left(\alpha \right)}^{\perp B}$$={\alpha }^{\perp B}$, thus $A\subseteq B$ is weak rigid. □

Theorem 6. 

If $A\subseteq B$ is an admissible weak r–extension such that $\left\{{\beta }^{\perp B}\right|\beta \in \mathrm{Con}\left(B\right)\}\subseteq \left\{C{g}_B\left(\alpha \right)\right|\alpha \in \mathrm{Con}\left(A\right)\}$ and both A and B satisfy (v) and (vi) from Condition 1, then $A\subseteq B$ is an m–extension and an r–extension and $\Gamma :\mathcal{M}in\left(B\right)\to \mathcal{M}in\left(A\right)$ is a homeomorphism.

Proof. 

Assume that $A\subseteq B$ is an admissible weak r–extension, so that $\Gamma$ is surjective by Corollary 3. We will apply Proposition 5.

Assume by absurdum that there exists a $\mu \in \mathrm{Min}\left(B\right)$ such that $\mu \cap {\nabla }_A\notin \mathrm{Min}\left(A\right)$, so that $\varphi ⊊\mu \cap {\nabla }_A$ for some $\varphi \in \mathrm{Min}\left(A\right)$. By the above, $\varphi =\epsilon \cap {\nabla }_A$ for some $\epsilon \in \mathrm{Min}\left(B\right)$, so that $\epsilon \cap {\nabla }_A⊊\mu \cap {\nabla }_A$, thus $\epsilon \ne \mu$, hence $\epsilon \setminus \mu \ne \varnothing$, that is $(x,y)\in \epsilon \setminus \mu$ for some $x,y\in B$.

We have $C{g}_B(x,y)⊈\mu$, hence there exists an $\alpha \in \mathrm{Con}\left(A\right)$ such that $\alpha ⊈\mu$ and $C{g}_B{(x,y)}^{\perp B}\subseteq {\alpha }^{\perp B}=C{g}_B{\left(\alpha \right)}^{\perp B}$ since $A\subseteq B$ is a weak r–extension. But $\alpha ⊈\mu$ implies $C{g}_B\left(\alpha \right)⊈\mu$, hence $C{g}_B{\left(\alpha \right)}^{\perp B}\subseteq \mu$. Since $C{g}_B(x,y)\subseteq \epsilon$, we have $C{g}_B{\left(\alpha \right)}^{\perp B}\supseteq C{g}_B{(x,y)}^{\perp B}⊈\epsilon$, thus $\alpha \subseteq C{g}_B\left(\alpha \right)\subseteq \epsilon$, hence $\alpha \subseteq \epsilon \cap {\nabla }_A=\varphi \subset \mu \cap {\nabla }_A\subseteq \mu$, hence $C{g}_B\left(\alpha \right)\subseteq \mu$, thus $C{g}_B{\left(\alpha \right)}^{\perp B}⊈\mu$, contradicting the above.

Therefore $A\subseteq B$ is an m–extension.

Now let $\mu ,\nu \in \mathrm{Min}\left(B\right)$ such that $\mu \cap {\nabla }_A=\nu \cap {\nabla }_A$. Assume by absurdum that $\mu \ne \nu$, so that $\mu \setminus \nu \ne \varnothing$, that is $(u,v)\in \mu \setminus \nu$ for some $u,v\in B$. Then $C{g}_B(u,v)⊈\nu$, thus, since $A\subseteq B$ is a weak r–extension, there exists a $\xi \in \mathrm{Con}\left(A\right)$ such that $\xi ⊈\nu$ and $C{g}_B{(u,v)}^{\perp B}\subseteq {\xi }^{\perp B}$. Since $C{g}_B(u,v)\subseteq \mu$, $C{g}_B{(u,v)}^{\perp B}⊈\mu$, hence, by Proposition 11.(ii), $C{g}_B\left({\xi }^{\perp A}\right)=C{g}_B{\left(\xi \right)}^{\perp B}={\xi }^{\perp B}⊈\mu$, thus $\xi \subseteq {\xi }^{\perp A\perp A}\subseteq C{g}_B\left({\xi }^{\perp A\perp A}\right)=C{g}_B{\left({\xi }^{\perp A}\right)}^{\perp B}\subseteq \mu$, hence $\xi \subseteq \mu \cap {\nabla }_A=\nu \cap {\nabla }_A\subseteq \nu$, contradicting the above. Therefore $\Gamma$ is injective and thus a homeomorphism by Theorem 5.

Finally, let $\mu \in \mathrm{Min}\left(B\right)$ and $\beta \in \mathrm{PCon}\left(B\right)$ such that $\beta ⊈\mu$, so that, since $A\subseteq B$ is a weak r–extension, there exists a $\gamma \in \mathrm{Con}\left(A\right)$ such that $\gamma ⊈\mu$ and ${\beta }^{\perp B}\subseteq {\gamma }^{\perp B}$. Then $(w,z)\in \gamma \setminus \mu$ for some $w,z\in A$, so that $C{g}_A(w,z)⊈\mu$ and $C{g}_A(w,z)\subseteq \gamma$, so that $C{g}_B\left(C{g}_A(w,z)\right)\subseteq C{g}_B\left(\gamma \right)$ and thus ${\beta }^{\perp B}\subseteq {\gamma }^{\perp B}=C{g}_B{\left(\gamma \right)}^{\perp B}\subseteq C{g}_B{\left(C{g}_A(w,z)\right)}^{\perp B}=C{g}_A{(w,z)}^{\perp B}$. Therefore $A\subseteq B$ is an r–extension. □

Theorem 7. 

If $A\subseteq B$ is an admissible weak $r^{*}$–extension such that $\left\{{\beta }^{\perp B}\right|\beta \in \mathrm{Con}\left(B\right)\}\subseteq \left\{C{g}_B\left(\alpha \right)\right|\alpha \in \mathrm{Con}\left(A\right)\}$ and both A and B satisfy (v) and (vi) from Condition 1, then $A\subseteq B$ is an m–extension and an $r^{*}$–extension and Γ is a homeomorphism with respect to the Stone topologies.

Proof. 

By adapting the proof of Theorem 6. □

Corollary 7. 

If $\left\{{\beta }^{\perp B}\right|\beta \in \mathrm{Con}\left(B\right)\}\subseteq \left\{C{g}_B\left(\alpha \right)\right|\alpha \in \mathrm{Con}\left(A\right)\}$, both A and B satisfy (v) and (vi) from Condition 1 and $A\subseteq B$ is admissible and either a weak r–extension or a weak $r^{*}$–extension, then $A\subseteq B$ is a weak rigid extension.

Corollary 8. 

If $A\subseteq B$ is admissible, $\left\{{\beta }^{\perp B}\right|\beta \in \mathrm{Con}\left(B\right)\}\subseteq \left\{C{g}_B\left(\alpha \right)\right|\alpha \in \mathrm{Con}\left(A\right)\}$ and both A and B satisfy (v) and (vi) from Condition 1, then the following are equivalent:

• $A\subseteq B$ is a weak rigid extension;
• $A\subseteq B$ is a weak r–extension;
• $A\subseteq B$ is a weak $r^{*}$–extension;
• $A\subseteq B$ is an r–extension;
• $A\subseteq B$ is an $r^{*}$–extension.

7. Conclusions

Commutator theory, developed for different kinds of varieties, with congruence–modular varieties as an important case [1], is a powerful tool for extending properties of concrete algebraic structures to universal algebras. Our paper illustrates this by generalizing some results from [5] on classes of ring extensions to universal algebras whose commutators satisfy certain conditions; this includes members of semidegenerate congruence–modular varieties, but also applies to more general cases. We start by studying some algebraic and topological properties of the minimal prime spectrum of an algebra whose commutator operation is commutative and distributive with respect to arbitrary joins, then use these results on the minimal prime spectrum to obtain characterizations for some classes of extensions of such algebras. Our results can be applied to many kinds of structures, for instance to generalize those results in [5] to non–commutative rings or semirings.

In future work we will study the preservation of these properties of extensions of universal algebras by the reticulation, look for classes of universal algebras to which other results from [5] can be generalized and study abstractions of our results, using commutator lattices and lattice morphisms preserving the commutator operations, in the manner from [25].