On the Equational Theory of Lattice-Based Algebras for Layered Graphs

Abstract

Layered algebras are introduced and used to express layered graphs. Layered graphs are considered to be a highly effective abstract tool to manage the difficulty in conceptualizing and reasoning regarding complex systems related to coding in email exchange and access control in security. In the present paper, we study the varieties of several classes of lattice-based layer algebras and show that all these varieties have decidable equational theory via a finite model property.

1. Introduction

A binary operation · on an algebra $\mathrm{A}$ with a carrier preordered set $(A,\le )$ is said to be residuated if there exist binary operations ∖ and / on A such that for any $a,b,c\in A$

$$a·b\le z\iff b\le a\setminus c\iff a\le c/b.$$

One refers to ∖ and / by the right and the left residual of ·, respectively.

A residuated Boolean algebra (r-algebra) is an algebraic structure $\mathrm{A}=(A,\wedge ,\vee ,\neg ,·,\setminus ,/)$ such that $(A,\wedge ,\vee ,\neg )$ is a Boolean algebra and · is a binary operation on A, where ∖ and / are its right and left residuals, respectively. The R-algebras were first introduced by Jósson and Tsinakis [1] as generalizations for relation algebras due to A.Tarski.

The algebraic structures studied under the name of layered algebras by Collinson et al. [2] are the same as those discussed here. The non-commutative and non-associative operation · in groupoids is used to capture the feature of layering in layered graphs. Let $G_1,G_2$ be directed graphs. The composition of graphs is defined $G_1·G_2$ if $V\left(G_1\right)\cap V\left(G_2\right)=⌀$, $G_1⇝G_2$, and $G_2⇝̸G_1$, where $V\left(G_i\right)$ is the set of all vectors of graph $G_i$ ($i\in \{1,2\}$) and $G_1⇝G_2$ means that $G_2$ is reachable from $G_1$. A graph G is said to be layered if there exist $G_1,G_2$ such that $G_i$ ($i\in \{1,2\}$) is a subgraph of G and $G=G_1·G_2$. The Boolean component and residuated duals of · are used to express properties in the layered graphs.

In the literature [2,3,4], the layered graph is considered to be a highly effective abstract tool to manage the difficulty in conceptualizing and reasoning about complex systems, including transport systems, IP stacks, email exchange, and access control in security. In order to describe the subgraph relation in directed graphs, Heyting variants of layered algebras were developed by Docherty and Pym in [4]. A Heyting layered algebra is a combination of a Heyting algebra and a residuated groupoid.

Boolean and Heyting layered algebras are both residuated lattice-ordered groupoids ($\mathbf{RLG}$s). This class of algebras was introduced and developed as mathematical tools of categorical grammars for natural language processing; see [5]. A residauted lattice ($\mathbf{RL}$) with a carrier preordered set $(A,\le )$ is an $\mathbf{RLG}$ in which · admits the associative property, i.e., $(a·b)·c=a·(b·c)$, for any $a,b,c\in A$. The algebras corresponding to logics for querying graphs [6] and bigraphs [7] are residuated lattices. Further, many structures that are well studied already, such as generalized Boolean algebras, Brouwerian algebras, relative Stone algebras, and ‘l-groups, are residuated lattices. The class of all residuated lattices will be denoted by $\mathcal{RL}$. Obviously, the residuated property can be captured by equations. Thus, $\mathcal{RL}$ and $\mathcal{RLG}$ are a finitely based variety.

For any algebraic structure $\mathrm{A}$ with domain A, an assignment in A is a function $\mu$:$\mathbf{Var}\to A$. Every assignment $\mu$ in A can be extended homomorphically to the term algebra. Let $\mu \left(t\right)$ be the element in A. An algebraic model is a pair ($A,\mu$) where A is an algebraic structure and $\mu$ is an assignment in A. We say that ($A,\mu$) satisfies an equation $t_1\approx t_2$ if $\mu \left(t_1\right)=\mu \left(t_2\right)$. We write $i{e}_i$ for the inequalities $t_i\le t_j$, where $i,j\ge 0$. We say that ($A,\mu$) satisfies a quasi-equation $i{e}_1\&\dots \&i{e}_n\Rightarrow i{e}_0$ if ($A,\mu$) satisfies $i{e}_0$ whenever it satisfies $i{e}_i$ $1\le i\le n$. We say an equation e is true in the classes of algebra $\mathcal{A}$ if for any $\mathrm{A}\in A$ and any assignment $\mu$, ($A,\mu$) satisfies e. A quasi-equation $e_1\&\dots \&i{e}_n\Rightarrow i{e}_0$ is said to be true in $\mathcal{A}$ if $e_0$ is true in $\mathcal{A}$ whenever each $e_i$ $1\le i\le n$ is true in $\mathcal{A}$.

A variety $\mathrm{V}$ is said to have a decidable equational theory if there is a computer algorithm that determines whether an equation is true or not in this variety in finite times. Let $\mathrm{V}$ be any finitely based variety. If $\mathrm{V}$ has the finite model property (FMP), then it has a decidable equational theory. If one replaces equation with quasi-equation, then $\mathrm{V}$ has a decidable quasi-equational theory. This problem for quasi-equation is referred to as the word problem for the classes of algebras. It is known that the strong finite model property (SFMP) is equivalent to the finite embeddability property (FEP) over $\mathcal{RLG}$ and $\mathcal{RL}$, which implies the decidability of quasi-equational theory [8].

We mention some FMP and decidability results of the equational and universal theories for classes of algebras closely related to $\mathcal{RLG}$ and $\mathcal{RL}$. The $\mathcal{RG}$ and $\mathcal{DLRG}$ have FEP, so the equational and quasi-equantional theories of them are both decidable (see M. Farulewski [9], W. Buszkowski [5], and Z. Hanikova and R. Horcik [10]). The same holds for the $\mathcal{BRG}$ and $\mathcal{HRG}$ [5] (also see M. Kaminski and N. Francez [11,12] and S. Docerty and D. Pym [4]). For $\mathcal{LRG}$, the undecidability of the universal theory is proved by K. Chvalovsky in [13], whereas the equational theory is proved to be decidable in [14]. The classes of $\mathcal{RL}$ and $\mathcal{RSG}$ have undecidability universal theory [15]; however, the equational theories are both decidable [14,16]. In [12], FMPs are proved for $\mathcal{DRL}$ and $\mathcal{HRL}$ via algebraic and relational semantics, respectively. It follows that the equational theories of the corresponding classes of algebras are decidable. The undecidability of the universal theory of (commutative) $\mathcal{DRL}$ is shown by N. Galatos [17]. Not all varieties of residuated lattices have decidable equational theory. Contractions $\mathcal{RL}$ (assuming $a·a=a$) and $\mathcal{BRL}$ have undecidable equational theories [8].

In the present paper, we consider the decision problem for equational theories of various classes of bounded $\mathcal{DLRG}$ and several classes of commutative and contractive $\mathcal{QBRL}$. We prove that all these classes of algebras have decidable equational theories via showing they have FMPs. The FMP and decidability results are extended to the classes of fusion of associative and non-associative algebras under the same lattice-based algebras, e.g., $\mathcal{QBRL}\oplus \mathcal{QBRLG}$. There are a number of reasons to investigate the equational theories of these classes of algebras. Firstly, by changing different lattices based in layer algebras, one can increase expressivity of graph properties. For example, a 4-valued and 3-valued sentence can be expressed in a layer graph when the bounded $\mathcal{DLRG}$ is a quasi-Boolean or Kleene algebra, respectively. Secondly, the associative and commutative multiplicative operation may help to reason about the decomposition of graphs into disjoint subgraphs [6,7]. Finally, there are also technical motivations. Although the FMP for bounded $\mathcal{DLRG}$, $\mathcal{HLRG}$, and $\mathcal{BLRG}$ follows from the FEP results, we provide direct proofs for the FMP of these classes of algebras. Further, our proofs cover some new classes of $\mathcal{DLRG}$, which have not been considered before, e.g., $\mathcal{QBLRG}$, $\mathcal{KLRG}$, and so on. Furthermore, we extend our results to several classes of commutative and contractive $\mathcal{QBRL}$, including $\mathcal{BRL}$.

We conclude the introduction by summarizing the contents of the paper. In Section 2, we recall some basic definitions and results of Boolean and Heyting residuated groupoids. In Section 3, we study various $\mathcal{RLG}$s. We prove that all these classes of algebras have finite model properties (FMPs), and their equation theories are decidable. Section 4 extends the results in Section 3 to several quasi-Boolean residuated algebras (including the Boolean one) in which the fusion · admits associative, commutative, and contractive aspects, i.e., $a·a=a$, for some a in the domain of algebras. Further, we argue that the same results hold for several classes of algebras that are fusions of some classes of algebras considered in Section 3 and Section 4. In the last section, we conclude our results and list some open problems that we hope will stimulate further research.

2. Layer Algebras

In this section, we recall the basic definitions and properties of (Heyting) layer algebras. These results are obtained from [5]. We assume familiarity with standard lattice theoretic notions, which can be found in [8,18].

Definition 1. 

A magma (groupoid) $\mathrm{M}$ (M,·) is a structure such that M is a set of elements and · is a binary operation on M; i.e., for any $a,b\in M$, $a·b\in M$. If · admits the associative property, i.e., $a·(b·c)=(a·b)·c$, then $\mathrm{M}$ is a semigroup.

Definition 2. 

([2]). A layer algebra, first introduced and named residuated algebra, is a structure ($M,·,\setminus ,/,\wedge ,\vee ,\neg ,1,0$) such that ($M,\wedge ,\vee ,\neg ,1,0$) is a Boolean algebra and $·,\setminus ,/$ are binary operations on carrier set M satisfying that for any $a,b,c\in M$

• (res) $a·b\le c$ iff $b\le a\setminus c$ iff $b\le c/a$

where ≤ is the lattice ordering.

The reduct $\mathrm{Rg}=(G,·,\setminus ,/,\le )$ of a layer algebra is a residuated groupoid. A Heyting layer algebra $\mathrm{H}=(H,·,\setminus ,/,\wedge ,\vee ,\to ,1,0)$ is a structure where ($H,\wedge ,\vee ,\to ,0,1$) is a Heyting algebra and $(H,·,\setminus ,/,\le )$ is a residuated groupoid with the lattice ordering ≤. Obviously, a layer algebra is a Heyting layer algebra. In the following, $\mathrm{A}$ is used to represent an A algebra, $\mathbf{A}$ is used to represent a class of A algebras, and $\mathcal{A}$ is used to represent all classes of A algebras. We follow the tradition from [5] and use Boolean (Heyting) residuated groupoids for (Heyting) layer algebras, respectively.

Example 1. 

The algebra in Figure 1 is an example of $\mathrm{BRG}s$.

Example 2. 

The algebra in Figure 2 is an example of $\mathrm{HRG}s$. One has $a\vee \neg a=a\ne 1$.

A layered magma is a structure (M, ·) with a partial binary operation · on a carrier set M. The operation is said to be contra-commutative if, for all $m,n\in M$, if $m·n$ is defined, then $n·m$ is undefined. Given a layered magma (M, ·), one constructs layered algebras by the following. Let $P\left(M\right)$ be the power set of M and $X,Y\in P\left(W\right)$. Define

• $X•Y=\{a·b|a\in X\&b\in Y,a·b\mathrm{is}\mathrm{defined}\}$.
• $X\setminus \setminus Y=\left\{c\right|\forall a\in X$ if $c·a$ is defined, and then $c·a\in Y\}$.
• $X//Y=\left\{c\right|\forall a\in X$ if $a·c$ is defined, and then $a·c\in Y\}$.

A layered magma can form a layered algebra by the following way.

Lemma 1. 

In a Heyting residuated groupoid $\mathrm{L}=(L,·,\setminus ,/,\wedge ,\vee ,\neg ,1,0)$, for any $a,b,c\in L$, if $a\le b$, then $a·c\le b·c$ and $c·a\le c·b$.

Corollary 1. 

In a Heyting residuated groupoid $\mathrm{L}=(L,·,\setminus ,/,\wedge ,\vee ,\neg ,1,0)$, for any $a,b,c,d\in L$, if $a\le b$ and $c\le d$, then $a·c\le b·d$.

Lemma 2. 

In a Heyting-residuated groupoid $\mathrm{L}=(L,·,\setminus ,/,\wedge ,\vee ,\neg ,1,0)$, let $A,B\subseteq L$, and $\bigvee A,\bigvee B,\bigwedge A,\bigwedge B$ exist. Then, the following hold

• ${\bigvee }_{a\in A,b\in B}a·b$ exits and $\bigvee A·\bigvee B={\bigvee }_{a\in A,b\in B}a·b$.
• for any $c\in L$, ${\bigwedge }_{a\in A}a\setminus c$ and ${\bigwedge }_{b\in B}c\setminus b$ exist and $\bigvee A\setminus c={\bigwedge }_{a\in A}a\setminus c$, $c\setminus \bigwedge B={\bigwedge }_{b\in B}c\setminus b$.

Recall that Boolean (Heyting) residuated groupoids NRGs (HRGs) form varieties, which are studied from logic point of view in [5]. Gentzen systems are present for the logics of BRGs and HRGs based on sequents, which are pairs of trees of formulas. Here, we describe the algebraic version of Gentzen system for BRGs and RRGs by adapting the denotation from [8]. Let $\mathbf{Var}$ be a set of countably many variables $x_1,x_2,\dots$ in the languages.

Definition 3. 

The set of terms $Te$ is defined inductively as follows:

$$Te\ni t::=x\in \mathbf{Var}\mid 1\mid 0\mid t\wedge t\mid t\vee t\mid t\to t\mid t·t\mid t\setminus t\mid t/t$$

Define $\neg t=t\to 0$.

Definition 4. 

A tree over a finite set of terms is a structure such that all leaves of tree are terms in T and connected by connective ·.

Example 3. 

Let $T=\{t_1,t_2,t_3\}$. Then, $(t_1·t_2)·(t_2·t_3)$ and $(t_1·t_3)·t_1$ are all trees over T.

Obviously, all tree are terms. In the following, we use $\overline{t},\overline{s},\overline{q},\dots$ to denote trees when emphasized.

Definition 5. 

A context is a structure generated from a tree $\overline{t}[-]$ with a designated position $[-]$ that can be filled with a tree. In particular, a single position $[-]$ is a context. Let $\overline{t}\left[\overline{t^{\prime }}\right]$ be a tree obtained from $\overline{t}[-]$ by substituting $\overline{t^{\prime }}$ for $[-]$.

Example 4. 

Let expression $\overline{t}[-]=(-·\neg t_1)·t_2\wedge t_3)$ be a context. If we replace the tree $\overline{t^{\prime }}=t_4\setminus t_5$ for the position − in $\overline{t}[-]$, then we obtain the term $\overline{t}\left[\overline{t^{\prime }}\right]=((t_4\setminus t_5)·\neg t_1)·t_2\wedge t_3$. If we replace the tree $t_4·t_5$ for the position − in $\overline{t}[-]$, then we obtain the term $\overline{t}\left[\overline{t^{\prime }}\right]=((t_4·t_5)·\neg t_1)·t_2\wedge t_3$.

Definition 6. 

An algebraic Gentzen system is a finite set $\mathbf{GS}$ of quasi-inequalities in the following form

$$t_{11}\le t_{12}\&\dots \&t_{n1}\le t_{n2}\Rightarrow t_{01}\le t_{02}$$

where $t_{ij}$ $0\le i,j\le n$ are terms.

Definition 7. 

We define ${\mathbf{GS}}_{\mathbf{HRG}}$ to be the finite set of quasi-inequalities given in the following

• $\Rightarrow t\le t$ (refl)
  $\Rightarrow t_1\wedge (t_2\vee t_3)\le (t_1\wedge t_2)\vee (t_1\wedge t_3)$ (dist)
  $\Rightarrow t\le 1$ (g1)
  $\Rightarrow 0\le t$ (l0)
• $t_1\le t_2\&\overline{t_3}\left[t_2\right]\le t_4\Rightarrow \overline{t_3}\left[t_1\right]\le t_4$ (tran)
• $t_1\le t_2\&t_3\le t_4\Rightarrow t_1·t_3\Rightarrow t_2·t_4$ (·R)
• $t_1\le t_2\&\overline{t_3}\left[t_4\right]\le t_5\Rightarrow \overline{t_3}[t_1·t_2\setminus t_4]\le t_5$ (∖L)
  $t_1·t_2\le t_3\Rightarrow t_2\le t_1\setminus t_3$ (∖R)
• $t_1\le t_2\&\overline{t_3}\left[t_4\right]\le t_5\Rightarrow \overline{t_3}[t_4/t_2·t_1]\le t_5$ (/L)
  $t_2·t_1\le t_3\Rightarrow t_2\le t_3/t_1$ (/R)
• $\overline{t_1}\left[t_2\right]\le t_3\&\overline{t_1}\left[t_4\right]\le t_3\Rightarrow \overline{t_1}[t_2\vee t_4]\le t_3$ (∨L)
  $t_1\le t_2\Rightarrow t_1\le t_2\vee t_3$ (∨R)
  $t_1\le t_2\Rightarrow t_1\le t_3\vee t_2$ (∨R)
• $\overline{t_1}\left[t_2\right]\le t_3\Rightarrow \overline{t_1}[t_2\wedge t_4]\le t_3$ (∧L)
  $\overline{t_1}\left[t_2\right]\le t_3\Rightarrow \overline{t_1}[t_4\wedge t_2]\le t_3$ (∧L)
  $t_1\le t_2\&t_1\le t_3\Rightarrow t_1\le t_2\wedge t_3$(∧R)
• $t_1\le t_2\&\overline{t_3}\left[t_4\right]\le t_5\Rightarrow \overline{t_3}[t_1\wedge t_2\to t_4]\le t_5$ (→L)
  $t_1\wedge t_2\le t_3\Rightarrow t_2\le t_1\setminus t_3$ (→R)

Define ${\mathbf{GS}}_{\mathbf{BRG}}={\mathbf{GS}}_{\mathbf{HRG}}\cup \{\Rightarrow 1\le t\vee \neg t\left(\mathrm{em}\right)\}$.

Recall that the definitions of rooted trees and Genzten are provable.

Definition 8. 

A rooted tree is a poset with a least element, called the root, and, for each element, the set of all elements below it is linearly ordered.

Definition 9. 

A proof-tree in a $\mathbf{GS}$ is a finite rooted tree in which each element is an inequality, which is an instance of a member of $\mathbf{GS}$.

Definition 10. 

An inequality $t_1\le t_2$ is said to be Gentzen-provable in $\mathbf{GS}$ denoted by ${⊢}_{\mathbf{GS}}{t}_1\le t_2$ if there exists a proof-tree in $\mathbf{GS}$ with this inequality as the root.

Let $\mathrm{T}$ be the terms of algebras defined in Definition 2. Write $\mathrm{T}\left(\mathrm{W}\right)$ for the term algebra of term T with variables in the set W. Let $\mu :W\mapsto A$, where A is the carrier domain of an $\mathrm{A}$-algebra. Obviously, $\mu$ can be extended to a unique homomorphism ${\mu }^{\prime }:T\mapsto H$.

Definition 11. 

A ${\mathbf{GS}}_{\mathbf{A}}$ is called sound with respect to all classes of algebra $\mathcal{A}$ if for any inequality $s\le t$. If ${⊢}_{{\mathbf{GS}}_{\mathbf{A}}}s\le t$, then, for any $\mathrm{A}\in \mathcal{A}$ with carrier domain A, there is a μ defined as above such that $\mu \left(s\right)\le \mu \left(t\right)$ holds in $\mathrm{A}$.

Definition 12. 

A ${\mathbf{GS}}_{\mathbf{A}}$ is called complete with respect to all classes of algebra $\mathcal{A}$ if for any inequality $s\le t$. If ${⊬}_{{\mathbf{GS}}_{\mathbf{A}}}s\le t$, then there is $\mathrm{A}\in \mathcal{A}$ with carrier domain A and a μ defined as above such that $\mu \left(s\right)\le \mu \left(t\right)$ fails in $\mathrm{A}$.

Theorem 1. 

${\mathbf{GS}}_{\mathbf{BRG}}$ and ${\mathbf{GS}}_{\mathbf{HRG}}$ are sound and complete with respect to $\mathcal{BRG}$ and $\mathcal{HRG}$.

The soundness can be shown by proving that all quasi-inequalities are valid in $\mathcal{BRG}$ ($\mathcal{HRG})$. The completeness follows directly from the proof of FMP in the next section. Details of proofs are included in [4,5,11,12].

Definition 13. 

Let $\mathrm{VA}$ be the variety of $\mathbf{A}$-algebras. VA is said to have the finite model property (FMP) if for any $s,t\in T$ there is an $\mathrm{A}$ and a μ, satisfying that $\mu \left(s\right)\le \mu \left(t\right)$ implies $\pi \left(s\right)\le \pi \left(t\right)$ for some finite ${\mathrm{A}}^{\prime }$ and π.

The FMP can be equivalently defined by the Genzten systems.

Definition 14. 

Let $\mathrm{VA}$ be a variety of $\mathbf{A}$-algebras and ${\mathbf{GS}}_{\mathbf{A}}$ be its corresponding Genzten system. VA is said to have finite model property if for any $s,t\in T$${⊬}_{{\mathbf{GS}}_{\mathbf{A}}}s\le t$; then, there is an $\mathrm{A}$ and a μ such that $\mu \left(s\right)\le \mu \left(t\right)$ does not hold in $\mathrm{A}$.

Theorem 2. 

The varieties of $\mathcal{BRG}$ and $\mathcal{HRG}$ have FMPs and decidable equational theory.

3. Bounded Distributive Lattice-Ordered Residuated Groupoid and Semigroup

Definition 15. 

A bounded distributive lattice-ordered residuated groupoid (DLRG) $\mathrm{A}=(A,·,\setminus ,/,\wedge ,\vee ,$ $1,0)$ is a structure where ($A,\wedge ,\vee ,0,1$) is a bounded distributive lattice and $(A,·,\setminus ,/,\le )$ is a residuated groupoid with lattice ordering ≤.

Proposition 1. 

The following equations are satisfied by every $DLRG$:

1 

$a·(b\vee c)=(a·b)\vee (a·c)$;

2 

$a\setminus (b\wedge c)\le (a\setminus c)\wedge (a\setminus c)$;

3 

$(a\wedge b)/c\le (a/c)\wedge (b/c)$.

The DLRG extensions are DLRGs enriched with unary operation ¬ or binary operation → satisfying some conditions in the following definition. In the following, we write DLR$G_n$s and DLR$G_i$s, meaning DLRGs enriched with ¬ and →, respectively.

Definition 16. 

In a DLRG with carrier domain A, the following conditions are considered for any $a,b,c\in A$:

• $\left(\mathrm{t}\right)$ $a\le b$ implies $\neg b\le \neg a$;
• $\left(\mathrm{dn}\right)$ $a=\neg \neg a$;
• $\left(\mathrm{ko}\right)(a\wedge \neg a)\vee b\le b\vee \neg b$;
• $\left(\mathrm{k}\right)(a\wedge \neg a)\vee b\le b$;
• $\left(\mathrm{b}\right)a\vee \neg a=1\&a\wedge \neg a=0$;
• $\left(\mathrm{h}\right)$ $a\wedge b\le c$ iff $b\le a\to c$.

Definition 17. 

Various extensions of DLRG based on conditions in Definition 16 are listed as follows:

• QBLRG: quasi-Boolean residuated groupoid is DLR$G_n$ s.t. (d), (t), and (dn) hold.
• KLRG: Kleene residuated groupoid is QBLRG s.t. (k) holds (this class of algebra is considered in [19]).
• KoLRG: Kleene ordered residuated groupoid is QBLRG s.t. (ko) holds (this class of algebra is considered in [20,21]).
• BRG: Boolean residuated groupoid is DLR$G_n$ s.t. (b) holds.
• HRG: Heyting residuated groupoid is DLR$G_i$ s.t. (d) and (h) hold.

Defining $·,\setminus ,/$ similar to Examples 2 and 3 on different lattices, one obtains different examples of the corresponding algebras. Here, we present an example of QBLRG. Others can be treated similarly.

Example 5. 

The algebra in Figure 3 is an example of QBLRG. One has $(a\wedge \neg a)\vee b=1\le ̸b=b\vee \neg b$, $(a\wedge \neg a)\vee b=1\le ̸b$ $a\vee \neg a\ne 1$, and $a\wedge \neg a\ne 0$.

Definition 18. 

We define ${\mathbf{GS}}_{\mathbf{DLRG}}$ to be the finite set of quasi-inequalities given in the following

• $\Rightarrow t\le t$ (refl)
  $\Rightarrow t_1\wedge (t_2\vee t_3)\le (t_1\wedge t_2)\vee (t_1\wedge t_3)$ (dist)
  $\Rightarrow t\le 1$ (g1)
  $\Rightarrow 0\le t$ (l0)
• $t_1\le t_2\&\overline{t_3}\left[t_2\right]\le t_4\Rightarrow \overline{t_3}\left[t_1\right]\le t_4$ (tran)
• $t_1\le t_2\&t_3\le t_4\Rightarrow t_1·t_3\Rightarrow t_2·t_4$ (·R)
• $t_1\le t_2\&\overline{t_3}\left[t_4\right]\le t_5\Rightarrow \overline{t_3}[t_1·t_2\setminus t_4]\le t_5$ (∖L)
  $t_1·t_2\le t_3\Rightarrow t_2\le t_1\setminus t_3$ (∖R)
• $t_1\le t_2\&\overline{t_3}\left[t_4\right]\le t_5\Rightarrow \overline{t_3}[t_4/t_2·t_1]\le t_5$ (/L)
  $t_2·t_1\le t_3\Rightarrow t_2\le t_3/t_1$ (/R)
• $\overline{t_1}\left[t_2\right]\le t_3\&\overline{t_1}\left[t_4\right]\le t_3\Rightarrow \overline{t_1}[t_2\vee t_4]\le t_3$ (∨L)
  $t_1\le t_2\Rightarrow t_1\le t_2\vee t_3$ (∨R)
  $t_1\le t_2\Rightarrow t_1\le t_3\vee t_2$ (∨R)
• $\overline{t_1}\left[t_2\right]\le t_3\Rightarrow \overline{t_1}[t_2\wedge t_4]\le t_3$ (∧L)
  $\overline{t_1}\left[t_2\right]\le t_3\Rightarrow \overline{t_1}[t_4\wedge t_2]\le t_3$ (∧L)
  $t_1\le t_2\&t_1\le t_3\Rightarrow t_1\le t_2\wedge t_3$(∧R)

Obviously, ${\mathbf{GS}}_{\mathbf{DLRG}}\cup \{(\to \mathrm{L}),(\to \mathrm{R})\}={\mathbf{GS}}_{\mathbf{HRG}}$. Define ${\mathbf{GS}}_{\mathbf{QBLRG}}$, ${\mathbf{GS}}_{\mathbf{KLRG}}$, ${\mathbf{GS}}_{\mathbf{KoLRG}}$ as follows:

• ${\mathbf{GS}}_{\mathbf{QBLRG}}={\mathbf{GS}}_{\mathbf{DLRG}}\cup \{s\le t\Rightarrow \neg t\le \neg s\left(trpo\right),\neg \neg s=s\left(doubn\right)\}$;
• ${\mathbf{GS}}_{\mathbf{KLRG}}={\mathbf{GS}}_{\mathbf{QBLRG}}\cup \{\Rightarrow (s\wedge \neg s)\vee t\le t\left(kleene\right)\}$;
• ${\mathbf{GS}}_{\mathbf{KoLRG}}={\mathbf{GS}}_{\mathbf{QBLRG}}\cup \{(s\wedge \neg s)\vee t\le t\vee \neg t\left(kleeno\right)\}$.

Definition 19. 

A residuated semigroup (RSG) $\mathrm{G}$ (G,$·,\setminus ,/$, ≤) is a structure such that ($G,·$) is a semigroup and $·,\setminus ,/$ are binary operations on G satisfying (res).

Definition 20. 

A commutative and contractive residuated semigroup (RSG) $\mathrm{G}$ (G,$·,\setminus ,/$, ≤) is an RSG such that ·, satisfying the following

• (com) $a·b=b·a$;
• (con) $a·a=a$.

for any $a,b\in G$.

Definition 21. 

An associative, commutative, and contractive residuated quasi-Boolean algebra $\mathrm{G}$ (G,$·,\setminus ,/$, $\wedge ,\vee ,\neg$) such that (G,$\wedge ,\vee ,\neg$) is a quasi-Boolean algebra and (G,$·,\setminus ,/,\le$) is a commutative and contractive residuated semigroup where ≤ is the lattice order. If (G,$\wedge ,\vee ,\neg$) is Kleene (order) or Boolean algebra, then $\mathrm{G}$ is an associative, commutative and contractive residuated Kleene (order) or Boolean algebra.

Proposition 2. 

The following equation are satisfied by every associative, commutative, and contractive residuated quasi-Boolean algebra.

1 

$a·\left(b{\to }_{\#}c\right)\le (a·b){\to }_{\#}c$;

2 

$a{\to }_{\#}b\le (a·c){\to }_{\#}(b·c)$;

3 

$\left(a{\to }_{\#}b\right)·\left(b{\to }_{\#}c\right)\le \left(a{\to }_{\#}c\right)$;

4 

$a{\to }_{\#}(b·c)\le \left(a{\to }_{\#}c\right){\to }_{\#}b$;

5 

$\left(a{\to }_{\#}c\right){\to }_{\#}b\le a{\to }_{\#}(b·c)$;

6 

$\left(a{\to }_{\#}a\right)·a\le a$;

7 

$a\le \left(a{\to }_{\#}a\right)·a$;

8 

$a{\to }_{\#}(b\wedge c)\le \left(a{\to }_{\#}b\right)\wedge \left(a{\to }_{\#}c\right)$;

9 

$\left(a{\to }_{\#}b\right)\wedge \left(a{\to }_{\#}c\right)\le a{\to }_{\#}(b\wedge c)$;

10 

$a{\to }_{\#}\left(b{\to }_{\#}c\right)\le b{\to }_{\#}\left(a{\to }_{\#}c\right)$;

11 

$b{\to }_{\#}\left(a{\to }_{\#}c\right)\le a{\to }_{\#}\left(b{\to }_{\#}c\right)$;

12 

$a{\to }_{\#}\left(a{\to }_{\#}b\right)\le a{\to }_{\#}b$;

13 

$a{\to }_{\#}b\le a{\to }_{\#}\left(a{\to }_{\#}b\right)$.

The proof refers to Proposition 1.4.4 in [8] and Lemma 11.

Let $\mathcal{CC}xRA$, where $x\in \{\mathcal{QB},\mathcal{K},\mathcal{K}o,\mathcal{B}\}$ are the classes of associative, commutative, and contractive residuated x algebras. Since the commutative holds in $\mathcal{CC}$x$\mathcal{RG}$, $a\setminus b=a/b$ for any $a,b\in G$, in the following, we write ${\to }_{\#}$, and we mean ∖ and /. and recall the definition of terms.

Definition 22. 

The set of terms $Te$ are defined inductively as follows:

$$Te\ni t::=x\in \mathbf{Var}\mid 1\mid 0\mid t\wedge t\mid \neg t\mid t·t\mid t{\to }_{\#}t$$

Define $t\vee t=\neg (\neg t\wedge \neg t)$.

Definition 23. 

We define ${\mathbf{GS}}_{\mathbf{CCQBRA}}$ to be the finite set of quasi-inequalities s.t. ${\mathbf{GS}}_{\mathbf{L}}\cup \{\left(t\right),\left(dn\right)\}\subseteq {\mathbf{GS}}_{\mathbf{AL}}$ and the following

• $\overline{t_1}[\overline{t_2}·\overline{t_3}]\le t_4\iff \overline{t_1}[\overline{t_2}·\overline{t_3}]\le t_4$ (com);
• $\overline{t_1}\left[\overline{t_2}\right]\le t_3\iff \overline{t_1}[\overline{t_2}·\overline{t_2}]\le t_3$ (con);
• $\overline{t_1}[(\overline{t_2}·\overline{t_3})·\overline{t_4}]\le t_5\iff \overline{t_1}[\overline{t_2}·(\overline{t_3}·\overline{t_4})]\le t_5$ (ass).

Define ${\mathbf{GS}}_{\mathbf{CCKRA}}={\mathbf{GS}}_{\mathbf{CCQBRA}}\cup \left\{\left(kln\right)\right\}$, ${\mathbf{GS}}_{\mathbf{CCKoRA}}={\mathbf{GS}}_{\mathbf{CCQBRA}}\cup \left\{\left(klno\right)\right\}$, and ${\mathbf{GS}}_{\mathbf{CCBRA}}={\mathbf{GS}}_{\mathbf{CCQBRA}}\cup \{\left(em\right),\Rightarrow a\wedge \neg a=1\left(lc\right)\}$

Let $\mathbf{GS}\in \{{\mathbf{GS}}_{\mathbf{DLRG}},{\mathbf{GS}}_{\mathbf{QBRG}},{\mathbf{GS}}_{\mathbf{KRG}},{\mathbf{GS}}_{\mathbf{KoRG}},{\mathbf{GS}}_{\mathbf{CCQBRA}},{\mathbf{GS}}_{\mathbf{CCKRA}},{\mathbf{GS}}_{\mathbf{CCKoRA}},{\mathbf{GS}}_{\mathbf{CCBRA}}\}$, and ${\mathcal{A}}_{\mathcal{GS}}$ be the classes of its corresponding algebras.

Proposition 3. 

${⊢}_{\mathbf{GS}}t·s\le q$ iff ${⊢}_{\mathbf{GS}}t\le s{\to }_{\#}q$ iff ${⊢}_{\mathbf{GS}}s\le t{\to }_{\#}q$.

Theorem 3. 

$\mathbf{GS}$ is sound and complete with respect to ${\mathcal{A}}_{\mathcal{GS}}$.

The soundness can be shown by proving that all quasi-inequalities are valid in ${\mathcal{A}}_{\mathcal{GS}}$. The completeness follows directly from the proof of FMP in the next section.

4. Finite Model Property and Decidability

In this section, we show the FMPs of all algebras considered in Section 2 and Section 3. Let $\mathbf{LG}\in \{{\mathbf{GS}}_{\mathbf{DLRG}},{\mathbf{GS}}_{\mathbf{QBRG}},{\mathbf{GS}}_{\mathbf{KRG}},{\mathbf{GS}}_{\mathbf{KoRG}},{\mathbf{GS}}_{\mathbf{HRG}},{\mathbf{GS}}_{\mathbf{BRG}}\}$. In the following, we always assume that T is a finite set of terms and $\{1,0\}\subseteq T$. Define $\neg T=\{\neg t|t\in T\}$, $d\left(T\right)$ as the $\wedge ,\vee$ closure of T and $d^{\prime }\left(T\right)$ as the $\wedge ,\vee ,\neg$ closure of T. Let $T^{\ast }$ be $d\left(T\right)$ when $\mathbf{LG}\in \{{\mathbf{GS}}_{\mathbf{DLRG}},{\mathbf{GS}}_{\mathbf{HRG}}\}$. Otherwise, $T^{\ast }=d^{\prime }\left(T\right)$. Let $T^{!}$ be the set of trees over $T^{\ast }$.

Definition 24. 

We define ${\le }_{\mathcal{T}}$ on $T^{!}$ as follows: for $\overline{t_1},\overline{t_2}\in T^{!}$, $\overline{t_1}{\le }_{\mathcal{T}}\overline{t_2}$ iff for term $s\in T^{\ast }$, if ${⊢}_{\mathbf{LG}}\overline{t_2}\le s\Rightarrow \overline{t_1}\le s$.

Let $\overline{t_1}{\approx }_{\mathcal{T}}\overline{t_2}$ be $\overline{t_1}{\le }_{\mathcal{T}}\overline{t_2}$ and $\overline{t_2}{\le }_{\mathcal{T}}\overline{t_1}$, and then ${\approx }_{\mathcal{T}}$ is an equivalence relation. Let ${\left[s\right]}_{\mathcal{T}}=\{\overline{t}\mid s{\approx }_{\mathcal{T}}\overline{t}\&\overline{t}\in T^{!}\}$ for any $s\in T^{\ast }$. Let $\left[T^{\ast }\right]=\{{\left[s\right]}_{\mathcal{T}}\mid s\in T^{\ast }\}$.

Definition 25. 

A term t is called a disjunction normal form with respect to the set of terms T if it is the disjunction of conjunction of some terms in $T\cup \neg T$.

Since distributive of lattices and the De Morgan properties are always assumed here, for any $s\in T^{\ast }$, there is a term $dn{f}_{\mathcal{T}}\left(s\right)$ in disjunction normal form such that $s{\approx }_{\mathcal{T}}dn{f}_{\mathcal{T}}\left(s\right)$. Clearly, $\left\{dn{f}_{\mathcal{T}}\left(s\right)\right|s\in T^{\ast }\}$ is finite since T is finite. Due to ${\left[s\right]}_{\mathcal{T}}={\left[dn{f}_{\mathcal{T}}\left(s\right)\right]}_{\mathcal{T}}$ and the number of ${\left[dn{f}_{\mathcal{T}}\left(s\right)\right]}_{\mathcal{T}}$ is finite, $\left[T^{\ast }\right]$ is finite.

Lemma 3. 

For any $\overline{t}\in T^{!}$, there is a $s\in dnf\left(T^{\ast }\right)$ such that $\overline{t}{\approx }_{\mathcal{T}}s$.

Proof. 

Let $s=\bigwedge \left\{dn{f}_{\mathcal{T}}\left(s_i\right)\right|{⊢}_{\mathbf{LG}}\overline{t}\le s_i,s_i\in T^{\ast }\}$. Assume that ${⊢}_{\mathbf{LG}}\overline{t}\le s^{\prime }$ for some $s^{\prime }\in T^{\ast }$. Then, there is an i such that $s_i=s^{\prime }$. Clearly, ${⊢}_{\mathbf{LG}}s\le s^{\prime }$. Thus, $s{\le }_{\mathcal{T}}\overline{t}$. In the opposite direction, since ${⊢}_{\mathbf{LG}}\overline{t}\le s_i$, by (∧R), one obtains ${⊢}_{\mathbf{LG}}\overline{t}\le s$. Suppose that ${⊢}_{\mathbf{LG}}s\le q$ for some $q\in T^{\ast }$. By applying (tran) to ${⊢}_{\mathbf{LG}}s\le q$ and ${⊢}_{\mathbf{LG}}t\le s$, one obtains ${⊢}_{\mathbf{LG}}t\le q$. Thus, $t{\le }_{\mathcal{T}}s$. □

Remark 1. 

Note that, for any $t\in T^{!}$, ${⊢}_{\mathbf{LG}}t\le 1$. Thus, $\left\{dn{f}_{\mathcal{T}}\left(s_i\right)\right|{⊢}_{\mathbf{LG}}t\le s_i,s_i\in T^{\ast }\}$ is not empty. Further, this set is finite. Hence, s always exists. If one considers the algebras without the greatest element 1, for instance $\mathbf{rg}s$ or $\mathbf{dlrg}s$ as in [5], then one needs more complex set of theoretic construction with the help of closure operator and interpolant proof theory, as in the result in [5].

Definition 26. 

Let $\mathcal{Q}=(\left[T^{\ast }\right],{\wedge }^{\ast },{\vee }^{\ast },{\neg }^{\ast },{·}^{\ast },{\setminus }^{\ast },{/}^{\ast },0^{\ast },1^{\ast })$ be the quotient algebra of $\left[T^{\ast }\right]$, where all operations are defined as follows: for any ${\left[s\right]}_{\mathcal{T}},{\left[q\right]}_{\mathcal{T}}\in \left[T^{\ast }\right]$,

• $\left(1\right)$ $1^{\ast }={\left[1\right]}_{\mathcal{T}}$;
• $\left(2\right)$ $0^{\ast }={\left[0\right]}_{\mathcal{T}}$;
• $\left(3\right)$ ${\left[s\right]}_{\mathcal{T}}{\wedge }^{\ast }{\left[q\right]}_{\mathcal{T}}={[s\wedge q]}_{\mathcal{T}}$;
• $\left(4\right)$ ${\left[s\right]}_{\mathcal{T}}{\vee }^{\ast }{\left[q\right]}_{\mathcal{T}}={[s\vee q]}_{\mathcal{T}}$;
• $\left(5\right)$ ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}={\left[w\right]}_{\mathcal{T}}$ s.t. $w{\approx }_{\mathcal{T}}s·q$;
• $\left(6\right)$ ${\left[s\right]}_{\mathcal{T}}{\setminus }^{\ast }{\left[q\right]}_{\mathcal{T}}={[w_1\vee \dots \vee w_n]}_{\mathcal{T}}$ s.t. ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[w_i\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}$ for any $i\in \{1,\dots ,n\}$;
• $\left(7\right)$ ${\left[s\right]}_{\mathcal{T}}{/}^{\ast }{\left[q\right]}_{\mathcal{T}}={[w_1\vee \dots \vee w_n]}_{\mathcal{T}}$ s.t. ${\left[w_i\right]}_{\mathcal{T}}{·}^{\ast }{\left[s\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}$ for any $i\in \{1,\dots ,n\}$;
• $\left(8\right)$ ${\neg }^{\ast }{\left[s\right]}_{\mathcal{T}}={[\neg s]}_{\mathcal{T}}$;

We define ${\left[s\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}$ as ${\left[s\right]}_{\mathcal{T}}{\wedge }^{\ast }{\left[q\right]}_{\mathcal{T}}={\left[s\right]}_{\mathcal{T}}$.

If $\mathbf{LG}={\mathbf{GS}}_{\mathbf{DLRG}}$, then $\mathcal{Q}$ is constructured as in 26 by excluding item (8).

Definition 27. 

Let $\mathcal{Q}=(\left[T^{\ast }\right],{\wedge }^{\ast },{\vee }^{\ast },{\to }^{\ast },{·}^{\ast },{\setminus }^{\ast },{/}^{\ast },0^{\ast },1^{\ast })$ be the quotient algebra of $\left[T^{\ast }\right]$, where all operations except ${\to }^{\ast }$ and ${\le }^{\ast }$ are defined as in Definition 26 and ${\to }^{\ast }$ is defined as follows.

• $\left(9\right)$ ${\left[s\right]}_{\mathcal{T}}{\to }^{\ast }{\left[q\right]}_{\mathcal{T}}={[w_1\vee \dots \vee w_n]}_{\mathcal{T}}$ s.t. ${\left[s\right]}_{\mathcal{T}}{\wedge }^{\ast }{\left[w_i\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}$ for any $i\in \{1,\dots ,n\}$;

Lemma 4. 

The following conditions are equivalent for all $s,q\in T^{\ast }$:

• $\left(1\right)$ $s{\le }_{\mathcal{T}}q$;
• $\left(2\right)$ ${⊢}_{\mathbf{LG}}s\le q$;
• $\left(3\right)$${\left[s\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}$

Proof. 

Let us show (1) implies (2). Let $s{\le }_{\mathcal{T}}q$. By (relf), ${⊢}_{\mathbf{LG}}q\le q$. Thus, by Definition 24, one obtains ${⊢}_{\mathbf{LG}}s\le q$.

Let us show (2) implies (3). It suffices to prove ${\left[s\right]}_{\mathcal{T}}{\wedge }^{\ast }{\left[q\right]}_{\mathcal{T}}={\left[s\right]}_{\mathcal{T}}$. By Definition 26. ${\left[s\right]}_{\mathcal{T}}{\wedge }^{\ast }{\left[q\right]}_{\mathcal{T}}={[s\wedge q]}_{\mathcal{T}}$. Assume that ${⊢}_{\mathbf{LG}}s\le q$. Clearly, by (relf) ${⊢}_{\mathbf{LG}}s\le s$. By (∧ R), one obtains ${⊢}_{\mathbf{LG}}s\le s\wedge q$. Obviously, $s{\le }_{\mathcal{T}}s\wedge q$. Conversely, ${⊢}_{\mathbf{LG}}s\wedge q\le s$. Thus, $s\wedge q{\le }_{\mathcal{T}}s$. Therefore, ${\left[s\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}$.

Let us show (3) implies (1). Assume that ${\left[s\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}$. Thus, ${\left[s\right]}_{\mathcal{T}}{\wedge }^{\ast }{\left[q\right]}_{\mathcal{T}}={[s\wedge q]}_{\mathcal{T}}={\left[s\right]}_{\mathcal{T}}$. Thus, $s\wedge q{\approx }_{\mathcal{T}}s$. Therefore, $s{\le }_{\mathcal{T}}s\wedge q$. Thus, ${⊢}_{\mathbf{LG}}s\le s\wedge q$. Clearly, ${⊢}_{\mathbf{LG}}s\wedge q\le q$. By (trans), ${⊢}_{\mathbf{LG}}s\le q$. Let ${⊢}_{\mathbf{LG}}q\le w$ for some $w\in T^{\ast }$. Thus, by (trans), ${⊢}_{\mathbf{LG}}s\le w$. Hence, $s{\le }_{\mathcal{T}}q$. □

Lemma 5. 

All the operations defined in Definition 26 are well defined.

Proof. 

$\wedge ,\vee$ (or $\wedge ,\vee ,\neg$) are well defined since $T^{\ast }$ is closed under these operations. We provide the proof for operation ·. Then, by the functional definition, $\setminus ,/$ (→) are all well defined. Let ${\left[s\right]}_{\mathcal{T}}={\left[q\right]}_{\mathcal{T}}$. It suffices to show ${\left[w\right]}_{\mathcal{T}}{·}^{\ast }{\left[s\right]}_{\mathcal{T}}={\left[w\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}$. Let ${\left[w\right]}_{\mathcal{T}}{·}^{\ast }{\left[s\right]}_{\mathcal{T}}=\left[w_1\right]$ s.t. $w_1{\approx }_{\mathcal{T}}w·s$ and ${\left[w\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}=\left[w_1\right]$ s.t. $w_2{\approx }_{\mathcal{T}}w·q$. Obviously, ${⊢}_{\mathbf{LG}}{w}_1\le w_1$. Then, by Definition 24, ${⊢}_{\mathbf{LG}}w·s\le w_1$. Clearly, ${⊢}_{\mathbf{LG}}q\le s$. Thus, by (trans), ${⊢}_{\mathbf{LG}}w·q\le w_1$. Then, by by Definition 24, ${⊢}_{\mathbf{LG}}{w}_2\le w_1$. So, by Lemma 4, ${\left[w_1\right]}_{\mathcal{T}}{\le }^{\ast }{\left[w_2\right]}_{\mathcal{T}}$. Similarly, one proves ${\left[w_2\right]}_{\mathcal{T}}{\le }^{\ast }{\left[w_1\right]}_{\mathcal{T}}$. Thus, ${\left[w\right]}_{\mathcal{T}}{·}^{\ast }{\left[s\right]}_{\mathcal{T}}={\left[w\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}$. Similarly, ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[w\right]}_{\mathcal{T}}={\left[q\right]}_{\mathcal{T}}{·}^{\ast }{\left[w\right]}_{\mathcal{T}}$ Hence, · is well defined. □

Lemma 6. 

For any ${\left[s\right]}_{\mathcal{T}},{\left[q\right]}_{T^{\ast }},{\left[w\right]}_{\mathcal{T}}\in \left[T^{\ast }\right]$ in $\mathcal{Q}$ according to Definitions 26 and 27, ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[w\right]}_{\mathcal{T}}$ iff ${\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[s\right]}_{\mathcal{T}}{\setminus }^{\ast }{\left[w\right]}_{\mathcal{T}}$ iff ${\left[s\right]}_{\mathcal{T}}{\le }^{\ast }{\left[w\right]}_{\mathcal{T}}/h^{\ast }{\left[q\right]}_{\mathcal{T}}$.

Proof. 

Let us show ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[w\right]}_{\mathcal{T}}$ implies ${\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[s\right]}_{\mathcal{T}}{\setminus }^{\ast }{\left[w\right]}_{\mathcal{T}}$. Let ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}={\left[r\right]}_{\mathcal{T}}$ and ${\left[s\right]}_{\mathcal{T}}{\setminus }^{\ast }{\left[w\right]}_{\mathcal{T}}={\left[u\right]}_{\mathcal{T}}={[\bigvee \left\{dn{f}_{\mathcal{T}}\left(u_i\right)\right|{\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[u_i\right]}_{\mathcal{T}}\le {\left[w\right]}_{\mathcal{T}}\}]}_{\mathcal{T}}$. Assume that ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[w\right]}_{\mathcal{T}}$. Then, there is a $u_i$ such that $u_i=q$. Then, by (refl) and (∨R), ${⊢}_{\mathbf{LG}}q\le u$. Thus, by Lemma 4, ${\left[q\right]}_{\mathcal{T}}\le {\left[u\right]}_{\mathcal{T}}$. Therefore, ${\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[s\right]}_{\mathcal{T}}{\setminus }^{\ast }{\left[w\right]}_{\mathcal{T}}$.

Let us show the opposite direction. Assume that ${\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[s\right]}_{\mathcal{T}}{\setminus }^{\ast }{\left[w\right]}_{\mathcal{T}}$. By Lemma 4, ${⊢}_{\mathbf{LG}}q\le u$ Thus, ${⊢}_{\mathbf{LG}}s·u_i\le w$. So, ${⊢}_{\mathbf{LG}}s·dn{f}_{\mathcal{T}}\left(u_i\right)\le w$. By (∨L), ${⊢}_{\mathbf{LG}}s·u\le w$. Hence, by (trans), ${⊢}_{\mathbf{LG}}s·q\le w$. Hence, ${⊢}_{\mathbf{LG}}r\le w$. So, ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[w\right]}_{\mathcal{T}}$.

Similarly, one proves ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[w\right]}_{\mathcal{T}}$ iff ${\left[s\right]}_{\mathcal{T}}{\le }^{\ast }{\left[w\right]}_{\mathcal{T}}/h^{\ast }{\left[q\right]}_{\mathcal{T}}$. □

Corollary 2. 

For any ${\left[s\right]}_{\mathcal{T}},{\left[q\right]}_{\mathcal{T}},{\left[w\right]}_{\mathcal{T}}\in \left[T^{\ast }\right]$ in $\mathcal{Q}$ according to Definition 27, ${\left[s\right]}_{\mathcal{T}}{\wedge }^{\ast }{\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[w\right]}_{\mathcal{T}}$ iff ${\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[s\right]}_{\mathcal{T}}{\to }^{\ast }{\left[w\right]}_{\mathcal{T}}$.

Lemma 7. 

For any ${\left[s\right]}_{\mathcal{T}},{\left[q\right]}_{\mathcal{T}},{\left[w\right]}_{\mathcal{T}}\in \left[T^{\ast }\right]$, the following hold

• $\left(1\right)$ ${\left[s\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}$ implies $\neg {\left[q\right]}_{\mathcal{T}}{\le }^{\ast }\neg {\left[s\right]}_{\mathcal{T}}$ if $\mathbf{LG}$ contains (QB);
• $\left(2\right)$ ${\left[s\right]}_{\mathcal{T}}{=}^{\ast }{\neg }^{\ast }{\neg }^{\ast }{\left[s\right]}_{\mathcal{T}}$ if $\mathbf{LG}$ contains (QB);
• $\left(3\right)\left({\left[s\right]}_{\mathcal{T}}{\wedge }^{\ast }{\neg }^{\ast }{\left[s\right]}_{\mathcal{T}}\right){\vee }^{\ast }{\left[q\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}{\vee }^{\ast }{\neg }^{\ast }{\left[q\right]}_{\mathcal{T}}$ if $\mathbf{LG}$ contains (Ko);
• $\left(4\right)\left({\left[s\right]}_{\mathcal{T}}{\wedge }^{\ast }{\neg }^{\ast }{\left[s\right]}_{\mathcal{T}}\right){\vee }^{\ast }b{\le }^{\ast }b$ if $\mathbf{LG}$ contains (K);
• $\left(5\right){\left[s\right]}_{\mathcal{T}}{\vee }^{\ast }{\neg }^{\ast }{\left[s\right]}_{\mathcal{T}}{=}^{\ast }{1}^{\ast }\&{\left[s\right]}_{\mathcal{T}}{\wedge }^{\ast }{\neg }^{\ast }{\left[s\right]}_{\mathcal{T}}=0^{\ast }$ if $\mathbf{LG}$ contains (B).

Proof. 

We only provide the details for (1). Others can be proved similarly. Assume that ${\left[s\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}$. By Lemma 4, ${⊢}_{\mathbf{LG}}s\le q$. Then, by (QB), one obtains ${⊢}_{\mathbf{LG}}\neg q\le \neg s$. Again, by Lemma 4, $\neg {\left[q\right]}_{\mathcal{T}}{\le }^{\ast }\neg {\left[s\right]}_{\mathcal{T}}$. □

Lemma 8. 

$\mathcal{Q}$ is a finite iRG where $i\in \{DL,QB,K,Ko,B,H\}$ if $\mathbf{LG}$ contains (i) axioms.

Proof. 

Immediately follows from Lemmas 6 and 7 and Corollary 2. □

Lemma 9. 

The following conditions hold for $\mathcal{Q}$: for any $s,q\in T$,

• $\left(1\right)$ If $s·q\in \mathcal{T}$, then ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}={[s·q]}_{\mathcal{T}}$.
• $\left(2\right)$ If $s\setminus q\in \mathcal{T}$, then ${\left[s\right]}_{\mathcal{T}}{\setminus }^{\ast }{\left[q\right]}_{\mathcal{T}}={[s\setminus q]}_{\mathcal{T}}$.
• $\left(3\right)$ If $s/q\in \mathcal{T}$, then ${\left[s\right]}_{\mathcal{T}}{/}^{\ast }{\left[q\right]}_{\mathcal{T}}={[s/q]}_{\mathcal{T}}$.
• $\left(4\right)$ If $s\to q\in \mathcal{T}$, then ${\left[s\right]}_{\mathcal{T}}{\to }^{\ast }{\left[q\right]}_{\mathcal{T}}={[s\to q]}_{\mathcal{T}}$.

Proof. 

Let us show (1). Let ${\left[s\right]}_{\mathcal{T}}{·}^{\ast }{\left[q\right]}_{\mathcal{T}}={\left[w\right]}_{\mathcal{T}}$. Then, $w{\approx }_{\mathcal{T}}s·q$. Thus, ${⊢}_{\mathbf{LG}}w\le s·q$ and ${⊢}_{\mathbf{LG}}s·q\le w$. By Lemma 4, ${\left[w\right]}_{\mathcal{T}}={[s·q]}_{\mathcal{T}}$.

Let us show (2). Let ${\left[s\right]}_{\mathcal{T}}{\setminus }^{\ast }{\left[q\right]}_{\mathcal{T}}={\left[w\right]}_{\mathcal{T}}$. Let $w=\bigvee \left\{w_i\right|{⊢}_{\mathbf{LG}}s·w_i\le q\}$. By (∨L), one obtains ${⊢}_{\mathbf{LG}}s·w\le q$. Thus, (∖R) ${⊢}_{\mathbf{LG}}w\le s\setminus q$. Furthermore, ${⊢}_{\mathbf{LG}}s·s\setminus \le q$. Thus, there is a $w_i=s\setminus q$ for some i. So, ${⊢}_{\mathbf{LG}}s\setminus q\le w$. □

Lemma 10. 

Let $T^{\ast }$ be set of terms generated from $s,q$ in Definition 22. If ${⊬}_{\mathbf{LG}}s\le q$, then there is ${\mathcal{A}}_{\mathcal{LG}}$ $\mathcal{Q}$ and an assignment σ, satisfying that ${⊭}_{\mathcal{Q}}{\left[s\right]}_{\mathcal{T}}{\le }^{\ast }{\left[q\right]}_{\mathcal{T}}$.

Proof. 

Recall that T is the smallest set containing all the terms in $Sub\left(s\right)\cup Sub\left(q\right)\cup \{0,1\}$. Let $T^{\ast }$ be defined as above. Clearly, $T^{\ast }$ is finitely based. Assume that ${⊬}_{\mathbf{LG}}s\le q$. Construct $\mathcal{Q}$ as in Definition 26 or Definition 27 with respect to $\mathbf{LG}$. Let $\sigma :\mathbf{Var}\left(T\right)\mapsto \left[T^{\ast }\right]$ such that $\sigma \left(p\right)={\left[p\right]}_{\mathcal{T}}$. $\sigma$ can be simply extended to $\hat{\sigma }$ satisfying that $\hat{\sigma }\left(w\right)={\left[w\right]}_{\mathcal{T}}$ for any $w\in T^{\ast }$. Assume that $\widehat{\sigma \left(s\right)}\le \widehat{\sigma \left(q\right)}$. Then, ${\left[s\right]}_{\mathcal{T}}\le {\left[q\right]}_{\mathcal{T}}$. By Lemma 4 and Lemma 16, one obtains ${⊢}_{\mathbf{LG}}s\le q$, which yields contradiction. □

Theorem 4. 

(FMP and Decidability). $\mathsf{LG}$ has FMP and thus is decidable.

Proof. 

Suppose ${⊬}_{\mathsf{LG}}s\le t$ By Lemma 10, there is a finite counter-algebra $\mathcal{Q}$ such that ${⊭}_{\mathcal{Q}}s\le t$. Thus, $⊭{\mathbf{A}}_{\mathbf{LG}}s\Rightarrow t$. □

Remark 2. 

The proof result can be extended to the Galois connection extensions of distributive lattices, De Morgan algebras, Heyting algebras, and Boolean algebras. The Galois operators are a pair of modal operators denoted by $\diamond ,\square$ that usually satisfy the following properties.

• $\diamond a\le b$ iff $a\le \square b$ (gs)

A pair of Galois connection operators are essentially a unary version of fusion · and its residual duals $\setminus ,/$. However, if we consider more complex algebraic structures, such as requiring the modal operators to satisfy transitivity like the 4 axiom, then the construction presented here will not hold, and more sophisticated constructions and analytical proofs will be necessary.

Let $\mathbf{AG}\in \{{\mathbf{GS}}_{\mathbf{CCQBRA}},{\mathbf{GS}}_{\mathbf{CCKRA}},$ ${\mathbf{GS}}_{\mathbf{CCKoRA}},{\mathbf{GS}}_{\mathbf{CCBRA}}\}$. Define T be a set of finite terms containing $\{0,1\}$. Let $c\left(T\right)$ be the $\wedge {\to }_{\#}$ closure of T. Let $T^{\prime }$ be the $\wedge ,\neg$ closure of $c\left(T\right)$ and $T^{?}$ be the set of trees over $T^{\prime }$.

Lemma 11. 

The following hold in $\mathbf{AG}$:

• $\left(1\right)$ ${⊢}_{\mathbf{AG}}s·t_1\le q\&s·t_2\le q\Rightarrow s·t_1\vee t_2\le q$;
• $\left(2\right)$ ${⊢}_{\mathbf{AG}}\Rightarrow s{\to }_{\#}(t\wedge q)=\left(s{\to }_{\#}t\right)\wedge \left(s{\to }_{\#}q\right)$;
• $\left(3\right)$ ${⊢}_{\mathbf{AG}}\Rightarrow s{\to }_{\#}\left(t{\to }_{\#}q\right)=t{\to }_{\#}\left(s{\to }_{\#}q\right)$;
• $\left(4\right)$ ${⊢}_{\mathbf{AG}}\Rightarrow s{\to }_{\#}\left(s{\to }_{\#}t\right)=s{\to }_{\#}t$.

Proof. 

Let us show (1). Let ${⊢}_{\mathbf{AG}}s·t_1\le q$ and ${⊢}_{\mathbf{AG}}s·t_2\le q$. By (${\to }_{\#}$), ${⊢}_{\mathbf{AG}}{t}_i\le s{\to }_{\#}q$. By (trpo), ${⊢}_{\mathbf{AG}}\neg \left(s{\to }_{\#}q\right)\le \neg t_1$ and ${⊢}_{\mathbf{AG}}\neg \left(s{\to }_{\#}q\right)\le \neg t_2$. By (∧R), ${⊢}_{\mathbf{AG}}\neg \left(s{\to }_{\#}q\right)\le \neg t_1\wedge \neg t_2$. By (trpo) (doubn) and (trans), ${⊢}_{\mathbf{AG}}\neg (\neg t_1\wedge \neg t_2)\le s{\to }_{\#}q$. Thus, ${⊢}_{\mathbf{AG}}s·\neg (\neg t_1\wedge \neg t_2)\le q$, whence ${⊢}_{\mathbf{AG}}s·t_1\vee t_2\le q$.

Let us show (2). Obviously, ${⊢}_{\mathbf{AG}}\Rightarrow s\le s$ and ${⊢}_{\mathbf{AG}}t\wedge q\le q$. By (${\to }_{\#}$L), ${⊢}_{\mathbf{AG}}\Rightarrow s·s{\to }_{\#}(t\wedge q)\le q$. Hence, ${⊢}_{\mathbf{AG}}\Rightarrow s{\to }_{\#}(t\wedge q)\le \left(s{\to }_{\#}q\right)$. Similarly, ${⊢}_{\mathbf{AG}}\Rightarrow s{\to }_{\#}(t\wedge q)\le \left(s{\to }_{\#}q\right)$. By (∧R), ${⊢}_{\mathbf{AG}}\Rightarrow s{\to }_{\#}(t\wedge q)\le \left(s{\to }_{\#}t\right)\wedge \left(s{\to }_{\#}q\right)$. For the other direction, clearly ${⊢}_{\mathbf{AG}}\Rightarrow s·s{\to }_{\#}t\le t$ and ${⊢}_{\mathbf{AG}}\Rightarrow s·s{\to }_{\#}q\le q$. By (∧R) ${⊢}_{\mathbf{AG}}\Rightarrow s·\left(s{\to }_{\#}t\right)\wedge \left(s{\to }_{\#}q\right)\le t\wedge q$. Hence, ${⊢}_{\mathbf{AG}}\Rightarrow \left(s{\to }_{\#}t\right)\wedge \left(s{\to }_{\#}q\right)\le s{\to }_{\#}(t\wedge q)$.

Let us show (3). Obviously, ${⊢}_{\mathbf{AG}}q\le q$. Then, by (${\to }_{\#}L$) twice, ${⊢}_{\mathbf{AG}}s·t·s{\to }_{\#}\left(t{\to }_{\#}q\right)\le q$. By (com), one obtains ${⊢}_{\mathbf{AG}}t·s·s{\to }_{\#}\left(t{\to }_{\#}q\right)\le q$. Therefore, by (${\to }_{\#}$R) twice, one obtains ${⊢}_{\mathbf{AG}}\Rightarrow s{\to }_{\#}\left(t{\to }_{\#}q\right)\le t{\to }_{\#}\left(s{\to }_{\#}q\right)$. The opposite direction can be proved similarly.

Let us show (4). Obviously, ${⊢}_{\mathbf{AG}}t\le t$. Then, by (${\to }_{\#}L$) twice, ${⊢}_{\mathbf{AG}}s·s·s{\to }_{\#}\left(s{\to }_{\#}t\right)\le t$. Then, by (con), one obtains ${⊢}_{\mathbf{AG}}s·s{\to }_{\#}\left(s{\to }_{\#}t\right)\le t$. Then, by (${\to }_{\#}$), one obtains ${⊢}_{\mathbf{AG}}s{\to }_{\#}\left(s{\to }_{\#}t\right)\le s{\to }_{\#}t$. For the opposite direction, since ${⊢}_{\mathbf{AG}}t\le t$, by (${\to }_{\#}$) ${⊢}_{\mathbf{AG}}s·s{\to }_{\#}t\le t$. By (con), ${⊢}_{\mathbf{AG}}s·s·s{\to }_{\#}t\le t$. By (${\to }_{\#}$R) twice, ${⊢}_{\mathbf{AG}}s{\to }_{\#}t\le s{\to }_{\#}\left(s{\to }_{\#}t\right)$. □

Lemma 12. 

For any $\overline{t_1},\overline{t_2}\in T^{?}$ and $s\in T^{\prime }$ if ${⊢}_{\mathbf{AG}}\overline{t_1}·\overline{t_2}\le s$, then there is a $w\in c\left(T\right)$ such that ${⊢}_{\mathbf{AG}}\overline{t_1}·\overline{t_2}\le w$ and ${⊢}_{\mathbf{AG}}w\le s$

Proof. 

We proceed the proof by induction on the derivation in $\mathbf{AG}$. Let $\overline{t_1}·\overline{t_2}\le s$ be the ended inequalities obtained by rule (R). Assume that $\overline{t_1}·\overline{t_2}$ is not introduced by (R). Then, the claim holds by induction hypothesis. For instance, suppose that $\overline{t_1}·\overline{t_2}\le s$ is obtained by $\overline{t_1}·\overline{t_2}\le q$ and $q\le s$ by (trans). Then, by induction hypothesis, there is a $w\in c\left(T\right)$ s.t. $\overline{t_1}·\overline{t_2}\le w$ and $w\le q$. By (trans), $w\le s$. Thus, w is the required term for $\overline{t_1}·\overline{t_2}$.

Assume that $\overline{t_1}·\overline{t_2}$ is introduced by (R). If (R) is (·R), then $s=s_1·s_2$ for some $s_1,s_2$. Clearly, $s\in T^{\prime }$. So, $s\in c\left(T\right)$. Hence, s is the required term for $\overline{t_1}·\overline{t_2}$. Otherwise, suppose that $\overline{t_1}·\overline{t_2}\le s$ is obtained by $\overline{t_1}·q\le s$ and $\overline{t_2}\le q$ by (trans). Then, by induction hypothesis, there is a $w\in c\left(T\right)$ s.t. $\overline{t_1}·q\le w$ and $w\le s$. Hence, by (trans), $\overline{t_1}·\overline{t_2}\le w$. So, w is the required term for $\overline{t_1}·\overline{t_2}$. □

According to Lemma 11, by omitting repetitions, $c\left(T\right)$ is finite up to equivalent in $\mathbf{AG}$ when T is finite. ${\le }_{{\mathcal{T}}^{\prime }}$ and ${\approx }_{{\mathcal{T}}^{\prime }}$ are defined similarly as in Definition 24 with respect to $\mathbf{AG}$. ${\left[s\right]}_{{\mathcal{T}}^{\prime }}$ for some $s\in T^{\prime }$; [c(T)] and $\left[T^{\prime }\right]$ are defined naturally. Since quasi-Boolean of lattice is assumed, then $\wedge ,\neg$ closure of terms is finite up to equivalent. Thus, both $\left[c\right(T\left)\right]$ and $\left[T^{\prime }\right]$ are finite.

Lemma 13. 

For any $\overline{t_1}·\overline{t_2}\in T^{?}$, there is $s\in c\left(T\right)$ such that $\overline{t}{\approx }_{\mathcal{T}}s$.

Let ${\mathcal{Q}}^{\prime }=(\left[T^{\prime }\right],{\wedge }^{\prime },{\neg }^{\prime },{·}^{\prime },{\to }_{\#}^{\prime },0^{\prime },1^{\prime })$ be the quotient algebra of $\left[T^{\prime }\right]$ and ${\le }^{\prime }$ defined as in Definition 26.

Lemma 14. 

The following conditions are equivalent for all $s,q\in T^{\ast }$:

• $\left(1\right)$ $s{\le }_{{\mathcal{T}}^{\prime }}q$;
• $\left(2\right)$ ${⊢}_{\mathbf{AG}}s\le q$;
• $\left(3\right)$${\left[s\right]}_{{\mathcal{T}}^{\prime }}{\le }^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}$

The proof is quite similar to Lemma 4. Consequently, all operations in ${\mathcal{Q}}^{\prime }$ are well defined. Define ${\left[s\right]}_{\mathcal{T}}{\vee }^{\prime }{\left[t\right]}_{\mathcal{T}}={\neg }^{\prime }\left({\neg }^{\prime }{\left[s\right]}_{\mathcal{T}}{\wedge }^{\prime }{\neg }^{\prime }{\left[t\right]}_{\mathcal{T}}\right)$.

Lemma 15. 

For any ${\left[s\right]}_{{\mathcal{T}}^{\prime }},{\left[q\right]}_{{\mathcal{T}}^{\prime }},{\left[w\right]}_{{\mathcal{T}}^{\prime }}\in \left[T^{\prime }\right]$ in ${\mathcal{Q}}^{\prime }$, the following hold:

• $\left(1\right)$ ${\left[s\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}{\le }^{\prime }{\left[w\right]}_{{\mathcal{T}}^{\prime }}$ iff ${\left[q\right]}_{{\mathcal{T}}^{\prime }}{\le }^{\prime }{\left[s\right]}_{{\mathcal{T}}^{\prime }}{\to }_{\#}^{\prime }{\left[w\right]}_{{\mathcal{T}}^{\prime }}$;
• $\left(2\right)$ ${\left[s\right]}_{{\mathcal{T}}^{\prime }}{\le }^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}$ implies ${\neg }^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}{\le }^{\prime }{\neg }^{\prime }{\left[s\right]}_{{\mathcal{T}}^{\prime }}$;
• $\left(3\right)$ ${\left[s\right]}_{{\mathcal{T}}^{\prime }}{=}^{\ast }{\neg }^{\prime }{\neg }^{\prime }{\left[s\right]}_{{\mathcal{T}}^{\prime }}$;
• $\left(4\right)\left({\left[s\right]}_{{\mathcal{T}}^{\prime }}{\wedge }^{\prime }{\neg }^{\prime }{\left[s\right]}_{{\mathcal{T}}^{\prime }}\right){\vee }^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}{\le }^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}{\vee }^{\prime }{\neg }^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}$ if $\mathbf{AG}$ contains (Ko);
• $\left(5\right)\left({\left[s\right]}_{\mathcal{T};}{\wedge }^{\prime }{\neg }^{\prime }{\left[s\right]}_{{\mathcal{T}}^{\prime }}\right){\vee }^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}{\le }^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}$ if $\mathbf{AG}$ contains (K);
• $\left(6\right){\left[s\right]}_{{\mathcal{T}}^{\prime }}{\vee }^{\prime }{\neg }^{\prime }{\left[s\right]}_{{\mathcal{T}}^{\prime }}{=}^{\prime }{1}^{\prime }\&{\left[s\right]}_{{\mathcal{T}}^{\prime }}{\wedge }^{\prime }{\neg }^{\prime }{\left[s\right]}_{{\mathcal{T}}^{\prime }}{=}^{\prime }{0}^{\prime }$ if $\mathbf{AG}$ contains (B);
• $\left(7\right)$ $\left({\left[s\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[t\right]}_{{\mathcal{T}}^{\prime }}\right){·}^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}={\left[s\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }\left({\left[t\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}\right)$;
• $\left(8\right)$ ${\left[s\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[t\right]}_{{\mathcal{T}}^{\prime }}={\left[t\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[s\right]}_{{\mathcal{T}}^{\prime }}$;
• $\left(9\right)$ ${\left[s\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[s\right]}_{{\mathcal{T}}^{\prime }}={\left[s\right]}_{{\mathcal{T}}^{\prime }}$.

Proof. 

The proof is quite similar to Lemmas 6 and 7 except (7)–(9). (8) and (9) are easy. We provide the details for (7).

Let $\left({\left[s\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[t\right]}_{{\mathcal{T}}^{\prime }}\right){·}^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}={\left[r_1\right]}_{{\mathcal{T}}^{\prime }}$ and ${\left[s\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }\left({\left[t\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}\right)={\left[r_2\right]}_{{\mathcal{T}}^{\prime }}$. Let ${\left[s\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[t\right]}_{{\mathcal{T}}^{\prime }}={\left[w_1\right]}_{{\mathcal{T}}^{\prime }}$ and ${\left[t\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}={\left[w_2\right]}_{{\mathcal{T}}^{\prime }}$. Hence, ${\left[r_1\right]}_{{\mathcal{T}}^{\prime }}={[w_1·q]}_{{\mathcal{T}}^{\prime }}$ and ${\left[r_2\right]}_{{\mathcal{T}}^{\prime }}={[s·w_2]}_{{\mathcal{T}}^{\prime }}$. It suffices to show ${\left[r_1\right]}_{{\mathcal{T}}^{\prime }}{=}^{\prime }{\left[r_2\right]}_{{\mathcal{T}}^{\prime }}$.

Let ${⊢}_{\mathbf{AG}}{r}_1\le w$. Then, ${⊢}_{\mathbf{AG}}{w}_1·q\le w$. Further, ${⊢}_{\mathbf{AG}}s·t\le w_1$. Thus, by (trans), ${⊢}_{\mathbf{AG}}(s·t)·q\le w$. By ($as{s}_1$), ${⊢}_{\mathbf{AG}}s·(t·q)\le w$. By Lemma, there is $e\in c\left(T\right)$ such that ${⊢}_{\mathbf{AG}}s·(t·q)\le e$ and ${⊢}_{\mathbf{AG}}e\le w$. Then, ${⊢}_{\mathbf{AG}}t·q\le s{\to }_{\#}e$, where $s{\to }_{\#}e\in c\left(T\right)$. Hence, ${⊢}_{\mathbf{AG}}{w}_2\le s{\to }_{\#}e$. Then, ${⊢}_{\mathbf{AG}}s·w_2\le e$. Thus, ${⊢}_{\mathbf{AG}}{r}_2\le e$. By (trans), ${⊢}_{\mathbf{AG}}{r}_2\le w$. By Lemma ${\left[r_1\right]}_{{\mathcal{T}}^{\prime }}{\le }^{\prime }{\left[r_2\right]}_{{\mathcal{T}}^{\prime }}$. Similarly, ${\left[r_2\right]}_{{\mathcal{T}}^{\prime }}{\le }^{\prime }{\left[r_1\right]}_{{\mathcal{T}}^{\prime }}$. □

Lemma 16. 

The following conditions hold for ${\mathcal{Q}}^{\prime }$: for any,

• $\left(1\right)$ If $s·q\in T^{\prime }$, then ${\left[s\right]}_{{\mathcal{T}}^{\prime }}{·}^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}={[s·q]}_{{\mathcal{T}}^{\prime }}$;
• $\left(2\right)$ If $s{\to }_{\#}^{\prime }q\in T^{\prime }$, then ${\left[s\right]}_{{\mathcal{T}}^{\prime }}{\to }_{\#}^{\prime }{\left[q\right]}_{{\mathcal{T}}^{\prime }}={\left[s{\to }_{\#}q\right]}_{{\mathcal{T}}^{\prime }}$.

Lemma 17. 

Let $T^{\prime }$ be set of terms generated from $s,q$. If ${⊬}_{\mathbf{AG}}s\le q$, then there is ${\mathcal{A}}_{\mathcal{AG}}$ ${\mathcal{Q}}^{\prime }$ and an assignment σ that is extended to ${\sigma }^{\prime }$ as above, satisfying ${⊭}_{{\mathcal{Q}}^{\prime }}\sigma \left(s\right){\le }^{\prime }\sigma \left(t\right)$.

Theorem 5 

(FMP and Decidability). $\mathbf{AG}$ has FMP and is thus decidable.

Remark 3. 

In the finiteness proof delineated above, Lemma 11 is contingent on the properties of contraction and commutativity. The demonstration of Lemma 12, which serves as the cornerstone for constructing finite-quotient algebras, inherently relies on the validity of Lemma 11. Consequently, our proof does not extend to scenarios where fusion · fails to satisfy contraction and commutativity but still adheres to associativity.

Moreover, this method does not work for the cases such that fusion · only admits associativity and weakening property too.

5. Conclusions

In this paper, we considered different lattice variants of layered algebras. We defined new graph-theoretical operations on layered graphs, thereby enhancing their expressivity. We demonstrated that any class of algebras belonging to $\{\mathcal{DLRG}, \mathcal{QBRG}, \mathcal{KRG},$ $\mathcal{K}$o$\mathcal{RG}, \mathcal{HRG}, \mathcal{BRG}, \mathcal{CCQBRA}, \mathcal{CCKRA},$ $\mathcal{CCK}$o$\mathcal{RA}, \mathcal{CCBRA}\}$ has FMP and decidable equational theory. We extended the results in [5] to more classes of algebras. We restricted the classes of algebras considered here to be bounded, which leads to a more simple proof. Moreover, by using special interpolation properties (Lemma 12), we expanded the FMP and decidability results to some classes of associative, commutative, and contractive algebras successfully.

Layered graph algebras offer a novel approach to analyzing complex systems that are prevalent in these fields. Specifically, they enable more rigorous verification of key properties, such as security and error resilience. In cryptography, this framework can be used to analyze and identify weaknesses in cryptographic systems that rely on layered security measures. In coding theory, it can help to design new error-correcting codes that are better suited for environments with limited resources or prone to communication disruptions. Essentially, this framework bridges algebraic theory with real-world problems to ensure secure and reliable data transmission.

For further research, we are also interested in the complexity problems closely related to the results presented in this paper. In [22], it is shown that the bounded distributive residuated groupoid (denoted $\mathcal{DLRG}$ here) has an EXPTIME-complete quasi-equational theory and thus the EXPTIME upper bound for the equational theory of $\mathcal{DLRG}$. The same complexity results for quasi-equational theory $\mathcal{BRG}$ are proved in [23], while $\mathcal{BRG}$ has a PSPACE-complete equational theory [24]. Lastly, the complexity of the equational theories of other algebras considered here, to the best of our knowledge, has not been settled.

Moreover, we are interested in exploring extended questions of the finite model property for layered graph algebras over non-distributive lattices. Utilizing logical methods, we can demonstrate the cut-elimination property for the sequent calculus system corresponding to the considered algebra. Based on this result, by proving that the derivation tree is finite, we can establish the decidability of the corresponding logic. Consequently, this allows us to address the decidability problem of equations for the corresponding non-distributive layered graph algebras. However, proving the finite model property is considerably more challenging. In [8], the most basic non-distributive layered graph algebras (under the name non-distributive lattice-ordered residuated groupoid) and some of their negation extensions were shown to possess the finite model property. Nevertheless, the finite model property for many extensions remains an open problem, such as non-distributive layered graph algebras with De Morgan negation, Boolean negation extensions, and others.

The strong finite model property is another issue that we intend to focus on in the future. It is known that this problem is closely related to the word problem for classes of algebras, a topic that has been extensively studied. Moreover, since the classes of algebras we investigate are varieties, they all satisfy the HSP (homomorphic image, subalgebra, and product) property. As indicated in [8], for a class of algebras satisfying this property, the strong finite model property implies that the class has the finite embeddability property. Therefore, the strong finite model property is also a subject of our forthcoming research endeavors.