Vague Congruences and Quotient Lattice Implication Algebras

Abstract

The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given.

1. Introduction

Artificial intelligence (AI) is a newly developed and highly comprehensive frontier science with rapid development, and it mainly studies how to simulate human intelligence behavior by machine. Intelligent information processing is an important direction in the field of AI. Classical mathematical logic is enough to deal with reasoning in traditional mathematical theory, while, in the real world, there are large numbers of problems with uncertainties. To simulate human intelligence behavior better, uncertainty reasoning becomes a key part of computational science and artificial intelligence, and its rationality should be based on foundation of a kind of scientific logic, which is called nonclassical logic [1–3]. Therefore, nonclassical logic has become a considerable formal tool for computer science and artificial intelligence to deal with fuzzy information and uncertain information. Many-valued logic is an extension and development of classical logic and has always been a crucial direction in nonclassical logic.

As an important many-valued logic, lattice-valued logic [4] has two prominent roles. One is to extend the chain-type truth-valued field of the current logics to some relatively general lattices. The other is that the incompletely comparable property of truth value characterized by the general lattice can more effectively reflect the uncertainty of human being’s thinking, judging, and decision. Hence, lattice-valued logic has become an important research field and strongly influenced the development of algebraic logic, computer science, and artificial intelligent technology. In order to provide a reliable logical foundation for uncertain information processing theory, especially for the fuzziness and the incomparability in uncertain information reasoning, Xu [5] combined algebraic lattice and implication algebra, proposed the concept of lattice implication algebras (LIAs for short), and discussed some properties. Since then, this logical algebra has been extensively investigated by several researchers [4–14].

In 1965, Zadeh introduced the concept of fuzzy set [15]. So far, this idea has been applied to other algebraic structures such as groups, semigroups, rings, modules, vector spaces, and topologies and widely used in many fields. Meanwhile, the deficiency of fuzzy sets is also attracting the researchers’ attention. For example, a fuzzy set is a single function, and it cannot express the evidence of supporting and opposing. For this reason, the concept of vague set [16] was introduced in 1993 by Gau and Buehrer. In a vague set 𝐴, there are two membership functions: a truth membership function 𝑡_𝐴 and a false membership function 𝑓_𝐴, where 𝑡_𝐴(𝑥) is a lower bound of the grade of membership of 𝑥 derived from the “evidence for 𝑥” and 𝑓_𝐴(𝑥) is a lower bound on the negation of 𝑥 derived from the “evidence against 𝑥” and 𝑡_𝐴(𝑥)+𝑓_𝐴(𝑥)≤1. Thus, the grade of membership in a vague set 𝐴 is a subinterval [𝑡_𝐴(𝑥),1−𝑓_𝐴(𝑥)] of [0,1]. The idea of vague sets is an extension of fuzzy sets so that the membership of every element can be divided into two aspects including supporting and opposing. In fact, the idea of vague set is the same with the idea of intuitionistic fuzzy set [17]; so, the vague set is equivalent to intuitionistic fuzzy set.

With the development of vague set theory, some structures of algebras corresponding to vague set have been studied. Biswas [18] initiated the study of vague algebras by studying vague groups. Eswarlal [19] studied the vague ideals and normal vague ideals in semirings. Kham et al. [20] studied the vague relation and its properties, and moreover intuitionistic fuzzy filters and intuitionistic fuzzy congruences in a residuated lattice were researched [10, 13, 21–24]. However, it is well known that lattice implication algebra is a special kind of residuated lattice; so, lattice implication algebras have some special properties which are not possessed by a general residuated lattice. Therefore, it is necessary to investigate the vague congruence relation in a lattice implication algebra. Moreover, quotient algebras are a basic tool for exploring the structures of algebras. There are close correlations among filters, congruences, and quotient algebras.

In this paper, we introduce the concept of the vague congruence relation in the lattice implication algebra. In Section 2, we recall some definitions and theorems which will be used in later sections of this paper. In Section 3, we propose the concepts of vague similarity relation based on vague relation and vague congruence relations, and furthermore some properties and equivalent characterizations of vague congruence relation are investigated. Moreover, we show that there is a one-to-one correspondence between the set of all vague filters and all vague congruences of a lattice implication algebra. In Section 4, we construct a new lattice implication algebra induced by vague congruences, and finally the homomorphism theorem is given.

2. Preliminaries

Definition 1 (see [5]). Let (𝐿,∨,∧,𝑂,𝐼) be a bounded lattice with an order-reversing involution ′, the greatest element 𝐼 and the smallest element 𝑂, and let →∶𝐿×𝐿→𝐿 be a mapping. ℒ=(𝐿,∨,∧,′,→,𝑂,𝐼) is called a lattice implication algebra if the following conditions hold for any 𝑥,𝑦,𝑧∈𝐿.(𝐼₁)𝑥→(𝑦→𝑧)=𝑦→(𝑥→𝑧).(𝐼₂)𝑥→𝑥=𝐼.(𝐼₃)𝑥→𝑦=𝑦^′→𝑥^′.(𝐼₄)𝑥→𝑦=𝑦→𝑥=𝐼 implies 𝑥=𝑦.(𝐼₅)(𝑥→𝑦)→𝑦=(𝑦→𝑥)→𝑥.(𝑙₁)(𝑥∨𝑦)→𝑧=(𝑥→𝑧)∧(𝑦→𝑧).(𝑙₂)(𝑥∧𝑦)→𝑧=(𝑥→𝑧)∨(𝑦→𝑧).

In this paper, denote ℒ as a lattice implication algebra (𝐿,∨,∧,′,→,𝑂,𝐼).

In the following, we will list some basic properties of lattice implication algebras, and they are useful for our subsequent work.

Theorem 2 (see [4]). Let ℒ be a lattice implication algebra. Then, for any 𝑥,𝑦,𝑧∈𝐿, the following conclusions hold.(1)If 𝐼→𝑥=𝐼, then 𝑥=𝐼.(2)𝐼→𝑥=𝑥 and 𝑥→𝑂=𝑥^′.(3)𝑂→𝑥=𝐼 and 𝑥→𝐼=𝐼.(4)(𝑥→𝑦)→((𝑦→𝑧)→(𝑥→𝑧))=𝐼.(5)(𝑥→𝑦)∨(𝑦→𝑥)=𝐼.(6)If 𝑥≤𝑦, then 𝑥→𝑧≥𝑦→𝑧 and 𝑧→𝑥≤𝑧→𝑦.(7)𝑥≤𝑦 if and only if 𝑥→𝑦=𝐼.(8)(𝑧→𝑥)→(𝑧→𝑦)=(𝑥∧𝑧)→𝑦=(𝑥→𝑧)→(𝑥→𝑦).(9)𝑥→(𝑦→𝑧)=(𝑥∨𝑦)→𝑧 if and only if 𝑥→(𝑦→𝑧)=𝑥→𝑧=𝑦→𝑧.(10)𝑧≤𝑦→𝑥 if and only if 𝑦≤𝑧→𝑥.

Definition 3 (see [12]). A nonempty subset 𝐹 of a lattice implication algebra ℒ is called a filter of ℒ if it satisfies the following.(F1)𝐼∈𝐹.(F2)(∀𝑥∈𝐹)(∀𝑦∈𝐿)(𝑥→𝑦∈𝐹⇒𝑦∈𝐹).

Definition 4 (see [16]). A vague set 𝐴 in the universe of discourse 𝑋 is characterized by two membership functions given by(1)a truth membership function 𝑡_𝐴∶𝑋→[0,1],(2)a false membership function 𝑓_𝐴∶𝑋→[0,1],where 𝑡_𝐴(𝑥) is a lower bound of the grade of membership of 𝑥 derived from the “evidence for 𝑥” and 𝑓_𝐴(𝑥) is a lower bound on the negation of 𝑥 derived from the “evidence against 𝑥” and 𝑡_𝐴(𝑥)+𝑓_𝐴(𝑥)≤1. Thus, the grade of membership of 𝑥 in the vague set 𝐴 is bounded by a subinterval [𝑡_𝐴(𝑥),1−𝑓_𝐴(𝑥)] of [0,1]. The vague set 𝐴 is written as follows: 𝐴=𝑥,𝑡_𝐴(𝑥),𝑓_𝐴(𝑥)∣𝑥∈𝑋, (1) where the interval [𝑡_𝐴(𝑥),1−𝑓_𝐴(𝑥)] is called the value of 𝑥 in the vague set 𝐴 and denoted by 𝑉_𝐴(𝑥).

Definition 5 (see [16]). A Vague set 𝐴 is contained in the other vague set 𝐵, 𝐴⊆𝐵 if and only if 𝑉_𝐴(𝑥)≤𝑉_𝐵(𝑥); that is, 𝑡_𝐴(𝑥)≤𝑡_𝐵(𝑥) and 1−𝑓_𝐴(𝑥)≤1−𝑓_𝐵(𝑥) for any 𝑥∈𝑋.

Definition 6 (see [16]). Let 𝐴 be a vague set of 𝑋 with the truth membership function 𝑡_𝐴 and the false membership function 𝑓_𝐴. For any 𝛼,𝛽∈[0,1], the (𝛼,𝛽)-cut of the vague set 𝐴 is a crisp subset 𝐴_(𝛼,𝛽) of the set 𝑋 by 𝐴_(𝛼,𝛽)=𝑥∈𝑋∣𝑉_𝐴(𝑥)≥𝛼,𝛽. (2) Obviously, 𝐴_(0,0)=𝑋.

Definition 7 (see [16]). Let 𝐴 be a vague set of 𝑋 with the truth membership function 𝑡_𝐴 and the false membership function 𝑓_𝐴. For any 𝛼∈[0,1], the (𝛼,𝛼)-cut of the vague set 𝐴 is a crisp subset 𝐴_𝛼=𝐴_(𝛼,𝛼).

Let 𝐴 be a vague set of 𝑋, 𝐴₀=𝑋; and if 𝛼≤𝛽, then 𝐴_𝛼⊆𝐴_𝛽 and 𝐴_(𝛼,𝛽)=𝐴_𝛼. Equivalently, we can define the 𝛼-cut 𝐴_𝛼 of 𝐴, 𝐴_𝛼={𝑥∈𝑋∣𝑡_𝐴(𝑥)≥𝛼}.

Let 𝐴 and 𝐵 be two vague sets of 𝑋; then, 𝐴=𝐵 if and only if 𝑉_𝐴(𝑥)=𝑉_𝐵(𝑥) for any 𝑥∈𝑋.

Let 𝐼[0,1] denote the family of all closed subintervals of [0,1]. If 𝐼₁=[𝑎₁,𝑏₁] and 𝐼₂=[𝑎₂,𝑏₂] are two elements of 𝐼[0,1], we call 𝐼₁≥𝐼₂ if 𝑎₁≥𝑎₂ and 𝑏₁≥𝑏₂. We define the term imax to mean the maximum of two intervals as follows: imax𝐼₁,𝐼₂=max𝑎₁,𝑎₂,max𝑏₁,𝑏₂. (3) Similarly, we can define the term imin of any two intervals.

Definition 8 (see [20]). Let 𝑋 and 𝑌 be two universes. A vague relation of the universe 𝑋 with the universe 𝑌 is a vague set of the Cartesian product 𝑋×𝑌.

Definition 9 (see [20]). Let 𝑋 and 𝑌 be two universes. A vague subset 𝑅 of discourse 𝑋×𝑌 is characterized by two membership functions given by the following:(1)a truth membership function 𝑡_𝑅∶𝑋×𝑌→[0,1],(2)a false membership function 𝑓_𝑅∶𝑋×𝑌→[0,1],where 𝑡_𝑅(𝑥,𝑦) is a lower bound of the grade of membership of (𝑥,𝑦) derived from the “evidence for (𝑥,𝑦)” and 𝑓_𝑅(𝑥,𝑦) is a lower bound on the negation of (𝑥,𝑦) derived from the “evidence against (𝑥,𝑦)” and 𝑡_𝑅(𝑥,𝑦)+𝑓_𝑅(𝑥,𝑦)≤1. Thus, the grade of membership of (𝑥,𝑦) in the vague set 𝑅 is bounded by a subinterval [𝑡_𝑅(𝑥,𝑦),1−𝑓_𝑅(𝑥,𝑦)] of [0,1]. The vague relation 𝑅 is written as follows: 𝑅=𝑥,𝑦,𝑡_𝑅𝑥,𝑦,𝑓_𝑅𝑥,𝑦∣𝑥,𝑦∈𝑋×𝑌, (4) where the interval [𝑡_𝑅(𝑥,𝑦),1−𝑓_𝑅(𝑥,𝑦)] is called the value of (𝑥,𝑦) in the vague relation 𝑅 and denoted by 𝑉_𝑅(𝑥,𝑦).

3. Equivalent Characterizations of Vague Congruence Relation

From Definitions 4 and 8, we can obtain the equivalent definition of vague relation.

Definition 10. Let 𝑋 be nonempty universe; the vague relation 𝑅 on 𝑋 is called vague similarity relation if 𝑅 satisfies the following conditions:(1)∀𝑥∈𝑋, 𝑉_𝑅(𝑥,𝑥)=sup_{𝑢,𝑣∈𝑋}𝑉_𝑅(𝑢,𝑣) (vague reflexivity);(2)∀𝑥,𝑦∈𝑋, 𝑉_𝑅(𝑥,𝑦)=𝑉_𝑅(𝑦,𝑥) (vague symmetric);(3)∀𝑥,𝑦,𝑧∈𝑋, imin{𝑉_𝑅(𝑥,𝑧),𝑉_𝑅(𝑧,𝑦)}≤𝑉_𝑅(𝑥,𝑦) (vague transitivity).

Remark 11. For the vague transitivity, ∀𝑥,𝑦,𝑧∈𝑋imin𝑉_𝑅(𝑥,𝑧),𝑉_𝑅𝑧,𝑦≤𝑉_𝑅⟺𝑥,𝑦∀𝑥,𝑦∈𝑋sup_𝑧∈𝑋imin𝑉_𝑅(𝑥,𝑧),𝑉_𝑅𝑧,𝑦≤𝑉_𝑅𝑥,𝑦. (5)

Definition 12. Let 𝑅 be a vague relation on 𝑋, and 𝜆=[𝜆₁,𝜆₂]∈𝐼[0,1], where 𝜆₁,𝜆₂∈[0,1] and 𝜆₁≤𝜆₂. If 𝑉_𝑅(𝑥,𝑦)≥𝜆 holds for any 𝑥,𝑦∈𝑋, then we call the set 𝑅_𝜆={(𝑥,𝑦)∈𝑋×𝑋∣𝑉_𝑅(𝑥,𝑦)≥𝜆} is 𝜆-level relation of 𝑅.

From Definition 12, any level relation 𝑅_𝜆 of vague relation 𝑅 on 𝑋 is a relation on 𝑋, whose characteristic function 𝜒_𝑅𝜆 is as follows: 𝜒_𝑅𝜆𝑥,𝑦=1,if𝑥,𝑦∈𝑅_𝜆,0,if𝑥,𝑦∉𝑅_𝜆. (6)

Theorem 13. Let 𝑅 be a vague relation on universe 𝑋; then, 𝑅 is vague similarity relation if and only if 𝑅_𝜆 is an equivalent relation on 𝑋, where 𝜆 is a close subinterval of [0,1].

Definition 14. Let 𝑅 be a vague relation on ℒ. 𝑅 is said to be a vague congruence relation on ℒ, if(1)𝑅 is a vague similarity relation on ℒ;(2)𝑉_𝑅(𝑥→𝑧,𝑦→𝑧)≥𝑉_𝑅(𝑥,𝑦) for any 𝑥,𝑦,𝑧∈𝐿.

Let 𝑅 be a vague relation on ℒ, for all close subintervals 𝑡=[𝑡₁,𝑡₂]∈𝐼[0,1]; the 𝑡-level relation and strong 𝑡-level subset of 𝐿, respectively, are defined as follows: 𝑅_𝑡=𝑥,𝑦∈𝐿×𝐿∣𝑉_𝑅,𝑅𝑥,𝑦≥𝑡_𝑡>=𝑥,𝑦∈𝐿×𝐿∣𝑉_𝑅𝑥,𝑦>𝑡. (7)

Definition 15. Let 𝑅 be a vague relation on nonempty set ℒ. 𝑅 satisfies the sup property if, for every subset 𝑇 of ℒ, there exists 𝑢,𝑣∈𝑇 such that sup_{𝑥,𝑦∈𝑇}𝑉_𝑅(𝑥,𝑦)=𝑉_𝑅(𝑢,𝑣).

Lemma 16. Let 𝑅 be a vague relation on ℒ; then, 𝑅_𝑡=⋂_{[0,0]≤𝑠<𝑡}𝑅_𝑡> and 𝑅_𝑡>=⋃_{𝑡<𝑠≤[1,1]}𝑅_𝑠, where 𝑡,𝑠∈𝐼[0,1].

Proof. Let 𝑅 be a vague relation on ℒ, 𝑡∈𝐼[0,1]; we have 𝑅_𝑡={(𝑥,𝑦)∈𝐿×𝐿∣𝑉_𝑅(𝑥,𝑦)≥𝑡}=⋂_{[0,0]≤𝑠<𝑡}{(𝑥,𝑦)∈𝐿×𝐿∣𝑉_𝑅(𝑥,𝑦)>𝑠}=⋂_{[0,0]≤𝑠<𝑡}𝑅_𝑡>. 𝑅_𝑡>={(𝑥,𝑦)∈𝐿×𝐿∣𝑉_𝑅(𝑥,𝑦)>𝑡}=⋃_{𝑡<𝑠≤[1,1]}{(𝑥,𝑦)∈𝐿×𝐿∣𝑉_𝑅(𝑥,𝑦)≥𝑠}=⋃_{𝑡<𝑠≤[1,1]}𝑅_𝑡.

Theorem 17. Let 𝑅 be a vague relation on ℒ that satisfies the sup property. Then, the following statements are equivalent:(1)𝑅 is a vague similarity relation on ℒ;(2)𝑅_𝑡(≠∅) is an equivalence relation on ℒ, for all 𝑡∈𝐼[0,1];(3)𝑅_𝑡>(≠∅) is an equivalence relation on ℒ, for all [0,0]≤𝑡<[1,1].

Proof. ((1)⇒(2)) It can be easily proved.
((2)⇒(3)) Let 𝑅_𝑡>≠∅, for all [0,0]≤𝑡<[1,1]. It follows by Lemma 16 that there exist 𝑠∈𝐼[0,1] and 𝑡<𝑠≤[1,1] such that 𝑅_𝑠≠∅. Since 𝑅_𝑠 is an equivalent relation, 𝑅_𝑠 is reflexive; that is, (𝑥,𝑥)∈𝑅_𝑠⊆𝑅_𝑡>. It follows that 𝑅_𝑡> is reflexive. Similarly, we can prove 𝑅_𝑡> is symmetric.
Now, we need to prove 𝑅_𝑡> is transitive. Suppose (𝑥,𝑦)∈𝑅_𝑡>, (𝑦,𝑧)∈𝑅_𝑡>; then, there exist 𝑠₁,𝑠₂∈𝐼[0,1] and 𝑡<𝑠₁≤[1,1], 𝑡<𝑠₂≤[1,1] such that (𝑥,𝑦)∈𝑅_𝑠1, (𝑦,𝑧)∈𝑅_𝑠2. Taking 𝛼=imin{𝑠₁,𝑠₂}, then 𝛼∈𝐼[0,1] and 𝑡<𝛼≤[1,1]. Therefore, (𝑥,𝑦)∈𝑅_𝑠1⊆𝑅_𝛼, (𝑦,𝑧)∈𝑅_𝑠2⊆𝑅_𝛼. By (2), it follows that 𝑅_𝛼 is an equivalent relation, then (𝑥,𝑧)∈𝑅_𝛼⊆𝑅_𝑡>. Hence, 𝑅_𝑡> is transitive. And so 𝑅_𝑡> is an equivalence relation on ℒ, for all [0,0]≤𝑡<[1,1].
((3)⇒(1)) Let sup_{𝑢,𝑣∈𝐿}(𝑉_𝑅(𝑢,𝑣))=𝑡∈𝐼[0,1]. Since 𝑅 satisfies sup property, then there exists 𝑎,𝑏∈𝐿 such that 𝑉_𝑅(𝑎,𝑏)=𝑡, which implies that 𝑅_𝑡≠∅; of course, 𝑅_𝑠>≠∅ for any 𝑠∈𝐼[0,1] and [0,0]≤𝑠<𝑡. Therefore, (𝑥,𝑥)∈𝑅_𝑠>, and so (𝑥,𝑥)∈⋂_{[0,0]≤𝑠<𝑡}=𝑅_𝑡. We have 𝑉_𝑅(𝑥,𝑥)≥𝑡≥𝑉_𝑅(𝑥,𝑥). Therefore, 𝑉_𝑅(𝑥,𝑥)=sup_{𝑢,𝑣∈𝐿}(𝑉_𝑅(𝑢,𝑣))=𝑡 which shows 𝑅_𝑡> is reflexive. Similarly, we can show that 𝑅_𝑡> is symmetric. Now, we show 𝑅_𝑡> is transitive. Let 𝑡=imin{𝑉_𝑅(𝑥,𝑧),𝑉_𝑅(𝑧,𝑦)}, for any 𝑥,𝑦∈𝐿 and some 𝑧∈𝐿; this implies that 𝑉_𝑅(𝑥,𝑧)≥𝑡, 𝑉_𝑅(𝑧,𝑦)≥𝑡. Therefore, (𝑥,𝑧),(𝑧,𝑦)∈𝑅_𝑡. By Lemma 16, we have (𝑥,𝑧),(𝑧,𝑦)∈𝑅_𝑠> for any [0,0]≤𝑠<𝑡; it follows from (3) that (𝑥,𝑦)∈𝑅_𝑠> for any [0,0]≤𝑠<𝑡. Hence, 𝑉_𝑅(𝑥,𝑦)≥𝑡=imin{𝑉_𝑅(𝑥,𝑧),𝑉_𝑅(𝑧,𝑦)}; so, (1) holds.

Theorem 18. Let 𝑅 be a vague relation on ℒ that satisfies the sup property. Then, the following statements are equivalent:(1)𝑅 is a vague congruence relation on ℒ;(2)𝑅_𝑡(≠∅) is a congruence relation on ℒ, for all 𝑡∈𝐼[0,1];(3)𝑅_𝑡>(≠∅) is a congruence relation on ℒ, for all [0,0]≤𝑡<[1,1].

Proof. (1)⇒(2) Let 𝑅_𝑡≠𝜙, for all 𝑡∈𝐼[0,1]. Since 𝑅 is a vague congruence relation ℒ, then 𝑅_𝑡≠𝜙 is an equivalence relation. Let (𝑥,𝑦)∈𝑅_𝑡; then, 𝑉_𝑅(𝑥,𝑦)≥𝑡. For any 𝑧∈𝐿, since 𝑅 is a vague congruence relation on ℒ, then 𝑉_𝑅(𝑥→𝑧,𝑦→𝑧)≥𝑉_𝑅(𝑥,𝑦)≥𝑡. That is, (𝑥→𝑧,𝑦→𝑧)∈𝑅_𝑡. Hence, 𝑅_𝑡≠𝜙 is a congruence, for all 𝑡∈𝐼[0,1].
(2)⇒(3) Let 𝑅_𝑡>≠𝜙, for 𝑡∈𝐼[0,1). By Theorem 13, we know that 𝑅_𝑡> is an equivalence relation on ℒ. Let (𝑥,𝑦)∈𝑅_𝑡; then there exist 𝑠∈𝐼[0,1] and 𝑡<𝑠≤[1,1] such that (𝑥,𝑦)∈𝑅_𝑠 by Lemma 16. Since 𝑅_𝑠 is a congruence relation on ℒ, then (𝑥→𝑧,𝑦→𝑧)∈𝑅_𝑠⊆𝑅_𝑡>. That is, (𝑥→𝑧,𝑦→𝑧)∈𝑅_𝑡>. Hence, 𝑅_𝑡> is a congruence on ℒ for 𝑡∈𝐼[0,1).
(3)⇒(1) We know that 𝑅 is a vague similarity relation on ℒ from Lemma 16.
Suppose that 𝑥,𝑦,𝑧∈𝐿 and 𝑉_𝑅(𝑥,𝑦)=𝑡, then (𝑥,𝑦)∈𝑅_𝑡=⋂_{[0,0]≤𝑠<𝑡}𝜌_𝑠> by Lemma 16. Since 𝑅_𝑠> is a congruence relation on ℒ, then (𝑥→𝑧,𝑦→𝑧)∈𝑅_𝑠>. Thus, (𝑥→𝑧,𝑦→𝑧)∈⋂_{[0,0]≤𝑠<𝑡}𝑅_𝑠>=𝑅_𝑡. That is, 𝑉_𝑅𝑡(𝑥→𝑧,𝑦→𝑧)≥𝑡=𝑉_𝑅(𝑥,𝑦). Hence, 𝑅 is a vague congruence relation on ℒ.

4. Quotient LIAs and Homomorphism Theorem

Let 𝑅 be a vague similarity relation on ℒ. For each 𝑎∈𝐿, we define a vague subset 𝑅^𝑎={(𝑥,[𝑡_𝑅^𝑎(𝑥),𝑓_𝑅𝑎(𝑥)]∣𝑥∈𝐿} of 𝐿, where 𝑉_𝑅^𝑎(𝑥)=𝑉_𝑅(𝑎,𝑥). for all 𝑥∈𝐿.

Proposition 19. Let 𝑅 be a vague congruence on ℒ; then, 𝑅^𝐼 is a vague filter of ℒ.

Proposition 20. Let 𝑅 be a vague congruence on ℒ. For any 𝑎,𝑏∈𝐿, we have 𝑅^𝑎=𝑅^𝑏𝑖𝑓𝑓𝑉_𝑅(𝑎,𝑏)=sup_{𝑦,𝑧∈𝐿}𝑉_𝑅𝑦,𝑧, (8)

Proof. Let 𝑅^𝑎=𝑅^𝑏; then, 𝑉_𝑅^𝑎=𝑉_𝑅^𝑏. Since 𝑅 is a vague congruence on ℒ, then 𝑉_𝑅(𝑎,𝑏)=𝑉_𝑅^𝑎(𝑏)=𝑉_𝑅^𝑏(𝑏)=𝑉_𝑅(𝑏,𝑏)=sup_{𝑦,𝑧∈𝐿}𝜌𝑦,𝑧. (9)
Conversely, assume 𝑉_𝑅(𝑎,𝑏)=sup_{𝑦,𝑧∈𝐿}𝜌(𝑦,𝑧). For any 𝑥∈𝐿, since 𝑉_𝑅^𝑎(𝑥)=𝑉_𝑅(𝑎,𝑥)≥sup_𝑚∈𝐿imin{𝑉_𝑅(𝑎,𝑚),𝑉_𝑅(𝑚,𝑥)}≥imin{𝑉_𝑅(𝑎,𝑏),𝑉_𝑅(𝑏,𝑥)}=𝑉_𝑅(𝑏,𝑥)=𝑉_𝑅^𝑏(𝑥), then 𝑉_𝑅^𝑎⊇𝑉_𝑅^𝑏. Similarly, we have 𝑉_𝑅^𝑎⊆𝑉_𝑅^𝑏. Hence, 𝑉_𝑅^𝑎=𝑉_𝑅^𝑏, and so 𝑅^𝑎=𝑅^𝑏.

Definition 21. Let 𝑅 be a vague congruence on a lattice implication algebra ℒ and let 𝑎∈𝐿. The vague subset 𝑅^𝑎 of ℒ is called a vague congruence class of 𝑅 containing 𝑎∈𝐿.

Denote 𝐿𝑅=𝑅^𝑎𝑎∈𝐿, (10) which is the set of vague congruence classes of ℒ w.r.t. 𝑅.

In 𝐿/𝑅, we define the operations as follows: 𝑅^𝑎⟶𝑅^𝑏=𝑅^𝑎→𝑏;𝑅^𝑎∪𝑅^𝑏=𝑅^𝑎∨𝑏;𝑅^𝑎∩𝑅^𝑏=𝑅^𝑎∧𝑏;𝑅^𝑎′=𝑅^𝑎′. (11)

Proposition 22. Let ℒ be a lattice implication algebra. The operations →,∪,∩,′ on 𝐿/𝑅 are well defined.

Proof. Suppose that 𝑅^𝑎=𝑅^𝑏 and 𝑅^𝑐=𝑅^𝑑. Then, we have that 𝑉_𝑅(𝑎,𝑏)=𝑉_𝑅(𝑐,𝑑)=sup_{𝑦,𝑧∈𝐿}𝑉_𝑅(𝑦,𝑧) by Proposition 20.
(1) By Theorem 18, we have 𝑉_𝑅(𝑎→𝑐, 𝑏→𝑑)≥imin{𝑉_𝑅(𝑎,𝑏),𝑉_𝑅(𝑐,𝑑)}=sup_{𝑦,𝑧∈𝐿}𝑉_𝑅(𝑦,𝑧). Obviously, 𝑉_𝑅(𝑎→𝑐, 𝑏→𝑑)≤sup_{𝑦,𝑧∈𝐿}𝑉_𝑅(𝑦,𝑧). Then, 𝑉_𝑅(𝑎→𝑐, 𝑏→𝑑)=sup_{𝑦,𝑧∈𝐿}𝑉_𝑅(𝑦,𝑧).
Hence, 𝑅^𝑎→𝑐=𝑅^𝑏→𝑑 by Proposition 20. By definition of the operation → on 𝐿/𝑅, we have 𝑅^𝑎→𝑅^𝑐=𝑅^𝑎→𝑐=𝑅^𝑏→𝑑=𝑅^𝑏→𝑅^𝑑.
Hence, the operation → is well defined.
(2) Since 𝑉_𝑅(𝑎∨𝑐,𝑏∨𝑑)≥sup_𝑥∈𝐿imin𝑉_𝑅(𝑎∨𝑐,𝑥),𝑉_𝑅(𝑉𝑥,𝑏∨𝑑)≥imin_𝑅(𝑎∨𝑐,𝑏∨𝑐),𝑉_𝑅(𝑉𝑏∨𝑐,𝑏∨𝑑)≥imin_𝑅(𝑎,𝑏),𝑉_𝑅(𝑐,𝑑)=sup_{𝑦,𝑧∈𝐿}𝑉_𝑅𝑦,𝑧, (12) then 𝑉_𝑅(𝑎∨𝑐,𝑏∨𝑑)≥sup_{𝑦,𝑧∈𝐿}𝑉_𝑅(𝑦,𝑧). Obviously, 𝑉_𝑅(𝑎∨𝑐,𝑏∨𝑑)≤sup_{𝑦,𝑧∈𝐿}𝑉_𝑅(𝑦,𝑧). Hence, 𝑉_𝑅(𝑎∨𝑐,𝑏∨𝑑)=sup_{𝑦,𝑧∈𝐿}𝑉_𝑅(𝑦,𝑧). That is, 𝑅^𝑎∨𝑐=𝑅^𝑏∨𝑑.
By Proposition 20 and the definition of the operation ∪ on 𝐿/𝑅, we have 𝑅^𝑎∪𝑅^𝑐=𝑅^𝑎∨𝑐=𝑅^𝑏∨𝑑=𝑅^𝑏∪𝑅^𝑑. That is, ∪ is well defined.
(3) Similar to (2).
(4) Since 𝑅^𝑎=𝑅^𝑏, then 𝑉_𝑅(𝑎,𝑏)=sup_{𝑦,𝑧∈𝐿}𝑉_𝑅(𝑦,𝑧). As 𝑉_𝑅(𝑎,𝑏)=𝑉_𝑅(𝑎^′,𝑏^′)=sup_{𝑦,𝑧∈𝐿}𝑉_𝑅(𝑦,𝑧), so 𝑅^𝑎′=𝑅^𝑏′. By the definition of ′, we have (𝑅^𝑎)^′=(𝑅^𝑏)^′.
Hence, the ′ is well defined.

Theorem 23. Let 𝑅 be a vague congruence on a lattice implication algebra ℒ. Then, (𝐿/𝑅,∪,∩,′,→) is a lattice implication algebra.

Proof. By Proposition 22, we know that the operations →,∪,∩,′ on 𝐿/𝑅 are well defined.
First, we prove that 𝐿/𝑅 is a complemented lattice with universal bounds 𝑂, 𝐼.
In fact, for the operation ∪, there exists 𝑅^𝑎∪𝑅^𝑏=𝑅^𝑎∨𝑏=𝑅^𝑏∨𝑎=𝑅^𝑏∪𝑅^𝑎. That means the law of commutativity for ∪ holds. Similarly, the associative and idempotent laws hold. Furthermore, we can get the commutative, associative, and idempotent laws for ∩ and the absorptive law for ∪ and ∩. Hence, (𝐿/𝑅,∪,∩) is a lattice with the partial order ⊆ as follows: 𝑅^𝑎⊆𝑅^𝑏iff𝑅^𝑎⟶𝑅^𝑏=𝑅^𝐼iff𝑎⟶𝑏=𝐼iff𝑎≤𝑏. (13) If 𝑅^𝑎⊆𝑅^𝑏, then 𝑎≤𝑏. Hence, 𝑏^′≤𝑎^′; that is, 𝑅^𝑏′⊆𝑅^𝑎′. Thus, (𝑅^𝑏)^′⊆(𝑅^𝑎)^′. Moreover, because ((𝑅^𝑎)^′)^′=(𝑅^𝑎′)^′=𝑅^(𝑎′)′=𝑅^𝑎, then ′ is an order-reversing involution on 𝐿/𝑅. The universal bounds 𝑂=𝑅^𝑂, 𝐼=𝑅^𝐼.
Sum up the above, (𝐿/𝑅,∪,∩,′) is a complemented lattice.
Second, we need to prove (𝐼₁)–(𝐼₅), (𝑙₁), (𝑙₂) hold for 𝐿/𝑅. Here, we only prove (𝑙₁); the others can be proved similarly. In fact, there exists 𝑅^𝑎∪𝑅^𝑏⟶𝑅^𝑐=𝑅^{(𝑎∨𝑏)→𝑐}=𝑅^{(𝑎→𝑐)∧(𝑏→𝑐)}=𝑅^𝑎→𝑐∩𝑅^𝑏→𝑐=𝑅^𝑎⟶𝑅^𝑐∩𝑅^𝑏⟶𝑅^𝑐. (14) This means (𝑙₁) holds.
Hence, (𝐿/𝑅,∪,∩,′,→) is a lattice implication algebra.

Proposition 24. Let 𝑅 be a vague congruence relation on a lattice implication algebra ℒ. The mapping 𝐿𝑓∶𝐿⟶𝑅 (15) defined by, for 𝑎∈𝐿, 𝑓(𝑎)=𝑅^𝑎 (16) is a lattice implication homomorphism.

Proof. Obviously, 𝑓 is well defined.
For any 𝑥,𝑦∈𝐿, 𝑓(𝑥→𝑦)=𝑅^𝑥→𝑦=𝑅^𝑥→𝑅^𝑦=𝑓(𝑥)→𝑓(𝑦). So, 𝑓 is an implication homomorphism. As 𝑓(𝑥^′)=𝑅^𝑥′=(𝑅^𝑥)^′=(𝑓(𝑥))^′, 𝑓 is a lattice implication homomorphism.

Lemma 25. Let 𝐴 be a vague filter of ℒ and let 𝑅_𝐴 be a vague congruence relation induced by 𝐴. Then, (𝑅_𝐴)^𝑎=(𝑅_𝐴)^𝑏 if and only if 𝑡_𝐴(𝑎→𝑏)=𝑡_𝐴(𝑏→𝑎)=𝑡_𝐴(𝐼) and 𝑓_𝐴(𝑎→𝑏)=𝑓_𝐴(𝑏→𝑎)=𝑓_𝐴(𝐼).

Proof. Let (𝑅_𝐴)^𝑎=(𝑅_𝐴)^𝑏; we have 𝑉_(𝑅𝐴)^𝑎(𝑎)=𝑉_(𝑅𝐴)^𝑏(𝑎), and so 𝑉_(𝑅𝐴)^𝑎(𝑎)=imin𝑉_𝐴(𝑎⟶𝑎),𝑉_𝐴(=𝑡𝑎⟶𝑎)_𝐴(𝐼),1−𝑓_𝐴(𝐼)=𝑉_(𝑅𝐴)^𝑏(𝑎)=imin𝑉_𝐴(𝑏⟶𝑎),𝑉_𝐴(=𝑡𝑎⟶𝑏)min_𝐴(𝑎⟶𝑏),𝑡_𝐴(𝑏⟶𝑎),min1−𝑓_𝐴(𝑎⟶𝑏),1−𝑓_𝐴(𝑏⟶𝑎). (17) Hence, min𝑡_𝐴(𝑎⟶𝑏),𝑡_𝐴(𝑏⟶𝑎)=𝑡_𝐴(𝐼),min1−𝑓_𝐴(𝑎⟶𝑏),1−𝑓_𝐴(𝑏⟶𝑎)=1−𝑓_𝐴(𝐼). (18) It follows that 𝑡_𝐴(𝑎⟶𝑏)=𝑡_𝐴(𝑏⟶𝑎)=𝑡_𝐴(𝐼),𝑓_𝐴(𝑎⟶𝑏)=𝑓_𝐴(𝑏⟶𝑎)=𝑓_𝐴(𝐼). (19)
Conversely, suppose that 𝑡_𝐴(𝑎⟶𝑏)=𝑡_𝐴(𝑏⟶𝑎)=𝑡_𝐴(𝐼),𝑓_𝐴(𝑎⟶𝑏)=𝑓_𝐴(𝑏⟶𝑎)=𝑓_𝐴(𝐼).
For any 𝑥∈𝐿, 𝑉_𝑅𝐴^𝑎(𝑥)=imin𝑉_𝐴(𝑎⟶𝑥),𝑉_𝐴(𝑉𝑥⟶𝑎)≥iminimin_𝐴(𝑎⟶𝑏),𝑉_𝐴(,𝑉𝑏⟶𝑥)imin_𝐴(𝑥⟶𝑏),𝑉_𝐴(𝑡𝑏⟶𝑎)≥iminmin_𝐴(𝑎⟶𝑏),𝑡_𝐴(𝑏⟶𝑥),min1−𝑓_𝐴(𝑎⟶𝑏),1−𝑓_𝐴(,𝑡𝑏⟶𝑥)min_𝐴(𝑥⟶𝑏),𝑡_𝐴(𝑏⟶𝑎),min1−𝑓_𝐴(𝑥⟶𝑏),1−𝑓_𝐴(𝑡𝑏⟶𝑎)=imin_𝐴(𝑏⟶𝑥),1−𝑓_𝐴(,𝑡𝑏⟶𝑥)_𝐴(𝑥⟶𝑏),1−𝑓_𝐴(𝑉𝑥⟶𝑏)=imin_𝐴(𝑏⟶𝑥),𝑉_𝐴(𝑥⟶𝑏)=𝑉_𝑅𝐴^𝑏(𝑥). (20) Similarly, we can show that 𝑉_𝑅𝐴^𝑏(𝑥)≥𝑉_𝑅𝐴^𝑎(𝑥). Therefore, 𝑉_𝑅𝐴^𝑏(𝑥)=𝑉_𝑅𝐴^𝑎(𝑥); hence, (𝑅_𝐴)^𝑎=(𝑅_𝐴)^𝑏.

Theorem 26. Let 𝐴 be a vague filter of ℒ. Define 𝑎≡_𝐴𝑏⟺𝑅_𝐴^𝑎=𝑅_𝐴^𝑏. (21) Then, ≡_𝐴 is a congruence relation on 𝐿.

Proof. The proof can be completed easily from Lemma 25.

Theorem 27. Let 𝐴 be a vague filter of ℒ and let 𝐿/𝑅_𝐴 be the corresponding quotient algebra. Then, the map 𝑓∶𝐿→𝐿/𝑅_𝐴 defined by 𝑓(𝑎)=(𝑅_𝐴)^𝑎, for any 𝑎∈𝐿, is a lattice implication homomorphism and 𝐷-ker(𝑓)=𝑈(𝑡_𝐴;𝑡_𝐴(𝐼)∩𝐿(𝑓_𝐴;𝑓_𝐴(𝐼)), where 𝐷-ker(𝑓)={𝑥∈𝐿∣𝑓(𝑥)=(𝑅_𝐴)^𝐼}.

Proof. The first result is obvious from Proposition 24. We only need to prove the second result, that is, 𝐷-ker𝑓=𝑈𝑡_𝐴;𝑡_𝐴(𝐼)∩𝐿𝑓_𝐴;𝑓_𝐴(𝐼). (22) In fact, 𝑓⟺𝑅𝑥∈𝐷-ker_𝐴𝑥=𝑓(𝑥)=𝑅_𝐴^𝐼⟺𝑡_𝐴(𝑥→𝐼)=𝑡_𝐴(𝐼→𝑥)=𝑡_𝐴(𝐼),𝑓_𝐴(𝑥→𝐼)=𝑓_𝐴(𝐼→𝑥)=𝑓_𝐴(𝐼)⟺𝑡_𝐴(𝑥)=𝑡_𝐴(𝐼),𝑓_𝐴(𝑥)=𝑓_𝐴(𝐼)⟺𝑥∈𝑈𝑡_𝐴;𝑡_𝐴(𝐼)∩𝐿𝑓_𝐴;𝑓_𝐴(𝐼). (23)

Corollary 28. Let 𝐴 be a vague filter of ℒ; then, 𝐿/𝑅_𝐴≅𝐿/≡_𝐴.

5. Conclusion

Congruence theory plays a very important role in the research of a logic algebra, through which we can obtain much information such as the internal structure and homomorphism image of the logic algebra and furthermore induce some new algebra with more simple structure. The aim of this paper is to develop the congruence relation on lattice implication algebras. First, the concept of vague similarity relation is introduced based on vague relation, and the concept of vague congruence relations is proposed, and properties and equivalent characterizations of vague congruence relation are investigated. Secondly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by vague congruences, and the homomorphism theorem is given. The methods in this paper can be used in 𝑀𝑉-algebras, BL-algebras, MTL-algebras, and so forth. Meanwhile, we hope that it will be of great use to provide theoretical foundation to design intelligent information processing systems.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 61175055), Sichuan Key Technology Research and Development Program (Grant no. 2011FZ0051), the Speciality Comprehensive Reform of Mathematics and Applied Mathematics of Ministry of Education (ZG0464), the Speciality Comprehensive Reform of Mathematics and Applied Mathematics of Ministry of Education (01249), the Scientific Research Project of Department of Education of Sichuan Province (14ZA0245), and the Scientific Research Project of Neijiang Normal University (13ZB05).