1. Introduction

In 1993, Y. XU [12] first established the lattice implication algebra by combining lattice and implication algebra. Z. M. Song and Y. Xu [11] studied the congruence relations on lattice implication algebras. Congruence relation is also one of the important tool when an algebra is studied. Lattice implication algebra is a special kind of residuated lattice. Zadeh [14] proposed the theory of fuzzy sets. In1993, W. L. Gau and D.J. Buehrer [5] introduced the concept of vague sets, which are more useful to evaluate the real life problems. The idea of vague sets is that the membership of every element can be classified into two parameters including supporting and opposing. Ranjit Biswas [4] initiated the study of vague algebra by studying vague groups. At first Ya Qin and Yi Liu [9] applied the concept of vague set theory to lattice implication algebras and introduced the notion of v- filter, and investigated some of their properties. Y. Lin et al. [8] studied the fuzzy congruence relations and properties on lattice implication algebras. Kham et al. [7] studied the vague relation and its properties on lattice implication algebras. Ch. Zhong and Y. Qin [15] studied the vague congruence relations and its properties on lattice implication algebras. Xiaoyan Qin et al. [10] studied the vague congruence and quotient lattice implication algebras. There are close correlations among ideals, congruence and quotient algebras. In [1], we introduced the concept of VLI – ideals on lattice implication algebras and studied properties of VLI – ideals.

In this paper, we investigate the further properties of VLI – ideals and the relation between VLI – ideals and vague congruence relations.

2. Preliminaries

Definition 2.1 [12]. Let be a complemented lattice with the universal bounds 0, I.

binary operation of L. is called a lattice implication algebra, if the following axioms hold,

Definition 2.2 [6]. Let A be a subset of a lattice implication algebra L. A is said to be an LI - ideal of L if it satisfies the following conditions:

Definition 2.3 [5]. A vague set A in the universal of discourse X is characterized by two membership functions given by:

(1) A truth membership function and

(2) A false membership function

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 The vague set The value of x in the vague set A denoted by defined by

Notation: Let I[0, 1] denote the family of all closed subintervals of [0, 1]. If I₁ = [a₁, b₁], I₂ = [a₂, b₂] are two elements of I[0, 1], we call I₁ ≥I₂ if a₁ ≥a ₂ and b₁ ≥b₂. We define the term imax to mean the maximum of two interval as

Similarly, we can define the term imin of any two intervals.

Definition 2.4 [1]. Let A be a vague set of a lattice implication algebra L. A is said to be a vague LI - ideal of L if it satisfies the following conditions:

Definition 2.5 [7]. 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

Definition 2.6 [7]. Let X and Y be two universes. A vague subset R of discourse is characterized by two membership functions given by:

(1) A truth membership function and

(2) A false membership function

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 subinterval [t_R(x, y), 1 – f_R(x, y)] of [0,1]. The vague relation R is written as

The value of (x, y) in the vague relation R denoted by V_R(x, y), defined by

Definition 2.7 [15]. Let X be a nonempty universe. A vague relation R on X is called vague similarity relation, if R satisfies the following conditions:

Remark 2.8 [15]. For the vague transitivity,

Definition 2.9 [15]. Let R be a vague relation on L. R is said to be a vague congruence relation on L, if

Theorem 2.10 [15]. Let R be a vague relation on L. Then for any R satisfies the following conditions:

3. Some Properties of VLI – ideals

Theorem 3.1: The vague set A of L is a VLI – ideal if and only if A satisfies the following conditions:

Proof: suppose A is a VLI – ideal of L. Obviously A satisfies the first condition.

It follows that,

Conversely suppose that the vague set A of L satisfies the inequalities (1) and (2).

Taking z = 0 in (2), we get

Hence A is a VLI – ideal of L.

Theorem 3.2: The vague set A of L is a VLI – ideal of L if and only if for any

Proof: Suppose A is a VLI – ideal of L.

Conversely suppose that the vague set A of L satisfies the condition

Then, we have

Let then

It follows that

So A is a VLI – ideal of L.

Corrolary 3.3: The vague set A of L is a VLI – ideal of L if and only if A satisfies the condition

where for a₁, a₂, ……….., a_n є L.

Definition 3.4: Let A be a VLI – ideal of L. Then for any the set A[a] defined by

Theorem 3.5: Let A be a VLI – ideal of L. Then A[a] is an ideal of L.

Proof: Let A be a VLI – ideal of L.

Obviously

Let such that

Therefore A[a] is an ideal of L.

Theorem 3.6: Let A be a vague set of L and A[a] be an ideal of L for any then

Proof: Let A[a] be an ideal of L for any

Let

Theorem 3.7: Let A be a vague set of L and A satisfies

Then A[a] is an ideal of L for any

Proof: Let A be a vague set of L and A satisfies the above conditions.

Therefore A[a] is an ideal of L.

4. Vague Congruence Relation Induced by VLI – ideals

Theorem 4.1: Let R be a vague congruence relation on L. Then, for all x, y, z є L,

Since R is a vague congruence relation,

Theorem 4.2: Let R be a vague congruence relation on L. Then the vague set

is a VLI – ideal of L.

Proof: (1) Let x є L, then

Therefore

(2) Let then

Therefore A_R is a vague LI – ideal of L. A_R is called a vague LI – ideal induced by a vague congruence relation R.

Definition 4.3: Let R be a similarity relation on L. For each a є L, we define a vague subset

on L, where

Theorem 4.4: Let R be a vague congruence relation on L. Then, R⁰ is vague LI – ideal of L.

Proof: It is obvious from the theorem 4.2.

Theorem 4.5: Let A be a vague LI – ideal of L and R_A be a vague relation on L defined by

Where

Then R_A is a vague congruence relation on L.

Proof: Let R_A be a vague relation on L defined by

Where

It is clear R_A is reflexive and symmetric.

For any we have

Therefore R_A is transitive.

Since A is order reversing, we have

It follows that,

Therefore R_A is a vague congruence relation on L. R_A is called a vague congruence relation induced by a vague LI – ideal A of L.

Theorem 4.6: Let the set of all vague LI – ideals of L denoted by I_V(L) and the set of all vague congruence relations of L denoted by C_V(L). Then I_V(L) is isomorphic to C_V(L).

Proof: We define the mappings f and g as follows:

and

for any and

Obviously these mappings are well defined.

For any and

For any we have

That is (g o f) is an identical mapping on I_V(L), which implies that f is injective.

Let then A_R is a vague LI – ideal of L.

Then is a congruence relation on L.

For any

For any we have

That implies (f o g) (R) = R.

That is f o g is an identical mapping on C_V(L) which implies that f is surjective.

So f is a bijection from I_V(L) to C_V(L).

Therefore I_V(L) is isomorphic to C_V(L).

Theorem 4.7: Let A be a vague LI – ideal of L and R_A be a vague congruence relation induced by A. Then for any if and only if

Proof: Let we have for all then

It follows that,

and

Similarly, we can prove that

Therefore for all hence

Theorem 4.8: Let A be a VLI – ideal of L. Define

Then, is a congruence relation on L.

Proof: The proof can be obtained from theorem 4.7.

Theorem 4.9: Let A be a VLI – ideal of L and let L/ R_A be the corresponding quotient algebra. Then, the map

defined by for any is a lattice implication homomorphism and

Proof: Since A is a VLI – ideal of L, R_A is a vague congruence relation induced by A.

By proposition 24 (10), is a lattice implication homomorphism.

5. Conclusions

Congruence theory and ideal theory play very important role in research of logic algebra. Through which we can much information such as quotient structure and homomorphic image of the logic algebra. The aim of this paper is to develop the vague congruence relation induced by VLI – ideals on lattice implication algebras. We study the concept of vague congruence induced by VLI – ideals. First, some properties of VLI – ideals are investigated. Next, some properties of vague congruence relation induced by VLI – ideals are discussed. We obtain one - to - one correspondence between the set of all VLI – ideals and the set of all vague congruence relations of lattice implication algebras. Lastly we obtain the homomorphism theorem on lattice implication algebra induced by vague congruence. We hope that it will be of great use to provide theoretical foundation to design for enriching corresponding many – valued logical system.