Abstract

The concept of intuitionistic fuzzy ideal of an intuitionistic fuzzy lattice is introduced, and its certain characterizations are provided. We defined the quotient (or residual) of ideals of an intuitionistic fuzzy sublattice and studied their properties.

1. Introduction

The concept of intuitionistic fuzzy sets was introduced by Atanassov [1, 2] as a generalization of that of fuzzy sets and it is a very effective tool to study the case of vagueness. Further many researches applied this notion in various branches of mathematics especially in algebra and defined intuitionistic fuzzy subgroups (IFG), intuitionistic fuzzy subrings (IFR), and intuitionistic fuzzy sublattice (IFL), and so forth. In the last five years there are so many articles appeared in this direction. Kim [3], Kim and Jun [4], Kim and Lee [5], introduced different types of IFI’s in Semigroups. Torkzadeh and Zahedi [6] defined intuitionistic fuzzy commutative hyper K-ideals, Akram and Dudek [7] defined intuitionistic fuzzy Lie ideals of Lie algebras, and Hur et al. [8] introduced intuitionistic fuzzy prime ideals of a Ring.

The concept of ideal of a fuzzy subring was introduced by Mordeson and Malik in [9]. After that N Ajmal and A.S Prajapathi introduced the concept of residual of ideals of an L-Ring in [10]. Motivated by this, in this paper we first defined the intuitionistic fuzzy ideal of an IFL and certain characterizations are given. Lastly we defined quotients (residuals) of ideals of an intuitionistic fuzzy sublattice and studied their properties.

2. Preliminaries

We recall the following definitions and results which will be used in the sequel. Throughout this paper stands for a lattice (,,) with zero element “0” and unit element “1”.

Definition 1 (see [1]). Let be a nonempty set. An intuitionistic fuzzy set [IFS] of is an object of the following form , where and define the degree of membership and the degree of non membership of the element , respectively, and , .

The set of all IFS’s on is denoted by IFS ().

Definition 2 (see [1]). If and are any two IFS of   then (i) and ;(ii) and ;(iii);(iv), where ;(v), where ;(vi)   ; (vii)    .

Definition 3 (see [11]). Let be a lattice and be an IFS of . Then is called an intuitionistic fuzzy sublattice [IFL] of if the following conditions are satisfied.(i); (ii);(iii);(iv).The set of all intuitionist fuzzy sublattices (IFL’s) of is denoted as IFL ().

Definition 4 (see [11]). An IFS of is called an intuitionistic fuzzy ideal (IFI) of if the following conditions are satisfied. (i); (ii);(iii);(iv),.
The set of all IFI’s of is denoted as IFI ().

Definition 5 (see [12]). Let , IFS (). Then we define an IFS (),(i), where (ii), where (iii)  where

Lemma 1 (see [13]). Let , , and be IFS (), then the following assertions hold.(1), , .(2).(3).(4).(5).(6) and .

Lemma 2 (see [13]). Let A be an IFL of L, then(1).(2).

3. Ideal of an Intuitionistic Fuzzy Lattice

In this section we define the ideal of an IFL, and give some characterization of these ideals in terms of operations on IFS (). We also used and to represent maximum and minimum, respectively, which is clear from the context.

Definition 6. Let be an IFL of and an IFS of with . Then is called an intuitionistic fuzzy ideal (IFI) of if the following conditions are satisfied.(i).(ii).(iii).(iv).If IFI of , then we write

Example 1. Consider the lattice under divisibility.
Let be an IFL of defined by , , , and be an IFS of L given by , , , . Clearly .

Definition 7. Let be an IFL and is also an IFL with . Then is called an intuitionistic fuzzy sublattice of .

Lemma 3. The intersection of two IFI’s of is again an IFI of .

Proof. Let , be IFI’s of . Then we can prove that is also an IFI of . Since and , we have Also Also Hence, is an IFI of .

Theorem 1. Let an IFL and an IFS of with . Then is an IFI of if and only if(1),(2),(3).

Proof. Suppose that conditions (1), (2), and (3) hold. Then we prove that is an IFI of .
We have Similarly Hence Also Similarly Hence So from (1), (2), (a), and (b) is an IFI of .
Conversely suppose is an IFI of . Then obviously conditions (1) and (2) holds. Also we have
So with Hence .

Theorem 2. Let be an IFL of and an IFS with . Then is an IFI of if and only if(1),(2),(3).

Proof. Suppose conditions (1), (2), and (3) holds. We prove is an IFI of .
We have Similarly, we can obtain Hence Also Similarly Hence So from (1), (2), (a.1), and (b.1) is an IFI of .
Conversely suppose that is an IFI of . Then obviously conditions (1) and (2) hold.
Let and , where .
We have Thus Also Thus Hence .

Theorem 3. Let A be an IFL of L and B, C are IFI’s of A. Then is an IFI of A.

Proof. We have (by [11, Theorem  5.2]).
And (by Lemma 1 and Theorem 1).
Hence is an IFI of .

4. Quotient of Ideals

Here first we define the residual of ideals of an IFL and prove that the residual of ideals is again an IFI of the IFL. Moreover we establish that it is the largest ideal with respect to some property on the operation .

Definition 8. Let be an IFL of and , be IFI’s of . Then the quotient (residual) of by denoted as is defined by

Theorem 4. Let be an IFL of and , are IFI’s of . Then the quotient is an IFI of . Also .

Proof. Let . Suppose , .  Then and are IFI’s of such that and . Then by Theorem 3   is an IFI of . So by Lemmas 1 and 2  . Thus . Now That is, Also Similarly Thus Now That is Also Similarly Thus From (24), (27), (29), and (32) is an IFI of .
Clearly .
Since is an IFI of , (by Theorem 1).
Since , by Lemma 1  . Hence . So .
Thus we have

Theorem 5. Let be an IFL and , be IFI’s of . Then is the largest IFI of with the property .

Proof. Let . We have Let such that .
Then So Hence Consequently Also So Hence Consequently Thus from (37) and (41) .
If is an ideal of such that then . So . Hence . Thus is the largest IFI of such that .

Theorem 6. Let be an IFL and , , be IFI’s of . Then the following holds.(1)If then and .(2)If then .(3).

Proof. (1) Let . Write and . If then and . Thus and hence . So .
Similarly, let and . If then . But . So . Thus and hence . So .
(2) Let . Since , we have , and . Thus and hence , since is an IFI of . Therefore .
(3) We have . So from (2) .

Corollary 1. Let be an IFL of and , and be IFI’s of . Then(1),(2),(3).

Proof. (1) Since , by Theorem 6 (2), .
(2) By Theorem 6 (3) . Since by Theorem 6 (2), .
(3) Since and . So and . Hence by Theorem 6 (2), .

Theorem 7. Let be an IFL of and , , are IFI’s of . Then

Proof. Since   by Theorem 6 (1) .
Hence
Let Then Similarly Now let and . Then and . Also IFI of .
So that Thus . So .
Hence So Hence From (43) and (48). This completes the proof.

Next, we denote the set of all IFI’s    of an IFL that satisfies the property by IFI (*). Then we have the following results.

Lemma 4. Let be an IFL of and , IFI (*). Then(1) and .(2).(3) and .

Proof. (1) We have So . Similarly .
(2) We have (by Theorem 3) and (by (1)). So by Theorem 6 (1),
Write and .
Let , then and . So (by Lemma 1). But , since . Hence and also . So By Lemma 1, . Therefore . So . Thus From (50) and (51) .
(3) We have and . So by Theorem 6 (2), .
Also we have and . Hence by Theorem 6 (2) .

Theorem 8. Let be an IFL of and IFI (*) and C any IFI of . Then .

Proof. We have and , .
So by Theorem 6 (1), and .
Therefore Let Then Similarly Now let and . Then and . Also IFI of , so that So . Hence . Thus .
So Therefore From (52) and (56) , hence the result.