Quotient of Ideals of an Intuitionistic Fuzzy Lattice

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.


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.

Preliminaries
We recall the following definitions and results which will be used in the sequel.Throughout this paper L stands for a lattice (L,∨,∧) with zero element "0" and unit element "1".Definition 1 (see [1]).Let X be a nonempty set.An intuitionistic fuzzy set [IFS] A of X is an object of the following form A = { x, μ A (x), ν A (x) | x ∈ X}, where μ A : X → [01] and ν A : X → [01] define the degree of membership and the degree of non membership of the element x ∈ X, respectively, and ∀x ∈ X, The set of all IFS's on X is denoted by IFS (X).
Definition 4 (see [11]).An IFS A of L is called an intuitionistic fuzzy ideal (IFI) of L if the following conditions are satisfied.
The set of all IFI's of L is denoted as IFI (L).

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 (L).We also used ∨ and ∧ to represent maximum and minimum, respectively, which is clear from the context.Definition 6.Let A be an IFL of L and B an IFS of L with B ⊆ A. Then B is called an intuitionistic fuzzy ideal (IFI) of A if the following conditions are satisfied.Then B is called an intuitionistic fuzzy sublattice of A.

Lemma 3. The intersection of two IFI's of A is again an IFI of A.
Proof.Let B, C be IFI's of A. Then we can prove that B ∩ C is also an IFI of A. Since B ⊆ A and C ⊆ A, we have Also Hence, B ∩ C is an IFI of A.

Theorem 1. Let A an IFL and B an IFS of L with B ⊆ A. Then B is an IFI of A if and only if
Proof.Suppose that conditions (1), (2), and (3) hold.Then we prove that B is an IFI of A. We have Similarly Hence Similarly Hence So from ( 1), ( 2), (a), and (b) B is an IFI of A.
Conversely suppose B is an IFI of A. Then obviously conditions (1) and ( 2) holds.Also we have Hence AB ⊆ B.

Theorem 2. Let A be an IFL of L and B an IFS with
Proof.Suppose conditions (1), (2), and (3) holds.We prove B is an IFI of A.
We have Similarly, we can obtain Hence Similarly Hence So from ( 1), ( 2), (a.1), and (b.1) B is an IFI of A.
Conversely suppose that B is an IFI of A. Then obviously conditions ( 1) and ( 2) hold.
Let z ∈ L and z = n i=1 (x i ∧ y i ), where Also Thus Hence

Theorem 3. Let A be an IFL of L and B, C are IFI's of A. Then B + C is an IFI of A.
Proof.We have (by [11,Theorem 5.2]).And (by Lemma 1 and Theorem 1).Hence B + C is an IFI of A.

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 .Theorem 4 .
Let A be an IFL of L and B, C be IFI's of A. Then the quotient (residual) of B by C denoted as B/C is defined by B/C = ∪{D/D A, DC ⊆ B}. (22) Let A be an IFL of L and B,C are IFI's of A. Then the quotient B/C is an IFI of A. Also B ⊆ B/C ⊆ A. Proof.Let η = {D/D A, DC ⊆ B}.Suppose D, D ∈ η.Then D and D are IFI's of A such that DC ⊆ B and D C ⊆ B. Then by Theorem 3 D + D is an IFI of A. So by Lemmas 1 and 2 (D + D )C ⊆ DC + D C ⊆ B + B = B. Thus D + D ∈ η.Now

ν
B (a i ∧ b i ), since B is an IFI of A ≤ n i=1 [ν B/C (a i ) ∧ ν C (b i )]./C (a i ) ∧ ν C (b i )]/x = n i=1 (a i ∧ b i ) 37) and (41) (B/C) • C ⊆ B. If D is an ideal of A such that D • C ⊆ B then DC ⊆ D • C ⊆ B. So D ∈ η.Hence D ⊆ B/C.Thus B/C is the largest IFI of A such that (B/C) • C ⊆ B.Theorem 6.Let A be an IFL and B, C, D be IFI's of A. Then the following holds.