Zadeh [1] introduced the notion of fuzzy sets in 1965. Now, they are one of the most serious and possible paths for the advancement of the set theory of introduced by Georg Cantor. Despite the doubts and critical remarks expressed by some of the most influential mathematical logic experts in the second half of the 1960s against fuzzy sets, fuzzy sets were firmly developed as a fruitful field of study as well as a method for evaluating various objects and procedures.

In 1986, Atanassov [2] introducedintroduced intuitionistic fuzzy sets. In many applications, the intuitionist fuzzy sets are important and useful fuzzy sets. Atanassov [3, 4] in 1994 and 1999 proved that the intuitionistic fuzzy sets contain the degree of affiliation and the degree of non-affiliation, and therefore, the intuitionistic fuzzy sets have become more relevant and applicable. In 2001 and 2004, Szmidt and Kacprzyk [5, 6] showed that intuitionist fuzzy sets are so useful in situations where it seems extremely difficult to define a problem through a membership function.

The idea of intuitionistic fuzzy topology was described by Atanassov [7] in 1988, and the basic idea of intuitionistic fuzzy points was studied by Coker and Demirci [8] in 1995. Kuratowski [9] first proposed the concept of an ideal topological space in 1966, and Vaidyanathaswamy [10] proposed in 1944. In an ideal topological space, they also introduced a local function. In 1990, Jankovic and Hamlett [11] introduced a new topology by introduce the operator in any ideal topological space from the original ideal topological spaces.

Mashhour et al. [12] in 1983 introduced supra topological spaces. The concept of intuitionist fuzzy supra topological space was introduced by Turanl [13] in 2001. In addition to some features of an ideal supra topological notion obtained by Kandil et al. [14] in 2015.

The purpose of this paper is to introduce the notion of ideals on intuitionistic fuzzy supra topological spaces. Also, present the notion of S is compatible with the intuitionistic fuzzy ideal I and investigation some properties of intuitionistic fuzzy supra topological spaces S with intuitionistic fuzzy ideal I. Moreover, introduce an intuitionistic fuzzy set operator Ψ_S and study its properties.

Definition 2.1( [15])

Let X ≠ ∅, an intuitionistic fuzzy set A is subject with the form A = {< x,μ_A(x), ν_A(x) >: x ∈ X}, where μ_A : X –→ [0, 1] and ν_A : X –→ [0, 1] define the degree of membership μ_A(x) and the degree of non-membership ν_A(x) for every x ∈ X to the set A, respectively, and 0 ⩽ μ_A(x) + ν_A(x) ≤ for every x ∈ X.

Definition 2.2 ( [15])

1_~ = {< x,1, 0 >: x ∈ X} and 0_~ = {< x, 0, 1 >: x ∈ X}.

Definition 2.3 ( [16])

Let A, B be an intuitionistic fuzzy sets, then we define

• A ⊆ B if and only if μ_A(x) ⩽ μ_B(x) and ν_A(x) ≥ ν_B(x) for every x ∈ X.

• A = B if and only if A ⊆ B and B ⊆ A.

• A^c = {< x,ν_A(x), μ_A(x) : x ∈X >}.

• A ∩ B = {< x,μ_A(x) ∧ μ_B(x), μ_A(x) ∨ μ_B(x) >}.

• A ∪ B = {< x,μ_A(x) ∨ μ_B(x), μ_A(x) ∧ μ_B(x) >}.

Definition 2.4 ( [8])

LetX ≠ ∅ and let x ∈ X. If α ∈ (0, 1] and β ∈ [0, 1) are two fixed real numbers such that α + β ⩽ 1, then, in the intuitionistic fuzzy set

is called an intuitionistic fuzzy point in X, where α denotes the degree of membership of x_(α,β), β is the degree of non-membership of x_(α,β), and x ∈ X is the support of x_(α,β).

Definition 2.5 ( [2])

A subclass S is called an intuitionistic fuzzy supra topology on X if 0_~, 1_~ ∈ S and S is closed under arbitrary unions (X, S), which is called an intuitionistic fuzzy supra topoloical topological space, the members of S are called intuitionistic fuzzy supra open sets. An intuitionistic fuzzy set A is an intuitionistic fuzzy supra closed if and only if its complement A^c is fuzzy supra open.

Definition 2.6 ( [2])

Let (X, S) be an intuitionistic fuzzy supra topological space and let A be an intuitionistic fuzzy set in X. ThenSubsequently, the intuitionistic fuzzy supra interior and the intuitionistic fuzzy supra closure of A in (X, S) is defined as

and

respectively.

Corollary 2.1

From Definition 2.6, Int^S(A) is a fuzzy supra open set, and Cl^S(A) is a fuzzy supra closed set.

Definition 2.7 ( [8])

Let A and B be two intuitionistic fuzzy sets in X. A is called quasi-coincident with B (written AqB) if and only if, there exists x ∈ X such that μ_A(x) > ν_B(x) or ν_A(x) < μ_B(x).

Definition 2.8 ( [8])

Let x_(α,β) an intuitionistic fuzzy point and let A an intuitionistic fuzzy set in X. We say that x_(α,β) quasi-coincident with A, denoted by x_(α,β)qA if and only if α > ν_A(x) orβ < μ_A(x).

Definition 2.9 ( [8])

Let x_(α,β) an intuitionistic fuzzy point and let A an intuitionistic fuzzy set in X. Let α and β are real numbers between 0 and 1. The intuitionistic fuzzy point x_(α,β) is called properly contained in A if and only if, α < μ_A(x) and β > ν_A(x).

Definition 2.10 ([8])

Let x_(α.β) an intuitionistic fuzzy point . Then, x_(α,β) ∈ A if α ≤ μ_A(x) and β ≥ ν_A(x).

Definition 2.11 ( [2])

Let (X, S) be an intuitionistic fuzzy supra topological space and let A ⊆ X. Then, A is the s-neighborhood of an intuitionistic fuzzy point x_(α,β) if there is U ∈ S with x_(α,β) ∈ U ⊆ A(x_(α,β)qU ⊆ A). The collection N(x_(α,β)) of all s-neighborhood of x_(α,β) is called the s-neighborhood system of x_(α,β).

Definition 2.12 ( [2])

Let (X, S₁) and (X, S₂) be two intuitionistic fuzzy supra topologies and let S₁ ⊆ S₂. Then, we say that S₂ is stronger than S₁ or S₁ is weaker than S₂.

Definition 2.13 ( [2])

Let (X, S) be an intuitionistic fuzzy supra topological space and let β ⊆ S. Then, β is called a base for the intuitionistic fuzzy supra topology S if every intuitionistic fuzzy supra open set U ∈ S is a union of members of β. Equivalently, β is an intuitionistic fuzzy supra-base for S if for any intuitionistic fuzzy point x_(α,β) ∈ U there exists B ∈ β with x_(α,β) ∈ B ⊆ U.

Definition 2.14 ( [2])

A mapping c : P(X) → P(X) is called an intuitionistic fuzzy supra closure operator if it satisfies the following axioms:

• c(0_~) = 0_~,

• A ⊆ c(A) for every A ⊆ X,

• c(A) ∪ c(B) ⊆ c(A ∪ B) for every A, B ⊆ X,

• c(c(A)) = c(A) for every A ⊆ X.

Theorem 2.1 ( [2])

Let X ≠ ∅ and let the mapping c : P(X) → P(X) be an intuitionistic fuzzy supra closure operator. Then, the collection S = {A ∈ P(X) : c(A^c) = A^c} is an intuitionistic fuzzy supra topology on X induced by the intuitionistic fuzzy supra closure operator c.

Definition 2.15 ( [15])

Let I be a non-empty collection of intuitionistic fuzzy sets of X is called intuitionistic fuzzy ideal on X if and only if

• A ∈ I and B ⊆ A, then B ∈ I,

• A ∈ I and B ∈ I, then A ∪ B ∈ I.

Definition 3.1

Let (X, S) be an intuitionistic fuzzy supra topological space. Then, an intuitionistic fuzzy ideal I on X is called an intuitionistic fuzzy ideal supra topological space and is denoted as (X, S, I).

Definition 3.2

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let A be an intuitionistic fuzzy set in X. Then, the intuitionistic fuzzy S-local function A^*S(I, S) of A is the union of all intuitionistic fuzzy point x_(α,β) such that if U ∈ N(x_(α,β)) and A^*S(I, S) = ∪{x_(α,β) ∈ X: A ∩U ∉ I, for every U ∈ N(x_(α,β))}. We will occasionally write A^*S for A^*S(I, S).

Example 3.1

The simplest intuitionistic fuzzy ideal on X isare {0_~} and P(X). Obviously, I = {0_~} ⇔ A^*S = Cl^S(A), for any A ⊆ X and I = P(X) ⇔ A^*S = 0_~.

Theorem 3.1

Let (X, S, I) be an intuitionistic fuzzy supra topological space and let A, B ⊆ X. Then,

• (1) $0_{\sim }^{*S}=0_{\sim }$,

• (2) If A ⊆ B, then A^*S ⊆ B^*S,

• (3) If I₁ ⊆ I₂, then A^*S(I₂) ⊆ A^*S(I₁),

• (4) A^*S = Cl^S(A^*S) ⊆ Cl^S(A),

• (5) (A^*S)^*S ⊆ A^*S,

• (6) A^*S is an intuitionistic fuzzy supra closed set,

• (7) A^*S ∪ B^*S ⊆ (A ∪ B)^*S,

• (8) (A ∩ B)^*S ⊆ A^*S ∩ B^*S,

• (9) If E ∈ I, then (A ∪ E)^*S = A^*S = (A − E)^*S,

• (10) If U ∈ S, then U ∩ A^*S = U ∩ (U ∩ A)^*S ⊆ (U ∩ A)^*S,

• (11) If E ∈ I, then E^*S = 0_~,

• (12) E ∈ I, then ${(1_{\sim }-E)}^{*S}=1_{\sim }^{*S}$.

Theorem 3.2

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let A ⊆ X. If M ∈ S, M ∩ A ∈ I, then M ∩ A^*S = 0_~.

Theorem 3.3

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let A ⊆ X. Then, (A ∪ A^*S)^*S ⊆ A^*S.

Theorem 3.4

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, the operator Cl^*S : P(X) –→ P(X) defined by Cl^*S(A) = A ∪ A^*S for any A ⊆ X, is an intuitionistic fuzzy supra closure operator and hence it generates an intuitionistic fuzzy supra topology S^*(I) = {A ∈ P(X) : Cl^*S(A^c) = A^c}, which is finer than S.

Example 3.2

For any intuitionistic fuzzy ideal on X if I = {0_~} ⇒ Cl^*S(A) = A ∪ A^*S = A ∪ Cl^S(A) = Cl^S(A) for every A ∈ P(X). So S^*({0_~}) = S, and if I = P(X) ⇒ Cl^*S(A) = A, because A^*S = 0_~ for every A ∈ P(X). So S^*(P(X)) is an intuitionistic fuzzy discrete supra topology on X. Since {0_~} and P(X) are the two extreme intuitionistic fuzzy ideals on X, therefore for any intuitionistic fuzzy ideal I on X, we have {0_~} ⊆ I ⊆ P(X). So we can conclude by Theorem 3.1.(2) S^*({0_~}) ⊆ S^*(I) ⊆ S^*(P(X)), i.e. S ⊆ S^*(I), for any intuitionistic fuzzy ideal I on X. In particular, we have for any two intuitionistic fuzzy ideals I₁ and I₂ on X, I₁ ⊆ I₂ ⇒ S^*(I₁) ⊆ S^*(I₂).

Theorem 3.5

Let S₁, S₂ be two intuitionistic fuzzy supra topologies on X. Then, for any intuitionistic fuzzy ideal I on X, S₁ ⊆ S₂ implies

• (1) A^*S(S₂, I) ⊆ A^*S(S₁, I) for every A ∈ P(X),

• (2) $S_1^{*}(I)\subseteq S_2^{*}(I)$.

Theorem 3.6

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, A is an intuitionistic fuzzy S^*- supra closed if and only if A^*S ⊆ A. Then, A = Cl^S(A^*S) = Cl^*S(A).

Theorem 3.7

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, the collection β(I, S) = {U − H: U ∈ S, H ∈ I} is a base for the intuitionistic fuzzy supra topology S^*(I).

Theorem 3.8

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, S ⊆ β(I, S) ⊆ S^*.

Theorem 3.9

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, if I = {0_~}, then S = β(I, S) = S^*(I).

Example 3.3

Let T be the intuitionistic fuzzy indiscrete supra topology on X, i.e. T = {0_~, 1_~}. So 1_~ is the only s-neighborhoods of x_(α,β). Now, x_(α,β) ∈ A^*S for an intuitionistic fuzzy set A if and only if for every U ∈ N(x_(α,β)), then U ∩A ∉ I. So A ∉ I. Therefore, A^*S = 1_~ if A ∉ I and A^*S = 0_~ if A ∈ I. This implies that we have Cl^*S(A) = A ∪ A^*S = 1_~ if A ∉ I and Cl^*S(A) = A if A ∈ I for any intuitionistic fuzzy set A of X. Hence, T^* = {M: M^c ∈ I}. Let S ∪T^*(I) be the supremum intuitionistic fuzzy supra topology of S and T^*(I), i.e. the smallest intuitionistic fuzzy supra topology generated by S ∪ T^*(I). Then, we have the following theorem.

Theorem 3.10

S^*(I) = S ∪ T^*(I).

Definition 4.1

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. We say the S is S-compatible with the intuitionistic fuzzy ideal I, denoted as S ~ I, if the following holds for every intuitionistic fuzzy set A in X, if for every x_(α,β) ∈ A, there exists U ∈ N(x_(α,β)) such that U ∩ A ∈ I, then A ∈ I.

Theorem 4.1

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space, and the following properties are equivalent;

• (1) S ~ I,

• (2) For every intuitionistic fuzzy set A in X, A ∩ A^*S = 0_~ implies that A ∈ I,

• (3) For every intuitionistic fuzzy set A in X, A – A^*S ∈ I,

• (4) For every intuitionistic fuzzy set A in X, if A contains no non-empty intuitionistic fuzzy subset B with B ⊆ B^*S, then A ∈ I.

Theorem 4.2

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. If S is S-compatible with I, then the following equivalent properties hold: ;

• (1) For every intuitionistic fuzzy set A in X, A ∩ A^*S = 0_~ implies that A^*S = 0_~,

• (2) For every intuitionistic fuzzy set A in X, (A – A^*S)^*S = 0_~.

Theorem 4.3

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space, the following properties are equivalent;

• (1) S ∩ I = 0_~,

• (2) $1_{\sim }^{*S}=1_{\sim }$.

Definition 5.1

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. An operator Ψ_S : P(X) → S is defined as follows for every intuitionistic fuzzy set A in X, Ψ_S(A) ={x_α,β intuitionistic fuzzy point: there exists M ∈ N^S(x_α,β) such that M – A ∈ I}. We observe that Ψ_S(A) = 1_~ – (1_~ – A)^*S. The behaviors of the operator Ψ_S has been discussed in the following theorem:

Theorem 5.1

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Let A and B be two intuitionistic fuzzy set in X. Then,

• (1) Ψ_S(A) is intuitionistic fuzzy supra open set.

• (2) Int^S(A) ⊆ Ψ_S(A).

• (3) If A ⊆ B, then Ψ_S(A) ⊆ Ψ_S(B).

• (4) Ψ_S(A ∩ B) ⊆ Ψ_S(A) ∩ Ψ_S(B).

• (5) Ψ_S(A) ∪ Ψ_S(B) ⊆ Ψ_S(A ∪ B).

• (6) If U ∈ S, then U ⊆ Ψ_S(U).

• (7) Ψ_S(A) ⊆ Ψ_S(Ψ_S(A)).

• (8) Ψ_S(A) = Ψ_S(Ψ_S(A)) if and only if (1_~ – A)^*S = ((1_~ –A)^*S)^*S.

• (9) If (A – B) ∪ (B – A) ∈ I, then Ψ_S(A) = Ψ_S(B).

• (10) If E ∈ I, then ${\mathrm{\Psi }}_S(E)=1_{\sim }-1_{\sim }^{*S}$.

• (11) If E ∈ I, then Ψ_S(A – E) = Ψ_S(A).

• (12) If E ∈ I, then Ψ_S(A ∪ E) = Ψ_S(A).

Theorem 5.2

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. If η = {A ∈ P(X) : A ⊆ Ψ_S(A)}. Then, η is an intuitionistic fuzzy supra topology for X.

Definition 5.2

An intuitionistic fuzzy ideal I in a space (X, S, I) is called an S-codense intuitionistic fuzzy ideal if S ∩ I = {0_~}. The following theorem is related to the S-codense intuitionistic fuzzy ideal.

Theorem 5.3

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let I be S-codense with S. Then, $1_{\sim }=1_{\sim }^{*S}$.

Definition 5.3

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. An intuitionistic fuzzy set A in X is called a Ψ_S-C-intuitionistic fuzzy set if A ⊆ Cl^S(Ψ_S(A)). The collection of all Ψ_S-C-intuitionistic fuzzy set in (X, S, I) is denoted by Ψ_S(X, S).

Theorem 5.4

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. If A ∈ S, then A ∈ Ψ_S(X, S).

Theorem 5.5

Let {A_α : α ∈Δ} be a collection of non-empty Ψ_S-C-intuitionistic fuzzy sets in the intuitionistic fuzzy ideal supra topological space (X, S, I); then ∪_α∈ΔA_α ∈ Ψ_S(X, S).