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## Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 93-100

Published online March 25, 2021

https://doi.org/10.5391/IJFIS.2021.21.1.93

© The Korean Institute of Intelligent Systems

## Ideals on Intuitionistic Fuzzy Supra Topological Spaces

Salzburger Straße 195, Linz, Austria

Correspondence to :

Received: November 21, 2020; Revised: January 29, 2021; Accepted: March 3, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

Keywords: Intuitionistic fuzzy-I-supra topology, Intuitionistic fuzzy s-local function, Intuitionistic fuzzy set operator ΨS

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 ΨS and study its properties.

### Definition 2.1( [15])

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

### Definition 2.2 ( [15])

1~ = {< x,1, 0 >: xX} and 0~ = {< x, 0, 1 >: xX}.

### Definition 2.3 ( [16])

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

• AB if and only if μA(x) ⩽ μB(x) and νA(x) ≥ νB(x) for every xX.

• A = B if and only if AB and BA.

• Ac = {< x,νA(x), μA(x) : x ∈X >}.

• AB = {< x,μA(x) ∧ μB(x), μA(x) ∨ μB(x) >}.

• AB = {< x,μA(x) ∨ μB(x), μA(x) ∧ μB(x) >}.

### Definition 2.4 ( [8])

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

$x(α,β)={:x∈X}$

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 xX 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 Ac 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

$IntS(A)=∪{U:U⊆A,U∈S},$

and

$ClS(A)=∩{F:A⊆F,Fc∈S},$

respectively.

### Corollary 2.1

From Definition 2.6, IntS(A) is a fuzzy supra open set, and ClS(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 xX such that μA(x) > νB(x) or νA(x) < μB(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 > νA(x) orβ < μA(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 < μA(x) and β > νA(x).

### Definition 2.10 ([8])

Let x(α.β) an intuitionistic fuzzy point . Then, x(α,β)A if αμA(x) and βνA(x).

### Definition 2.11 ( [2])

Let (X, S) be an intuitionistic fuzzy supra topological space and let AX. Then, A is the s-neighborhood of an intuitionistic fuzzy point x(α,β) if there is US with x(α,β)UA(x(α,β)qUA). 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 S1S2. 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 βS. Then, β is called a base for the intuitionistic fuzzy supra topology S if every intuitionistic fuzzy supra open set US 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 Bβ with x(α,β)BU.

### 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~,

• Ac(A) for every AX,

• c(A) ∪ c(B) ⊆ c(AB) for every A, BX,

• c(c(A)) = c(A) for every AX.

### 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 = {AP(X) : c(Ac) = Ac} 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

• AI and BA, then BI,

• AI and BI, then ABI.

### 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 UN(x(α,β)) and A*S(I, S) = ∪{x(α,β)X: AUI, for every UN(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 = ClS(A), for any AX and I = P(X) ⇔ A*S = 0~.

### Theorem 3.1

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

• (1) $0∼*S=0∼$,

• (2) If AB, then A*SB*S,

• (3) If I1I2, then A*S(I2) ⊆ A*S(I1),

• (4) A*S = ClS(A*S) ⊆ ClS(A),

• (5) (A*S)*SA*S,

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

• (7) A*SB*S ⊆ (AB)*S,

• (8) (AB)*SA*SB*S,

• (9) If EI, then (AE)*S = A*S = (AE)*S,

• (10) If US, then UA*S = U ∩ (UA)*S ⊆ (UA)*S,

• (11) If EI, then E*S = 0~,

• (12) EI, then $(1∼-E)*S=1∼*S$.

Proof

(1) Clear from the definition of intuitionistic fuzzy S-local function .

(2) Since AB, let x(α,β)A*S, then AUI for every UN(x(α,β)). By hypothesis, we obtainget BUI, then x(α,β)B. Therefore, A*SB*S.

(3) Cleary, I1I2 implies A*S(I2) ⊆ A*S(I1), as there may be other intuitionistic fuzzy sets that which belong to I2 so that for an ituitionistic fuzzy point x(α,β)A*S(I1) but x(α,β)A*S(I2).

(4) Since {0~} ⊆ I for any intuitionistic fuzzy ideal on X, therefore by (3) and Example 3.1. A*S(I) ⊆ A*S({0~}) = ClS(A), for any intuitionistic fuzzy set A in X. Suppose, x1(α,β)ClS(A*S) such that for every UN(x1(α,β)), A*SU ≠ 0~ there exists x2(α,β)A*SU such that for every VN(x2(α,β)), then AVI. Since UVN(x2(α,β)), then A ∩ (UV ) ∉ I, which leads to AUI for every UN(x1(α,β)), therefore x1(α,β)A*S; and so ClS(A*S) ⊆ A*S while the other inclusion follows directly. Hence, A*S = ClS(A*S) ⊆ ClS(A).

(5) From (4), (A*S)*SClS(A*S) = A*S.

(6) Clear from (4).

(7) We know that AAB and BAB. Then, from (2), A*S ⊆ (AB)*S and B*S ⊆ (AB)*S. Hence, A*SB*S ⊆ (AB)*S.

(8) We know that (AB) ⊆ A and (AB) ⊆ B. Then, from (2), (AB)*SA*S and (AB)*SA*S. Hence, (AB)*SA*SB*S.

(9) Since A ⊆ (AE), then from (2) A*S ⊆ (AE)*S. Let x(α,β) ∈ (AE)*S. Then, for every UN(x(α,β)) such that U ∩ (AE) ∉ I. This implies that UAI(if possible suppose that UAI. Again, UEE implies UEI and hence U ∩ (AE) ∈ I, contradiction). Hence, x(α,β)A*S and (AE)*SA*S then (AE)*S = A*S.

Since (AE) ⊆ A, then from (2), (AE)*SA*S. For the reverse inclusion, let x(α,β)A*S. We claim that x(α,β) ∈ (AE)*S, if not, then there is UN(x(α,β)) such that U ∩ (AE) ∈ I. Given that EI, then E ∪ (U ∩ (AE)) ∈ I. This implies that E ∪ (UA) ∈ I. So, UAI, a contradiction to the fact that x(α.β)A*S. Hence, A*S ⊆ (AE)*S. Then, A*S = (AE)*S; therefore, (AE)*S = A*S = (AE)*S.

(10) Since VAA, then from (2), (VA)*SA*S. So V ∩ (VA)*SVA*S.

(11) Clear from the definition of intuitionistic fuzzy S-local function.

(12) Clear from proof (9).

### Theorem 3.2

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let AX. If MS, MAI, then MA*S = 0~.

Proof

Let x(α,β)MA*S. Then, x(α,β)M and x(α,β)A*S implies UAI for every UN(x(α,β)). Since x(α,β)MS, then MA*SI.

### Theorem 3.3

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let AX. Then, (AA*S)*SA*S.

Proof

Let x(α,β)A*S. Then, there exists UN(x(α,β)) such that UAIUA*S = 0~ (By Theorem 3.2.). Hence, U ∩ (AA*S) = (UA) ∪ (UA*S) = UAI. Therefore, x(α,β) ∉ (AA*S)*S. Hence, (AA*S)*SA*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) = AA*S for any AX, is an intuitionistic fuzzy supra closure operator and hence it generates an intuitionistic fuzzy supra topology S*(I) = {AP(X) : Cl*S(Ac) = Ac}, which is finer than S.

Proof

(1) By Theorem 3.1.(1), $0∼*S=0∼$, we have Cl*S(0~) = 0~.

(2) Clear ACl*S for every intuitionistic fuzzy set A.

(3) Let A and, B be two intuitionistic fuzzy sets. Then, Cl*S(A) ∪ Cl*S(B) = (AA*S) ∪ (BB*S) = (AB) ∪ (A*SB*S) ⊆ (AB) ∪ (AB)*S = Cl*S(AB) (by Theorem 3.1 (7). Hence, Cl*S(A)∪Cl*S(B) ⊆ Cl*S(AB).

(4) Let A be any intuitionistic fuzzy set. Since, by (2), ACl*S(A), then Cl*S(A) ⊆ Cl*S(Cl*S(A)). On the other hand, Cl*S(Cl*S(A)) = Cl*S(AA*S) = (AA*S)∪(AA*S)*SAA*SA*S = Cl*S(A) (by Theorem 3.3), it follows that Cl*S(Cl*S(A)) ⊆ Cl*S(A). Hence, Cl*S(Cl*S(A)) = Cl*S(A). Consequently, Cl*S(A) is an intuitionistic fuzzy supra closure operator. Also, it is also easy to show that the collection S*(I) = {AP(X) : Cl*S(Ac) = Ac} is an intuitionistic fuzzy supra topology on X, which is called the intuitionistic fuzzy supra topology induced by the intuitionistic fuzzy supra closure operator.

### Example 3.2

For any intuitionistic fuzzy ideal on X if I = {0~} ⇒ Cl*S(A) = AA*S = AClS(A) = ClS(A) for every AP(X). So S*({0~}) = S, and if I = P(X) ⇒ Cl*S(A) = A, because A*S = 0~ for every AP(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~} ⊆ IP(X). So we can conclude by Theorem 3.1.(2) S*({0~}) ⊆ S*(I) ⊆ S*(P(X)), i.e. SS*(I), for any intuitionistic fuzzy ideal I on X. In particular, we have for any two intuitionistic fuzzy ideals I1 and I2 on X, I1I2S*(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, S1S2 implies

• (1) A*S(S2, I) ⊆ A*S(S1, I) for every AP(X),

• (2) $S1*(I)⊆S2*(I)$.

Proof

(1) Since every S1 s-neighborhood of any intuitionistic fuzzy point x(α,β) is also ana S2 s-neighborhood of x(α,β). Therefore, A*S(S2, I) ⊆ A*S(S1, I).

(2) Clearly, $S1*(I)⊆S2*(I)$ as A*S(S2, I) ⊆ A*S(S1, 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*SA. Then, A = ClS(A*S) = Cl*S(A).

Proof

Clear.

### Theorem 3.7

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, the collection β(I, S) = {UH: US, HI} is a base for the intuitionistic fuzzy supra topology S*(I).

Proof

Let US*(I) and x(α,β)U. Then, Uc is an intuitionistic fuzzy S*-supra closed set suchso that Cl*S(Uc) = Uc, and hence (Uc)*SUc. Then, x(α,β) ∉ (Uc)*s, and so there, exists VN(x(α,β)) such that VUcI. putting H = VUc, then x(α,β)H, andHI. Thus, x(α,β)VH = VHc = V ∩(VHc)c = V ∩(V cU) = VUU. Hence, x(α,β)VHU, where VHβ(I, S). Hence, U is denotes the union of the sets in β(I, S).

### Theorem 3.8

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

Proof

Let US. Then, U = U – 0~β(I, S). Hence, Sβ(I, S). Now, let Gβ(I, S), then there exists US and HI such that G = UH. Then, Cl*S(Gc) = Cl*S(UH)c = (UH)c∪((UH)c)*S = (UcH)∪(UcH)*S. But HI, and then by Theorem 3.1.(8), (UcH)*S = (Uc)*S; and so, Cl*S(UH)c = UcH ∪(Uc)*SUcH. Hence, Cl*S(UH)cUcH = (UH)c, but (UH)cCl*S(UH)c. Hence, Cl*S(UH)c = (UH)c. Therefore, UHS*(I). Hence, β(I, S) ⊆ S*(I). Consequently, Sβ(I, S) ⊆ S*(I).

### 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).

Proof

It follows from Theorem 3.8.

### 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 UN(x(α,β)), then UAI. So AI. Therefore, A*S = 1~ if AI and A*S = 0~ if AI. This implies that we have Cl*S(A) = AA*S = 1~ if AI and Cl*S(A) = A if AI for any intuitionistic fuzzy set A of X. Hence, T* = {M: McI}. Let ST*(I) be the supremum intuitionistic fuzzy supra topology of S and T*(I), i.e. the smallest intuitionistic fuzzy supra topology generated by ST*(I). Then, we have the following theorem.

### Theorem 3.10

S*(I) = ST*(I).

Proof

Follows from the fact that β forms a basis for S*(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 UN(x(α,β)) such that UAI, then AI.

### 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, AA*S = 0~ implies that AI,

• (3) For every intuitionistic fuzzy set A in X, AA*SI,

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

Proof

(1) ⇒ (2) The proof is obvious.

(2) ⇒ (3) For any intuitionistic fuzzy set A in X, AA*SA, and (AA*S) ∩ (AA*S)*S ⊆ (AA*S) ∩ A*S = 0~. By (2), we obtain AA*SI.

(3) ⇒ (4) By (3), for every intuitionistic fuzzy set A in X, AA*SI. Let AA*S = EI; then A = E∪(AA*S) and by Theorem 3.1.(6) A*SE*S ∪ (AA*S)*S = (AA*S)*S and AA*SA then (AA*S)*SA*S therefore A*S = (AA*S)*S, we have AA*S = A∩(AA*S)*S ⊆ (AA*S)*S and AA*SA. By the assumption AA*S = 0~, and hence A = AA*SI.

(4) ⇒ (1) Let an intuitionistic fuzzy set in X and assume that for every x(α,β), there exists UN(x(α,β)) such that UAI. Then, AA*S = 0~. Suppose that A contains B such that BB*S. Then, B = BB*SAA*S = 0~. Therefore, A contains no non-empty subset B with BB*S. Hence AI.

### 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, AA*S = 0~ implies that A*S = 0~,

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

Proof

First, we show that (1) holds if S is S-compatible with I. Let A be any intuitionistic fuzzy set in X and AA*S = 0~. By Theorem 4.1. AI; then, A*S = 0~.

(1) ⇒ (2) Assume that for every intuitionistic fuzzy set A in X, AA*S = 0~ implies that A*S = 0~. Let B = AA*S, then BB*S = (AA*S)∩(AA*S)*S = (A∩(A*S)c)∩ (A ∩ (A*S)c)*S) subseteq((A astScap)(A*S)c) ∩ (A*S) ∩ ((A*S)c)*S) = 0~. By (1), we have B*S = 0~. Hence, (AA*S)*S = 0~.

(2) ⇒ (1) Assume that for every intuitionistic fuzzy set A in X, AA*S = 0~, and let B = AA*S, then A = B ∪ (AA*S) = B ∪ 0~ = B, then A*S = B*S = (AA*S)*S = 0~.

### Theorem 4.3

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

• (1) SI = 0~,

• (2) $1∼*S=1∼$.

Proof

(1) ⇒ (2) Let SI = 0~. Then $1∼*S=ClS(1∼)=1∼$.

$(2)⇒(1) 1∼=1∼*S={x(α,β)∈X:U∩1∼=U∉I, for every u∈N(x(α,β))}$. Hence ClS(S) ∩ I = 0~.

### Definition 5.1

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. An operator ΨS : P(X) → S is defined as follows for every intuitionistic fuzzy set A in X, ΨS(A) ={xα,β intuitionistic fuzzy point: there exists MNS(xα,β) such that MAI}. We observe that ΨS(A) = 1~ – (1~A)*S. The behaviors of the operator ΨS 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) ΨS(A) is intuitionistic fuzzy supra open set.

• (2) IntS(A) ⊆ ΨS(A).

• (3) If AB, then ΨS(A) ⊆ ΨS(B).

• (4) ΨS(AB) ⊆ ΨS(A) ∩ ΨS(B).

• (5) ΨS(A) ∪ ΨS(B) ⊆ ΨS(AB).

• (6) If US, then U ⊆ ΨS(U).

• (7) ΨS(A) ⊆ ΨSS(A)).

• (8) ΨS(A) = ΨSS(A)) if and only if (1~A)*S = ((1~A)*S)*S.

• (9) If (AB) ∪ (BA) ∈ I, then ΨS(A) = ΨS(B).

• (10) If EI, then $ΨS(E)=1∼-1∼*S$.

• (11) If EI, then ΨS(AE) = ΨS(A).

• (12) If EI, then ΨS(AE) = ΨS(A).

Proof

(1) Since (1~A)*S is an intuitionistic fuzzy supra closed set, then 1~–(1~A)*S is an intuitionistic fuzzy supra open set. Hence, ΨS(A) is an intuitionistic fuzzy supra open set.

(2) From the definition of the ΨS operator, ΨS(A) = 1~ – (1~A)*S. Then, 1~ClS(1~A) ⊆ 1~ –(1~A)*S = ΨS(A), from Theorem 3.1.(4). Hence, IntS(A) ⊆ ΨS(A).

(3) Let AB. Then, (1~B) ⊆ (1~A). ThenSubsequently, from Theorem 3.1.(2), (1~B)*S ⊆ (1~A)*S. Therefore, ΨS(A) ⊆ ΨS(B).

(4) We have ABA and ABB. Then from (3), ΨS(AB) ⊆ ΨS(A) ∩ ΨS(B).

(5) We have AAB and BAB. Then, from (3), ΨS(A) ∪ ΨS(B) ⊆ ΨS(AB).

(6) Let US. Then, (1~U) beis an intuitionistic fuzzy supra closed set, and hence ClS(1~U) = (1~U). Then, (1~U)*SClS(1~U) = (1~U). Hence, U ⊆ 1~ – (1~U)*S, and so U ⊆ ΨS(U).

(7) From (2), ΨS(A)∈S, and from (6), ΨS(A)⊆ΨSS(A)).

(8) Let ΨS(A) = ΨSS(A)). Then 1~ – (1~A)*S = ΨS(1~–(1~A)*S) = 1~–(1~–(1~–(1~A)*S)*S) = 1~ – ((1~A)*S)*S. Therefore, (1~A)*S = ((1~A)*S)*S. Conversely, suppose that (1~A)*S = ((1~A)*S)*S holds. Then, 1~–(1~A)*S = 1~–((1~A)*S)*S and 1~–(1~A)*S = 1~–(1~–(1~–(1~A)*S))*S = 1~ – (1~ – ΨS(A))*S. Hence, ΨS(A) = ΨSS(A)).

(9) Let (AB) ∪ (BA) ∈ I, and let AB = E1, BA = E2. We observe that E1, E2I by heredity, and B = (AE1) ∪ E2. Thus, ΨS(A) = ΨS(AE1) = ΨS((AE1) ∪ E2) = ΨS(B).

(10) By Theorem 3.1.(9), we obtain if EI, then $ΨS(E)=1∼-1∼*S$.

(11) This follows from Theorem 3.1.(9), and ΨS(AE) = 1~ – (1~ – (AE))*S = 1~ – ((1~A) ∪ E)*S = 1~ – (1~A)*S = ΨS(A).

(12) This follows from Theorem 3.1.(9), and ΨS(AE) = 1~ – (1~ – (AE))*S = 1~ – ((1~A) – E)*S = 1~ – (1~A)*S = ΨS(A).

### Theorem 5.2

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

Proof

Let η = {AP(X) : A ⊆ ΨS(A)}. By Theorem 3.1.(1), $0∼*S=0∼$ and $ΨS(1∼)=1∼-(1∼-1∼)*S=1∼-0∼*S=1∼$. Moreover, ΨS(0~) = 1~ – (1~ – 0~)*S = 1~ – 1~ = 0~. Therefore, we observe obtain that 0~ ⊆ ΨS(0~) and 1~ ⊆ ΨS(1~) = 1~, and thus 0~, 1~η. Now, if {Aα : α ∈Δ} ⊆ η, then Aα ⊆ ΨS(Aα) ⊆ ΨS(∪Aα) for every α; and hence, ∪Aα ⊆ ΨS(∪Aα). This shows that η is an intuitionistic fuzzy supra topology.

### Definition 5.2

An intuitionistic fuzzy ideal I in a space (X, S, I) is called an S-codense intuitionistic fuzzy ideal if SI = {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, $1∼=1∼*S$.

Proof

It is obvious.

### 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 ΨS-C-intuitionistic fuzzy set if AClSS(A)). The collection of all ΨS-C-intuitionistic fuzzy set in (X, S, I) is denoted by ΨS(X, S).

### Theorem 5.4

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

Proof

From Theorem 5.1.(6) it follows that S ⊆ ΨS(X, S).

### Theorem 5.5

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

Proof

For each α ∈ Δ,

$Aα⊆ClS(ΨS(Aα))⊆ClS(ΨS(∪α∈ΔAα)).$

This implies that ∪α∈ΔAαClSS(∪α∈ΔAα)). Thus ∪α∈ΔAα ∈ ΨS(X, S).

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Fadhil Abbas is an Ph.D. student at Johannes Kepler University, Linz, Austria since March 2018. He worked as an Assistant at the University of Telafer. He completed his Master’s degree from Gazi University in Fuzzy Topology in 2011. His current research interests include fuzzy topology and intuitionistic fuzzy topology, etc. He has has published more than three articles.

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 93-100

Published online March 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.1.93

## Ideals on Intuitionistic Fuzzy Supra Topological Spaces

Salzburger Straße 195, Linz, Austria

Received: November 21, 2020; Revised: January 29, 2021; Accepted: March 3, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

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

Keywords: Intuitionistic fuzzy-I-supra topology, Intuitionistic fuzzy s-local function, Intuitionistic fuzzy set operator ΨS

### 1. Introduction

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 ΨS and study its properties.

### Definition 2.1( [15])

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

### Definition 2.2 ( [15])

1~ = {< x,1, 0 >: xX} and 0~ = {< x, 0, 1 >: xX}.

### Definition 2.3 ( [16])

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

• AB if and only if μA(x) ⩽ μB(x) and νA(x) ≥ νB(x) for every xX.

• A = B if and only if AB and BA.

• Ac = {< x,νA(x), μA(x) : x ∈X >}.

• AB = {< x,μA(x) ∧ μB(x), μA(x) ∨ μB(x) >}.

• AB = {< x,μA(x) ∨ μB(x), μA(x) ∧ μB(x) >}.

### Definition 2.4 ( [8])

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

$x(α,β)={:x∈X}$

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 xX 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 Ac 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

$IntS(A)=∪{U:U⊆A,U∈S},$

and

$ClS(A)=∩{F:A⊆F,Fc∈S},$

respectively.

### Corollary 2.1

From Definition 2.6, IntS(A) is a fuzzy supra open set, and ClS(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 xX such that μA(x) > νB(x) or νA(x) < μB(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 > νA(x) orβ < μA(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 < μA(x) and β > νA(x).

### Definition 2.10 ([8])

Let x(α.β) an intuitionistic fuzzy point . Then, x(α,β)A if αμA(x) and βνA(x).

### Definition 2.11 ( [2])

Let (X, S) be an intuitionistic fuzzy supra topological space and let AX. Then, A is the s-neighborhood of an intuitionistic fuzzy point x(α,β) if there is US with x(α,β)UA(x(α,β)qUA). 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 S1S2. 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 βS. Then, β is called a base for the intuitionistic fuzzy supra topology S if every intuitionistic fuzzy supra open set US 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 Bβ with x(α,β)BU.

### 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~,

• Ac(A) for every AX,

• c(A) ∪ c(B) ⊆ c(AB) for every A, BX,

• c(c(A)) = c(A) for every AX.

### 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 = {AP(X) : c(Ac) = Ac} 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

• AI and BA, then BI,

• AI and BI, then ABI.

### 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 UN(x(α,β)) and A*S(I, S) = ∪{x(α,β)X: AUI, for every UN(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 = ClS(A), for any AX and I = P(X) ⇔ A*S = 0~.

### Theorem 3.1

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

• (1) $0∼*S=0∼$,

• (2) If AB, then A*SB*S,

• (3) If I1I2, then A*S(I2) ⊆ A*S(I1),

• (4) A*S = ClS(A*S) ⊆ ClS(A),

• (5) (A*S)*SA*S,

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

• (7) A*SB*S ⊆ (AB)*S,

• (8) (AB)*SA*SB*S,

• (9) If EI, then (AE)*S = A*S = (AE)*S,

• (10) If US, then UA*S = U ∩ (UA)*S ⊆ (UA)*S,

• (11) If EI, then E*S = 0~,

• (12) EI, then $(1∼-E)*S=1∼*S$.

Proof

(1) Clear from the definition of intuitionistic fuzzy S-local function .

(2) Since AB, let x(α,β)A*S, then AUI for every UN(x(α,β)). By hypothesis, we obtainget BUI, then x(α,β)B. Therefore, A*SB*S.

(3) Cleary, I1I2 implies A*S(I2) ⊆ A*S(I1), as there may be other intuitionistic fuzzy sets that which belong to I2 so that for an ituitionistic fuzzy point x(α,β)A*S(I1) but x(α,β)A*S(I2).

(4) Since {0~} ⊆ I for any intuitionistic fuzzy ideal on X, therefore by (3) and Example 3.1. A*S(I) ⊆ A*S({0~}) = ClS(A), for any intuitionistic fuzzy set A in X. Suppose, x1(α,β)ClS(A*S) such that for every UN(x1(α,β)), A*SU ≠ 0~ there exists x2(α,β)A*SU such that for every VN(x2(α,β)), then AVI. Since UVN(x2(α,β)), then A ∩ (UV ) ∉ I, which leads to AUI for every UN(x1(α,β)), therefore x1(α,β)A*S; and so ClS(A*S) ⊆ A*S while the other inclusion follows directly. Hence, A*S = ClS(A*S) ⊆ ClS(A).

(5) From (4), (A*S)*SClS(A*S) = A*S.

(6) Clear from (4).

(7) We know that AAB and BAB. Then, from (2), A*S ⊆ (AB)*S and B*S ⊆ (AB)*S. Hence, A*SB*S ⊆ (AB)*S.

(8) We know that (AB) ⊆ A and (AB) ⊆ B. Then, from (2), (AB)*SA*S and (AB)*SA*S. Hence, (AB)*SA*SB*S.

(9) Since A ⊆ (AE), then from (2) A*S ⊆ (AE)*S. Let x(α,β) ∈ (AE)*S. Then, for every UN(x(α,β)) such that U ∩ (AE) ∉ I. This implies that UAI(if possible suppose that UAI. Again, UEE implies UEI and hence U ∩ (AE) ∈ I, contradiction). Hence, x(α,β)A*S and (AE)*SA*S then (AE)*S = A*S.

Since (AE) ⊆ A, then from (2), (AE)*SA*S. For the reverse inclusion, let x(α,β)A*S. We claim that x(α,β) ∈ (AE)*S, if not, then there is UN(x(α,β)) such that U ∩ (AE) ∈ I. Given that EI, then E ∪ (U ∩ (AE)) ∈ I. This implies that E ∪ (UA) ∈ I. So, UAI, a contradiction to the fact that x(α.β)A*S. Hence, A*S ⊆ (AE)*S. Then, A*S = (AE)*S; therefore, (AE)*S = A*S = (AE)*S.

(10) Since VAA, then from (2), (VA)*SA*S. So V ∩ (VA)*SVA*S.

(11) Clear from the definition of intuitionistic fuzzy S-local function.

(12) Clear from proof (9).

### Theorem 3.2

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let AX. If MS, MAI, then MA*S = 0~.

Proof

Let x(α,β)MA*S. Then, x(α,β)M and x(α,β)A*S implies UAI for every UN(x(α,β)). Since x(α,β)MS, then MA*SI.

### Theorem 3.3

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let AX. Then, (AA*S)*SA*S.

Proof

Let x(α,β)A*S. Then, there exists UN(x(α,β)) such that UAIUA*S = 0~ (By Theorem 3.2.). Hence, U ∩ (AA*S) = (UA) ∪ (UA*S) = UAI. Therefore, x(α,β) ∉ (AA*S)*S. Hence, (AA*S)*SA*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) = AA*S for any AX, is an intuitionistic fuzzy supra closure operator and hence it generates an intuitionistic fuzzy supra topology S*(I) = {AP(X) : Cl*S(Ac) = Ac}, which is finer than S.

Proof

(1) By Theorem 3.1.(1), $0∼*S=0∼$, we have Cl*S(0~) = 0~.

(2) Clear ACl*S for every intuitionistic fuzzy set A.

(3) Let A and, B be two intuitionistic fuzzy sets. Then, Cl*S(A) ∪ Cl*S(B) = (AA*S) ∪ (BB*S) = (AB) ∪ (A*SB*S) ⊆ (AB) ∪ (AB)*S = Cl*S(AB) (by Theorem 3.1 (7). Hence, Cl*S(A)∪Cl*S(B) ⊆ Cl*S(AB).

(4) Let A be any intuitionistic fuzzy set. Since, by (2), ACl*S(A), then Cl*S(A) ⊆ Cl*S(Cl*S(A)). On the other hand, Cl*S(Cl*S(A)) = Cl*S(AA*S) = (AA*S)∪(AA*S)*SAA*SA*S = Cl*S(A) (by Theorem 3.3), it follows that Cl*S(Cl*S(A)) ⊆ Cl*S(A). Hence, Cl*S(Cl*S(A)) = Cl*S(A). Consequently, Cl*S(A) is an intuitionistic fuzzy supra closure operator. Also, it is also easy to show that the collection S*(I) = {AP(X) : Cl*S(Ac) = Ac} is an intuitionistic fuzzy supra topology on X, which is called the intuitionistic fuzzy supra topology induced by the intuitionistic fuzzy supra closure operator.

### Example 3.2

For any intuitionistic fuzzy ideal on X if I = {0~} ⇒ Cl*S(A) = AA*S = AClS(A) = ClS(A) for every AP(X). So S*({0~}) = S, and if I = P(X) ⇒ Cl*S(A) = A, because A*S = 0~ for every AP(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~} ⊆ IP(X). So we can conclude by Theorem 3.1.(2) S*({0~}) ⊆ S*(I) ⊆ S*(P(X)), i.e. SS*(I), for any intuitionistic fuzzy ideal I on X. In particular, we have for any two intuitionistic fuzzy ideals I1 and I2 on X, I1I2S*(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, S1S2 implies

• (1) A*S(S2, I) ⊆ A*S(S1, I) for every AP(X),

• (2) $S1*(I)⊆S2*(I)$.

Proof

(1) Since every S1 s-neighborhood of any intuitionistic fuzzy point x(α,β) is also ana S2 s-neighborhood of x(α,β). Therefore, A*S(S2, I) ⊆ A*S(S1, I).

(2) Clearly, $S1*(I)⊆S2*(I)$ as A*S(S2, I) ⊆ A*S(S1, 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*SA. Then, A = ClS(A*S) = Cl*S(A).

Proof

Clear.

### Theorem 3.7

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, the collection β(I, S) = {UH: US, HI} is a base for the intuitionistic fuzzy supra topology S*(I).

Proof

Let US*(I) and x(α,β)U. Then, Uc is an intuitionistic fuzzy S*-supra closed set suchso that Cl*S(Uc) = Uc, and hence (Uc)*SUc. Then, x(α,β) ∉ (Uc)*s, and so there, exists VN(x(α,β)) such that VUcI. putting H = VUc, then x(α,β)H, andHI. Thus, x(α,β)VH = VHc = V ∩(VHc)c = V ∩(V cU) = VUU. Hence, x(α,β)VHU, where VHβ(I, S). Hence, U is denotes the union of the sets in β(I, S).

### Theorem 3.8

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

Proof

Let US. Then, U = U – 0~β(I, S). Hence, Sβ(I, S). Now, let Gβ(I, S), then there exists US and HI such that G = UH. Then, Cl*S(Gc) = Cl*S(UH)c = (UH)c∪((UH)c)*S = (UcH)∪(UcH)*S. But HI, and then by Theorem 3.1.(8), (UcH)*S = (Uc)*S; and so, Cl*S(UH)c = UcH ∪(Uc)*SUcH. Hence, Cl*S(UH)cUcH = (UH)c, but (UH)cCl*S(UH)c. Hence, Cl*S(UH)c = (UH)c. Therefore, UHS*(I). Hence, β(I, S) ⊆ S*(I). Consequently, Sβ(I, S) ⊆ S*(I).

### 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).

Proof

It follows from Theorem 3.8.

### 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 UN(x(α,β)), then UAI. So AI. Therefore, A*S = 1~ if AI and A*S = 0~ if AI. This implies that we have Cl*S(A) = AA*S = 1~ if AI and Cl*S(A) = A if AI for any intuitionistic fuzzy set A of X. Hence, T* = {M: McI}. Let ST*(I) be the supremum intuitionistic fuzzy supra topology of S and T*(I), i.e. the smallest intuitionistic fuzzy supra topology generated by ST*(I). Then, we have the following theorem.

### Theorem 3.10

S*(I) = ST*(I).

Proof

Follows from the fact that β forms a basis for S*(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 UN(x(α,β)) such that UAI, then AI.

### 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, AA*S = 0~ implies that AI,

• (3) For every intuitionistic fuzzy set A in X, AA*SI,

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

Proof

(1) ⇒ (2) The proof is obvious.

(2) ⇒ (3) For any intuitionistic fuzzy set A in X, AA*SA, and (AA*S) ∩ (AA*S)*S ⊆ (AA*S) ∩ A*S = 0~. By (2), we obtain AA*SI.

(3) ⇒ (4) By (3), for every intuitionistic fuzzy set A in X, AA*SI. Let AA*S = EI; then A = E∪(AA*S) and by Theorem 3.1.(6) A*SE*S ∪ (AA*S)*S = (AA*S)*S and AA*SA then (AA*S)*SA*S therefore A*S = (AA*S)*S, we have AA*S = A∩(AA*S)*S ⊆ (AA*S)*S and AA*SA. By the assumption AA*S = 0~, and hence A = AA*SI.

(4) ⇒ (1) Let an intuitionistic fuzzy set in X and assume that for every x(α,β), there exists UN(x(α,β)) such that UAI. Then, AA*S = 0~. Suppose that A contains B such that BB*S. Then, B = BB*SAA*S = 0~. Therefore, A contains no non-empty subset B with BB*S. Hence AI.

### 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, AA*S = 0~ implies that A*S = 0~,

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

Proof

First, we show that (1) holds if S is S-compatible with I. Let A be any intuitionistic fuzzy set in X and AA*S = 0~. By Theorem 4.1. AI; then, A*S = 0~.

(1) ⇒ (2) Assume that for every intuitionistic fuzzy set A in X, AA*S = 0~ implies that A*S = 0~. Let B = AA*S, then BB*S = (AA*S)∩(AA*S)*S = (A∩(A*S)c)∩ (A ∩ (A*S)c)*S) subseteq((A astScap)(A*S)c) ∩ (A*S) ∩ ((A*S)c)*S) = 0~. By (1), we have B*S = 0~. Hence, (AA*S)*S = 0~.

(2) ⇒ (1) Assume that for every intuitionistic fuzzy set A in X, AA*S = 0~, and let B = AA*S, then A = B ∪ (AA*S) = B ∪ 0~ = B, then A*S = B*S = (AA*S)*S = 0~.

### Theorem 4.3

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

• (1) SI = 0~,

• (2) $1∼*S=1∼$.

Proof

(1) ⇒ (2) Let SI = 0~. Then $1∼*S=ClS(1∼)=1∼$.

$(2)⇒(1) 1∼=1∼*S={x(α,β)∈X:U∩1∼=U∉I, for every u∈N(x(α,β))}$. Hence ClS(S) ∩ I = 0~.

### Definition 5.1

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. An operator ΨS : P(X) → S is defined as follows for every intuitionistic fuzzy set A in X, ΨS(A) ={xα,β intuitionistic fuzzy point: there exists MNS(xα,β) such that MAI}. We observe that ΨS(A) = 1~ – (1~A)*S. The behaviors of the operator ΨS 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) ΨS(A) is intuitionistic fuzzy supra open set.

• (2) IntS(A) ⊆ ΨS(A).

• (3) If AB, then ΨS(A) ⊆ ΨS(B).

• (4) ΨS(AB) ⊆ ΨS(A) ∩ ΨS(B).

• (5) ΨS(A) ∪ ΨS(B) ⊆ ΨS(AB).

• (6) If US, then U ⊆ ΨS(U).

• (7) ΨS(A) ⊆ ΨSS(A)).

• (8) ΨS(A) = ΨSS(A)) if and only if (1~A)*S = ((1~A)*S)*S.

• (9) If (AB) ∪ (BA) ∈ I, then ΨS(A) = ΨS(B).

• (10) If EI, then $ΨS(E)=1∼-1∼*S$.

• (11) If EI, then ΨS(AE) = ΨS(A).

• (12) If EI, then ΨS(AE) = ΨS(A).

Proof

(1) Since (1~A)*S is an intuitionistic fuzzy supra closed set, then 1~–(1~A)*S is an intuitionistic fuzzy supra open set. Hence, ΨS(A) is an intuitionistic fuzzy supra open set.

(2) From the definition of the ΨS operator, ΨS(A) = 1~ – (1~A)*S. Then, 1~ClS(1~A) ⊆ 1~ –(1~A)*S = ΨS(A), from Theorem 3.1.(4). Hence, IntS(A) ⊆ ΨS(A).

(3) Let AB. Then, (1~B) ⊆ (1~A). ThenSubsequently, from Theorem 3.1.(2), (1~B)*S ⊆ (1~A)*S. Therefore, ΨS(A) ⊆ ΨS(B).

(4) We have ABA and ABB. Then from (3), ΨS(AB) ⊆ ΨS(A) ∩ ΨS(B).

(5) We have AAB and BAB. Then, from (3), ΨS(A) ∪ ΨS(B) ⊆ ΨS(AB).

(6) Let US. Then, (1~U) beis an intuitionistic fuzzy supra closed set, and hence ClS(1~U) = (1~U). Then, (1~U)*SClS(1~U) = (1~U). Hence, U ⊆ 1~ – (1~U)*S, and so U ⊆ ΨS(U).

(7) From (2), ΨS(A)∈S, and from (6), ΨS(A)⊆ΨSS(A)).

(8) Let ΨS(A) = ΨSS(A)). Then 1~ – (1~A)*S = ΨS(1~–(1~A)*S) = 1~–(1~–(1~–(1~A)*S)*S) = 1~ – ((1~A)*S)*S. Therefore, (1~A)*S = ((1~A)*S)*S. Conversely, suppose that (1~A)*S = ((1~A)*S)*S holds. Then, 1~–(1~A)*S = 1~–((1~A)*S)*S and 1~–(1~A)*S = 1~–(1~–(1~–(1~A)*S))*S = 1~ – (1~ – ΨS(A))*S. Hence, ΨS(A) = ΨSS(A)).

(9) Let (AB) ∪ (BA) ∈ I, and let AB = E1, BA = E2. We observe that E1, E2I by heredity, and B = (AE1) ∪ E2. Thus, ΨS(A) = ΨS(AE1) = ΨS((AE1) ∪ E2) = ΨS(B).

(10) By Theorem 3.1.(9), we obtain if EI, then $ΨS(E)=1∼-1∼*S$.

(11) This follows from Theorem 3.1.(9), and ΨS(AE) = 1~ – (1~ – (AE))*S = 1~ – ((1~A) ∪ E)*S = 1~ – (1~A)*S = ΨS(A).

(12) This follows from Theorem 3.1.(9), and ΨS(AE) = 1~ – (1~ – (AE))*S = 1~ – ((1~A) – E)*S = 1~ – (1~A)*S = ΨS(A).

### Theorem 5.2

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

Proof

Let η = {AP(X) : A ⊆ ΨS(A)}. By Theorem 3.1.(1), $0∼*S=0∼$ and $ΨS(1∼)=1∼-(1∼-1∼)*S=1∼-0∼*S=1∼$. Moreover, ΨS(0~) = 1~ – (1~ – 0~)*S = 1~ – 1~ = 0~. Therefore, we observe obtain that 0~ ⊆ ΨS(0~) and 1~ ⊆ ΨS(1~) = 1~, and thus 0~, 1~η. Now, if {Aα : α ∈Δ} ⊆ η, then Aα ⊆ ΨS(Aα) ⊆ ΨS(∪Aα) for every α; and hence, ∪Aα ⊆ ΨS(∪Aα). This shows that η is an intuitionistic fuzzy supra topology.

### Definition 5.2

An intuitionistic fuzzy ideal I in a space (X, S, I) is called an S-codense intuitionistic fuzzy ideal if SI = {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, $1∼=1∼*S$.

Proof

It is obvious.

### 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 ΨS-C-intuitionistic fuzzy set if AClSS(A)). The collection of all ΨS-C-intuitionistic fuzzy set in (X, S, I) is denoted by ΨS(X, S).

### Theorem 5.4

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

Proof

From Theorem 5.1.(6) it follows that S ⊆ ΨS(X, S).

### Theorem 5.5

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

Proof

For each α ∈ Δ,

$Aα⊆ClS(ΨS(Aα))⊆ClS(ΨS(∪α∈ΔAα)).$

This implies that ∪α∈ΔAαClSS(∪α∈ΔAα)). Thus ∪α∈ΔAα ∈ ΨS(X, S).

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