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

Fadhil Abbas

Salzburger Straße 195, Linz, Austria

**Correspondence to : **

Fadhil Abbas (fadhilhaman@gmail.com)

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 Ψ_{S} and study its properties.

**Keywords**: Intuitionistic fuzzy-I-supra topology, Intuitionistic fuzzy s-local function, Intuitionistic fuzzy set operator Ψ_{S}

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}

Let _{A}_{A}_{A}_{A}_{A}_{A}_{A}_{A}

1_{~} = {< _{~} = {<

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

A ⊆B if and only ifμ (_{A}x ) ⩽μ (_{B}x ) andν (_{A}x ) ≥ν (_{B}x ) for everyx ∈X .A =B if and only ifA ⊆B andB ⊆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 ) >}.

Let

is called an intuitionistic fuzzy point in X, where _{(}_{α,β}_{)}, _{(}_{α,β}_{)}, and _{(}_{α,β}_{)}.

A subclass S is called an intuitionistic fuzzy supra topology on X if 0_{~}, 1_{~} ∈ ^{c}

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.

From Definition 2.6, ^{S}^{S}

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 _{A}_{B}_{A}_{B}

Let _{(}_{α,β}_{)} an intuitionistic fuzzy point and let A an intuitionistic fuzzy set in X. We say that _{(}_{α,β}_{)} quasi-coincident with A, denoted by _{(}_{α,β}_{)}qA if and only if _{A}_{A}

Let _{(}_{α,β}_{)} an intuitionistic fuzzy point and let A an intuitionistic fuzzy set in X. Let _{(}_{α,β}_{)} is called properly contained in A if and only if, _{A}_{A}

Let _{(}_{α.β}_{)} an intuitionistic fuzzy point . Then, _{(}_{α,β}_{)} ∈ _{A}_{A}

Let (X, S) be an intuitionistic fuzzy supra topological space and let _{(}_{α,β}_{)} if there is _{(}_{α,β}_{)} ∈ _{(}_{α,β}_{)}_{(}_{α,β}_{)}) of all s-neighborhood of _{(}_{α,β}_{)} is called the s-neighborhood system of _{(}_{α,β}_{)}.

Let (_{1}) and (_{2}) be two intuitionistic fuzzy supra topologies and let _{1} ⊆ _{2}. Then, we say that _{2} is stronger than _{1} or _{1} is weaker than _{2}.

Let (X, S) be an intuitionistic fuzzy supra topological space and let _{(}_{α,β}_{)} ∈ _{(}_{α,β}_{)} ∈

A mapping

c (0_{~}) = 0_{~},A ⊆c (A ) for everyA ⊆X ,c (A ) ∪c (B ) ⊆c (A ∪B ) for everyA, B ⊆X ,c (c (A )) =c (A ) for everyA ⊆X .

Let ^{c}^{c}

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 andB ⊆A , thenB ∈I ,A ∈I andB ∈I , thenA ∪B ∈I .

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).

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 ^{*}^{S}_{(}_{α,β}_{)} such that if _{(}_{α,β}_{)}) and ^{*}^{S}_{(}_{α,β}_{)} ∈ _{(}_{α,β}_{)})}. We will occasionally write ^{*}^{S}^{*}^{S}

The simplest intuitionistic fuzzy ideal on X isare {0_{~}} and _{~}} ⇔ ^{*}^{S}^{S}^{*}^{S}_{~}.

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

(1)

${0}_{\sim}^{*S}={0}_{\sim}$ ,(2) If

A ⊆B , thenA ^{*} ⊆^{S}B ^{*} ,^{S}(3) If

I _{1}⊆I _{2}, thenA ^{*} (^{S}I _{2}) ⊆A ^{*} (^{S}I _{1}),(4)

A ^{*} =^{S}Cl (^{S}A ^{*} ) ⊆^{S}Cl (^{S}A ),(5) (

A ^{*} )^{S}^{*} ⊆^{S}A ^{*} ,^{S}(6)

A ^{*} is an intuitionistic fuzzy supra closed set,^{S}(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 , thenU ∩A ^{*} =^{S}U ∩ (U ∩A )^{*} ⊆ (^{S}U ∩A )^{*}^{S}, (11) If

E ∈I , thenE ^{*} = 0^{S}_{~},(12)

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

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

(2) Since _{(}_{α,β}_{)} ∈ ^{*}^{S}_{(}_{α,β}_{)}). By hypothesis, we obtainget _{(}_{α,β}_{)} ∈ ^{*}^{S}^{*}^{S}

(3) Cleary, _{1} ⊆ _{2} implies ^{*}^{S}_{2}) ⊆ ^{*}^{S}_{1}), as there may be other intuitionistic fuzzy sets that which belong to _{2} so that for an ituitionistic fuzzy point _{(}_{α,β}_{)} ∈ ^{*}^{S}_{1}) but _{(}_{α,β}_{)} ∉ ^{*}^{S}_{2}).

(4) Since {0_{~}} ⊆ ^{*}^{S}^{*}^{S}_{~}}) = ^{S}_{1(}_{α,β}_{)} ∈ ^{S}^{*}^{S}_{1(}_{α,β}_{)}), ^{*}^{S}_{~} there exists _{2(}_{α,β}_{)} ∈ ^{*}^{S}_{2(}_{α,β}_{)}), then _{2(}_{α,β}_{)}), then _{1(}_{α,β}_{)}), therefore _{1(}_{α,β}_{)} ∈ ^{*}^{S}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{S}^{*}^{S}^{S}

(5) From (4), (^{*}^{S}^{*}^{S}^{S}^{*}^{S}^{*}^{S}

(6) Clear from (4).

(7) We know that ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

(8) We know that (^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

(9) Since ^{*}^{S}^{*}^{S}^{(}^{α,β}^{)} ∈ (^{*}^{S}_{(}_{α,β}_{)}) such that _{(}_{α,β}_{)} ∈ ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

Since (^{*}^{S}^{*}^{S}_{(}_{α,β}_{)} ∈ ^{*}^{S}_{(}_{α,β}_{)} ∈ (^{*}^{S}_{(}_{α,β}_{)}) such that _{(}_{α.β}_{)} ∈ ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

(10) Since ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

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

(12) Clear from proof (9).

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let ^{*}^{S}_{~}.

Let _{(}_{α,β}_{)} ∈ ^{*}^{S}_{(}_{α,β}_{)} ∈ _{(}_{α,β}_{)} ∈ ^{*}^{S}_{(}_{α,β}_{)}). Since _{(}_{α,β}_{)} ∈ ^{*}^{S}

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let ^{*}^{S}^{*}^{S}^{*}^{S}

Let _{(}_{α,β}_{)} ∉ ^{*}^{S}_{(}_{α,β}_{)}) such that ^{*}^{S}_{~} (By Theorem 3.2.). Hence, ^{*}^{S}^{*}^{S}_{(}_{α,β}_{)} ∉ (^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, the operator ^{*}^{S}^{*}^{S}^{*}^{S}^{*}(^{*}^{S}^{c}^{c}

(1) By Theorem 3.1.(1), ^{*}^{S}_{~}) = 0_{~}.

(2) Clear ^{*}^{S}

(3) Let A and, B be two intuitionistic fuzzy sets. Then, ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

(4) Let A be any intuitionistic fuzzy set. Since, by (2), ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}(^{*}^{S}^{c}^{c}

For any intuitionistic fuzzy ideal on X if _{~}} ⇒ ^{*}^{S}^{*}^{S}^{S}^{S}^{*}({0_{~}}) = ^{*}^{S}^{*}^{S}_{~} for every ^{*}(_{~}} and P(X) are the two extreme intuitionistic fuzzy ideals on X, therefore for any intuitionistic fuzzy ideal I on X, we have {0_{~}} ⊆ ^{*}({0_{~}}) ⊆ ^{*}(^{*}(^{*}(_{1} and _{2} on X, _{1} ⊆ _{2} ⇒ ^{*}(_{1}) ⊆ ^{*}(_{2}).

Let _{1}, _{2} be two intuitionistic fuzzy supra topologies on X. Then, for any intuitionistic fuzzy ideal I on X, _{1} ⊆ _{2} implies

(1)

A ^{*} (^{S}S _{2},I ) ⊆A ^{*} (^{S}S _{1},I ) for everyA ∈P (X ),(2)

${S}_{1}^{*}(I)\subseteq {S}_{2}^{*}(I)$ .

(1) Since every _{1} s-neighborhood of any intuitionistic fuzzy point _{(}_{α,β}_{)} is also ana _{2} s-neighborhood of _{(}_{α,β}_{)}. Therefore, ^{*}^{S}_{2}, ^{*}^{S}_{1},

(2) Clearly, ^{*}^{S}_{2}, ^{*}^{S}_{1},

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

Clear.

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, the collection ^{*}(

Let ^{*}(_{(}_{α,β}_{)} ∈ ^{c}^{*}-supra closed set suchso that ^{*}^{S}^{c}^{c}^{c}^{*}^{S}^{c}_{(}_{α,β}_{)} ∉ (^{c}^{*}^{s}_{(}_{α,β}_{)}) such that ^{c}^{c}_{(}_{α,β}_{)} ∉ _{(}_{α,β}_{)} ∈ ^{c}^{c}^{c}^{c}_{(}_{α,β}_{)} ∈

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

Let _{~} ∈ ^{*}^{S}^{c}^{*}^{S}^{c}^{c}^{c}^{*}^{S}^{c}^{c}^{*}^{S}^{c}^{*}^{S}^{c}^{*}^{S}^{*}^{S}^{c}^{c}^{c}^{*}^{S}^{c}^{*}^{S}^{c}^{c}^{c}^{c}^{*}^{S}^{c}^{*}^{S}^{c}^{c}^{*}(^{*}(^{*}(

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

It follows from Theorem 3.8.

Let T be the intuitionistic fuzzy indiscrete supra topology on X, i.e. _{~}, 1_{~}}. So 1_{~} is the only s-neighborhoods of _{(}_{α,β}_{)}. Now, _{(}_{α,β}_{)} ∈ ^{*}^{S}_{(}_{α,β}_{)}), then ^{*}^{S}_{~} if ^{*}^{S}_{~} if ^{*}^{S}^{*}^{S}_{~} if ^{*}^{S}^{*} = {^{c}^{*}(^{*}(^{*}(

^{*}(^{*}(

Follows from the fact that ^{*}(

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 _{(}_{α,β}_{)} ∈ _{(}_{α,β}_{)}) such that

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 ^{*} = 0^{S}_{~}implies thatA ∈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 ^{*} , then^{S}A ∈I .

(1) ⇒ (2) The proof is obvious.

(2) ⇒ (3) For any intuitionistic fuzzy set A in X, ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}_{~}. By (2), we obtain ^{*}^{S}

(3) ⇒ (4) By (3), for every intuitionistic fuzzy set A in X, ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}_{~}, and hence ^{*}^{S}

(4) ⇒ (1) Let an intuitionistic fuzzy set in X and assume that for every _{(}_{α,β}_{)}, there exists _{(}_{α,β}_{)}) such that ^{*}^{S}_{~}. Suppose that A contains B such that ^{*}^{S}^{*}^{S}^{*}^{S}_{~}. Therefore, A contains no non-empty subset B with ^{*}^{S}

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 ^{*} = 0^{S}_{~}implies thatA ^{*} = 0^{S}_{~},(2) For every intuitionistic fuzzy set A in X, (

A –A ^{*} )^{S}^{*} = 0^{S}_{~}.

First, we show that (1) holds if S is S-compatible with I. Let A be any intuitionistic fuzzy set in X and ^{*}^{S}_{~}. By Theorem 4.1. ^{*}^{S}_{~}.

(1) ⇒ (2) Assume that for every intuitionistic fuzzy set A in X, ^{*}^{S}_{~} implies that ^{*}^{S}_{~}. Let ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{c}^{*}^{S}^{c}^{*}^{S}^{s}^{*}^{S}^{c}^{*}^{S}^{*}^{S}^{c}^{*}^{S}_{~}. By (1), we have ^{*}^{S}_{~}. Hence, (^{*}^{S}^{*}^{S}_{~}.

(2) ⇒ (1) Assume that for every intuitionistic fuzzy set A in X, ^{*}^{S}_{~}, and let ^{*}^{S}^{*}^{S}_{~} = ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}_{~}.

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

(1)

S ∩I = 0_{~},(2)

${1}_{\sim}^{*S}={1}_{\sim}$ .

(1) ⇒ (2) Let _{~}. Then

^{S}_{~}.

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. An operator Ψ_{S}_{S}_{α,β}^{S}_{α,β}_{S}_{~} – (1_{~} – ^{*}^{S}_{S}

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 , thenU ⊆ Ψ (_{S}U ).(7) Ψ

(_{S}A ) ⊆ Ψ (Ψ_{S} (_{S}A )).(8) Ψ

(_{S}A ) = Ψ (Ψ_{S} (_{S}A )) if and only if (1_{~}–A )^{*} = ((1^{S}_{~}–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 ).

(1) Since (1_{~} – ^{*}^{S}_{~}–(1_{~}–^{*}^{S}_{S}

(2) From the definition of the Ψ_{S}_{S}_{~} – (1_{~} –^{*}^{S}_{~} –^{S}_{~} –_{~} –(1_{~} –^{*}^{S}_{S}^{S}_{S}

(3) Let _{~} – _{~} – _{~} – ^{*}^{S}_{~} – ^{*}^{S}_{S}_{S}

(4) We have _{S}_{S}_{S}

(5) We have _{S}_{S}_{S}

(6) Let _{~} – ^{S}_{~} – _{~} – _{~} – ^{*}^{S}^{S}_{~} – _{~} – _{~} – (1_{~} – ^{*}^{S}_{S}

(7) From (2), Ψ_{S}_{S}_{S}_{S}

(8) Let Ψ_{S}_{S}_{S}_{~} – (1_{~} – ^{*}^{S}_{S}_{~}–(1_{~}–^{*}^{S}_{~}–(1_{~}–(1_{~}–(1_{~}–^{*}^{S}^{*}^{S}_{~} – ((1_{~} – ^{*}^{S}^{*}^{S}_{~} – ^{*}^{S}_{~} – ^{*}^{S}^{*}^{S}_{~} – ^{*}^{S}_{~} – ^{*}^{S}^{*}^{S}_{~}–(1_{~}–^{*}^{S}_{~}–((1_{~}–^{*}^{S}^{*}^{S}_{~}–(1_{~}–^{*}^{S}_{~}–(1_{~}–(1_{~}–(1_{~}–^{*}^{S}^{*}^{S}_{~} – (1_{~} – Ψ_{S}^{*}^{S}_{S}_{S}_{S}

(9) Let (_{1}, _{2}. We observe that _{1}, _{2} ∈ _{1}) ∪ _{2}. Thus, Ψ_{S}_{S}_{1}) = Ψ_{S}_{1}) ∪ _{2}) = Ψ_{S}

(10) By Theorem 3.1.(9), we obtain if

(11) This follows from Theorem 3.1.(9), and Ψ_{S}_{~} – (1_{~} – (^{*}^{S}_{~} – ((1_{~} – ^{*}^{S}_{~} – (1_{~} – ^{*}^{S}_{S}

(12) This follows from Theorem 3.1.(9), and Ψ_{S}_{~} – (1_{~} – (^{*}^{S}_{~} – ((1_{~} – ^{*}^{S}_{~} – (1_{~} – ^{*}^{S}_{S}

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. If _{S}

Let _{S}_{S}_{~}) = 1_{~} – (1_{~} – 0_{~})^{*}^{S}_{~} – 1_{~} = 0_{~}. Therefore, we observe obtain that 0_{~} ⊆ Ψ_{S}_{~}) and 1_{~} ⊆ Ψ_{S}_{~}) = 1_{~}, and thus 0_{~}, 1_{~} ∈ _{α}_{α}_{S}_{α}_{S}_{α}_{α}_{S}_{α}

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

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let I be S-codense with S. Then,

It is obvious.

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. An intuitionistic fuzzy set A in X is called a Ψ_{S}^{S}_{S}_{S}_{S}

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. If _{S}

From Theorem 5.1.(6) it follows that _{S}

Let {_{α}_{S}_{α}_{∈Δ}_{α}_{S}

For each

This implies that ∪_{α}_{∈Δ}_{α}^{S}_{S}_{α}_{∈Δ}_{α}_{α}_{∈Δ}_{α}_{S}

No potential conflict of interest relevant to this article was reported.

- Zadeh, LA (1965). Fuzzy sets. Information and Control.
*8*, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X - Atanassov, KT (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems.
*20*, 87-96. https://doi.org/10.1016/S0165-0114(86)80034-3 - Atanassov, KT (1994). New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems.
*61*, 137-142. - Atanasov, KT (1999). Intuitionistic Fuzzy Sets: Theory and Applications. Berlin, Germany: Springr
- Szmidt, E, and Kacprzyk, J (2001). Intuitionistic fuzzy sets in some medical applications. Notes on IFS.
*7*, 58-64. - Szmidt, E, and Kacprzyk, J (2004). Medical diagnostic reasoning using a similarity measure for intuitionistic fuzzy sets. Notes on IFS.
*10*, 61-69. - Atanassov, K 1988. Review and new results on intuitionistic fuzzy sets., Mathematical Foundations of Artificial Intelligence Seminar, Sofia, Bulgaria.
- Coker, D, and Demirci, M (1995). On intuitionistic fuzzy points. Notes on IFS.
*1*, 79-84. - Kuratowski, K (1966). Topology (Volume I). New York, NY: Academic Press
- Vaidyanathaswamy, R (1944). The localisation theory in settopology. Proceedings of the Indian Academy of Sciences-Section A.
*20*, 51-61. https://doi.org/10.1007/BF03048958 - Jankovic, D, and Hamlett, TR (1990). New topologies from old via ideals. The American Mathematical Monthly.
*97*, 295-310. https://doi.org/10.1080/00029890.1990.11995593 - Mashhour, AS, Allam, AA, Mahmoud, FS, and Khedr, FH (1983). On supra topological spaces. Indian Journal of Pure and Applied Mathematics.
*14*, 502-510. - Turanl, N 1999. On intuitionistic fuzzy supra topological spaces., Proceedings of International Conference on Modeling and Simulation, Spain, pp.69-77.
- Kandil, A, Tantawy, OA, El-Sheikh, SA, and Hazza, SA (2015). New supra topologies from old via ideals. Journal of New Theory.
*2015*, 1-5. - Atanassov, KT (1984). Intuitionistic fuzzy sets. VII ITKR’s Session, Sofia (deposed in Central Sci-Technical Library of Bulg Acad of Sci, 1697/84). Sofia, Bulgaria: Bulgarian Academy of Sciences
- Riecan, B, and Atanassov, KT (2008). Some properties of two operations over intuitionistic fuzzy sets. Notes Intuit Fuzzy Sets.
*14*, 6-10.

E-mail: fadhilhaman@gmail.com

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

Copyright © The Korean Institute of Intelligent Systems.

Fadhil Abbas

Salzburger Straße 195, Linz, Austria

**Correspondence to:**Fadhil Abbas (fadhilhaman@gmail.com)

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 Ψ_{S} and study its properties.

**Keywords**: Intuitionistic fuzzy-I-supra topology, Intuitionistic fuzzy s-local function, Intuitionistic fuzzy set operator Ψ_{S}

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}

Let _{A}_{A}_{A}_{A}_{A}_{A}_{A}_{A}

1_{~} = {< _{~} = {<

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

A ⊆B if and only ifμ (_{A}x ) ⩽μ (_{B}x ) andν (_{A}x ) ≥ν (_{B}x ) for everyx ∈X .A =B if and only ifA ⊆B andB ⊆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 ) >}.

Let

is called an intuitionistic fuzzy point in X, where _{(}_{α,β}_{)}, _{(}_{α,β}_{)}, and _{(}_{α,β}_{)}.

A subclass S is called an intuitionistic fuzzy supra topology on X if 0_{~}, 1_{~} ∈ ^{c}

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.

From Definition 2.6, ^{S}^{S}

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 _{A}_{B}_{A}_{B}

Let _{(}_{α,β}_{)} an intuitionistic fuzzy point and let A an intuitionistic fuzzy set in X. We say that _{(}_{α,β}_{)} quasi-coincident with A, denoted by _{(}_{α,β}_{)}qA if and only if _{A}_{A}

Let _{(}_{α,β}_{)} an intuitionistic fuzzy point and let A an intuitionistic fuzzy set in X. Let _{(}_{α,β}_{)} is called properly contained in A if and only if, _{A}_{A}

Let _{(}_{α.β}_{)} an intuitionistic fuzzy point . Then, _{(}_{α,β}_{)} ∈ _{A}_{A}

Let (X, S) be an intuitionistic fuzzy supra topological space and let _{(}_{α,β}_{)} if there is _{(}_{α,β}_{)} ∈ _{(}_{α,β}_{)}_{(}_{α,β}_{)}) of all s-neighborhood of _{(}_{α,β}_{)} is called the s-neighborhood system of _{(}_{α,β}_{)}.

Let (_{1}) and (_{2}) be two intuitionistic fuzzy supra topologies and let _{1} ⊆ _{2}. Then, we say that _{2} is stronger than _{1} or _{1} is weaker than _{2}.

Let (X, S) be an intuitionistic fuzzy supra topological space and let _{(}_{α,β}_{)} ∈ _{(}_{α,β}_{)} ∈

A mapping

c (0_{~}) = 0_{~},A ⊆c (A ) for everyA ⊆X ,c (A ) ∪c (B ) ⊆c (A ∪B ) for everyA, B ⊆X ,c (c (A )) =c (A ) for everyA ⊆X .

Let ^{c}^{c}

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 andB ⊆A , thenB ∈I ,A ∈I andB ∈I , thenA ∪B ∈I .

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).

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 ^{*}^{S}_{(}_{α,β}_{)} such that if _{(}_{α,β}_{)}) and ^{*}^{S}_{(}_{α,β}_{)} ∈ _{(}_{α,β}_{)})}. We will occasionally write ^{*}^{S}^{*}^{S}

The simplest intuitionistic fuzzy ideal on X isare {0_{~}} and _{~}} ⇔ ^{*}^{S}^{S}^{*}^{S}_{~}.

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

(1)

${0}_{\sim}^{*S}={0}_{\sim}$ ,(2) If

A ⊆B , thenA ^{*} ⊆^{S}B ^{*} ,^{S}(3) If

I _{1}⊆I _{2}, thenA ^{*} (^{S}I _{2}) ⊆A ^{*} (^{S}I _{1}),(4)

A ^{*} =^{S}Cl (^{S}A ^{*} ) ⊆^{S}Cl (^{S}A ),(5) (

A ^{*} )^{S}^{*} ⊆^{S}A ^{*} ,^{S}(6)

A ^{*} is an intuitionistic fuzzy supra closed set,^{S}(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 , thenU ∩A ^{*} =^{S}U ∩ (U ∩A )^{*} ⊆ (^{S}U ∩A )^{*}^{S}, (11) If

E ∈I , thenE ^{*} = 0^{S}_{~},(12)

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

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

(2) Since _{(}_{α,β}_{)} ∈ ^{*}^{S}_{(}_{α,β}_{)}). By hypothesis, we obtainget _{(}_{α,β}_{)} ∈ ^{*}^{S}^{*}^{S}

(3) Cleary, _{1} ⊆ _{2} implies ^{*}^{S}_{2}) ⊆ ^{*}^{S}_{1}), as there may be other intuitionistic fuzzy sets that which belong to _{2} so that for an ituitionistic fuzzy point _{(}_{α,β}_{)} ∈ ^{*}^{S}_{1}) but _{(}_{α,β}_{)} ∉ ^{*}^{S}_{2}).

(4) Since {0_{~}} ⊆ ^{*}^{S}^{*}^{S}_{~}}) = ^{S}_{1(}_{α,β}_{)} ∈ ^{S}^{*}^{S}_{1(}_{α,β}_{)}), ^{*}^{S}_{~} there exists _{2(}_{α,β}_{)} ∈ ^{*}^{S}_{2(}_{α,β}_{)}), then _{2(}_{α,β}_{)}), then _{1(}_{α,β}_{)}), therefore _{1(}_{α,β}_{)} ∈ ^{*}^{S}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{S}^{*}^{S}^{S}

(5) From (4), (^{*}^{S}^{*}^{S}^{S}^{*}^{S}^{*}^{S}

(6) Clear from (4).

(7) We know that ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

(8) We know that (^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

(9) Since ^{*}^{S}^{*}^{S}^{(}^{α,β}^{)} ∈ (^{*}^{S}_{(}_{α,β}_{)}) such that _{(}_{α,β}_{)} ∈ ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

Since (^{*}^{S}^{*}^{S}_{(}_{α,β}_{)} ∈ ^{*}^{S}_{(}_{α,β}_{)} ∈ (^{*}^{S}_{(}_{α,β}_{)}) such that _{(}_{α.β}_{)} ∈ ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

(10) Since ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

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

(12) Clear from proof (9).

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let ^{*}^{S}_{~}.

Let _{(}_{α,β}_{)} ∈ ^{*}^{S}_{(}_{α,β}_{)} ∈ _{(}_{α,β}_{)} ∈ ^{*}^{S}_{(}_{α,β}_{)}). Since _{(}_{α,β}_{)} ∈ ^{*}^{S}

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let ^{*}^{S}^{*}^{S}^{*}^{S}

Let _{(}_{α,β}_{)} ∉ ^{*}^{S}_{(}_{α,β}_{)}) such that ^{*}^{S}_{~} (By Theorem 3.2.). Hence, ^{*}^{S}^{*}^{S}_{(}_{α,β}_{)} ∉ (^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, the operator ^{*}^{S}^{*}^{S}^{*}^{S}^{*}(^{*}^{S}^{c}^{c}

(1) By Theorem 3.1.(1), ^{*}^{S}_{~}) = 0_{~}.

(2) Clear ^{*}^{S}

(3) Let A and, B be two intuitionistic fuzzy sets. Then, ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}

(4) Let A be any intuitionistic fuzzy set. Since, by (2), ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}(^{*}^{S}^{c}^{c}

For any intuitionistic fuzzy ideal on X if _{~}} ⇒ ^{*}^{S}^{*}^{S}^{S}^{S}^{*}({0_{~}}) = ^{*}^{S}^{*}^{S}_{~} for every ^{*}(_{~}} and P(X) are the two extreme intuitionistic fuzzy ideals on X, therefore for any intuitionistic fuzzy ideal I on X, we have {0_{~}} ⊆ ^{*}({0_{~}}) ⊆ ^{*}(^{*}(^{*}(_{1} and _{2} on X, _{1} ⊆ _{2} ⇒ ^{*}(_{1}) ⊆ ^{*}(_{2}).

Let _{1}, _{2} be two intuitionistic fuzzy supra topologies on X. Then, for any intuitionistic fuzzy ideal I on X, _{1} ⊆ _{2} implies

(1)

A ^{*} (^{S}S _{2},I ) ⊆A ^{*} (^{S}S _{1},I ) for everyA ∈P (X ),(2)

${S}_{1}^{*}(I)\subseteq {S}_{2}^{*}(I)$ .

(1) Since every _{1} s-neighborhood of any intuitionistic fuzzy point _{(}_{α,β}_{)} is also ana _{2} s-neighborhood of _{(}_{α,β}_{)}. Therefore, ^{*}^{S}_{2}, ^{*}^{S}_{1},

(2) Clearly, ^{*}^{S}_{2}, ^{*}^{S}_{1},

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

Clear.

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. Then, the collection ^{*}(

Let ^{*}(_{(}_{α,β}_{)} ∈ ^{c}^{*}-supra closed set suchso that ^{*}^{S}^{c}^{c}^{c}^{*}^{S}^{c}_{(}_{α,β}_{)} ∉ (^{c}^{*}^{s}_{(}_{α,β}_{)}) such that ^{c}^{c}_{(}_{α,β}_{)} ∉ _{(}_{α,β}_{)} ∈ ^{c}^{c}^{c}^{c}_{(}_{α,β}_{)} ∈

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

Let _{~} ∈ ^{*}^{S}^{c}^{*}^{S}^{c}^{c}^{c}^{*}^{S}^{c}^{c}^{*}^{S}^{c}^{*}^{S}^{c}^{*}^{S}^{*}^{S}^{c}^{c}^{c}^{*}^{S}^{c}^{*}^{S}^{c}^{c}^{c}^{c}^{*}^{S}^{c}^{*}^{S}^{c}^{c}^{*}(^{*}(^{*}(

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

It follows from Theorem 3.8.

Let T be the intuitionistic fuzzy indiscrete supra topology on X, i.e. _{~}, 1_{~}}. So 1_{~} is the only s-neighborhoods of _{(}_{α,β}_{)}. Now, _{(}_{α,β}_{)} ∈ ^{*}^{S}_{(}_{α,β}_{)}), then ^{*}^{S}_{~} if ^{*}^{S}_{~} if ^{*}^{S}^{*}^{S}_{~} if ^{*}^{S}^{*} = {^{c}^{*}(^{*}(^{*}(

^{*}(^{*}(

Follows from the fact that ^{*}(

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 _{(}_{α,β}_{)} ∈ _{(}_{α,β}_{)}) such that

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 ^{*} = 0^{S}_{~}implies thatA ∈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 ^{*} , then^{S}A ∈I .

(1) ⇒ (2) The proof is obvious.

(2) ⇒ (3) For any intuitionistic fuzzy set A in X, ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}_{~}. By (2), we obtain ^{*}^{S}

(3) ⇒ (4) By (3), for every intuitionistic fuzzy set A in X, ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}_{~}, and hence ^{*}^{S}

(4) ⇒ (1) Let an intuitionistic fuzzy set in X and assume that for every _{(}_{α,β}_{)}, there exists _{(}_{α,β}_{)}) such that ^{*}^{S}_{~}. Suppose that A contains B such that ^{*}^{S}^{*}^{S}^{*}^{S}_{~}. Therefore, A contains no non-empty subset B with ^{*}^{S}

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 ^{*} = 0^{S}_{~}implies thatA ^{*} = 0^{S}_{~},(2) For every intuitionistic fuzzy set A in X, (

A –A ^{*} )^{S}^{*} = 0^{S}_{~}.

First, we show that (1) holds if S is S-compatible with I. Let A be any intuitionistic fuzzy set in X and ^{*}^{S}_{~}. By Theorem 4.1. ^{*}^{S}_{~}.

(1) ⇒ (2) Assume that for every intuitionistic fuzzy set A in X, ^{*}^{S}_{~} implies that ^{*}^{S}_{~}. Let ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}^{c}^{*}^{S}^{c}^{*}^{S}^{s}^{*}^{S}^{c}^{*}^{S}^{*}^{S}^{c}^{*}^{S}_{~}. By (1), we have ^{*}^{S}_{~}. Hence, (^{*}^{S}^{*}^{S}_{~}.

(2) ⇒ (1) Assume that for every intuitionistic fuzzy set A in X, ^{*}^{S}_{~}, and let ^{*}^{S}^{*}^{S}_{~} = ^{*}^{S}^{*}^{S}^{*}^{S}^{*}^{S}_{~}.

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

(1)

S ∩I = 0_{~},(2)

${1}_{\sim}^{*S}={1}_{\sim}$ .

(1) ⇒ (2) Let _{~}. Then

^{S}_{~}.

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. An operator Ψ_{S}_{S}_{α,β}^{S}_{α,β}_{S}_{~} – (1_{~} – ^{*}^{S}_{S}

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 , thenU ⊆ Ψ (_{S}U ).(7) Ψ

(_{S}A ) ⊆ Ψ (Ψ_{S} (_{S}A )).(8) Ψ

(_{S}A ) = Ψ (Ψ_{S} (_{S}A )) if and only if (1_{~}–A )^{*} = ((1^{S}_{~}–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 ).

(1) Since (1_{~} – ^{*}^{S}_{~}–(1_{~}–^{*}^{S}_{S}

(2) From the definition of the Ψ_{S}_{S}_{~} – (1_{~} –^{*}^{S}_{~} –^{S}_{~} –_{~} –(1_{~} –^{*}^{S}_{S}^{S}_{S}

(3) Let _{~} – _{~} – _{~} – ^{*}^{S}_{~} – ^{*}^{S}_{S}_{S}

(4) We have _{S}_{S}_{S}

(5) We have _{S}_{S}_{S}

(6) Let _{~} – ^{S}_{~} – _{~} – _{~} – ^{*}^{S}^{S}_{~} – _{~} – _{~} – (1_{~} – ^{*}^{S}_{S}

(7) From (2), Ψ_{S}_{S}_{S}_{S}

(8) Let Ψ_{S}_{S}_{S}_{~} – (1_{~} – ^{*}^{S}_{S}_{~}–(1_{~}–^{*}^{S}_{~}–(1_{~}–(1_{~}–(1_{~}–^{*}^{S}^{*}^{S}_{~} – ((1_{~} – ^{*}^{S}^{*}^{S}_{~} – ^{*}^{S}_{~} – ^{*}^{S}^{*}^{S}_{~} – ^{*}^{S}_{~} – ^{*}^{S}^{*}^{S}_{~}–(1_{~}–^{*}^{S}_{~}–((1_{~}–^{*}^{S}^{*}^{S}_{~}–(1_{~}–^{*}^{S}_{~}–(1_{~}–(1_{~}–(1_{~}–^{*}^{S}^{*}^{S}_{~} – (1_{~} – Ψ_{S}^{*}^{S}_{S}_{S}_{S}

(9) Let (_{1}, _{2}. We observe that _{1}, _{2} ∈ _{1}) ∪ _{2}. Thus, Ψ_{S}_{S}_{1}) = Ψ_{S}_{1}) ∪ _{2}) = Ψ_{S}

(10) By Theorem 3.1.(9), we obtain if

(11) This follows from Theorem 3.1.(9), and Ψ_{S}_{~} – (1_{~} – (^{*}^{S}_{~} – ((1_{~} – ^{*}^{S}_{~} – (1_{~} – ^{*}^{S}_{S}

(12) This follows from Theorem 3.1.(9), and Ψ_{S}_{~} – (1_{~} – (^{*}^{S}_{~} – ((1_{~} – ^{*}^{S}_{~} – (1_{~} – ^{*}^{S}_{S}

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. If _{S}

Let _{S}_{S}_{~}) = 1_{~} – (1_{~} – 0_{~})^{*}^{S}_{~} – 1_{~} = 0_{~}. Therefore, we observe obtain that 0_{~} ⊆ Ψ_{S}_{~}) and 1_{~} ⊆ Ψ_{S}_{~}) = 1_{~}, and thus 0_{~}, 1_{~} ∈ _{α}_{α}_{S}_{α}_{S}_{α}_{α}_{S}_{α}

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

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space and let I be S-codense with S. Then,

It is obvious.

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. An intuitionistic fuzzy set A in X is called a Ψ_{S}^{S}_{S}_{S}_{S}

Let (X, S, I) be an intuitionistic fuzzy ideal supra topological space. If _{S}

From Theorem 5.1.(6) it follows that _{S}

Let {_{α}_{S}_{α}_{∈Δ}_{α}_{S}

For each

This implies that ∪_{α}_{∈Δ}_{α}^{S}_{S}_{α}_{∈Δ}_{α}_{α}_{∈Δ}_{α}_{S}

- Zadeh, LA (1965). Fuzzy sets. Information and Control.
*8*, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X - Atanassov, KT (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems.
*20*, 87-96. https://doi.org/10.1016/S0165-0114(86)80034-3 - Atanassov, KT (1994). New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems.
*61*, 137-142. - Atanasov, KT (1999). Intuitionistic Fuzzy Sets: Theory and Applications. Berlin, Germany: Springr
- Szmidt, E, and Kacprzyk, J (2001). Intuitionistic fuzzy sets in some medical applications. Notes on IFS.
*7*, 58-64. - Szmidt, E, and Kacprzyk, J (2004). Medical diagnostic reasoning using a similarity measure for intuitionistic fuzzy sets. Notes on IFS.
*10*, 61-69. - Atanassov, K 1988. Review and new results on intuitionistic fuzzy sets., Mathematical Foundations of Artificial Intelligence Seminar, Sofia, Bulgaria.
- Coker, D, and Demirci, M (1995). On intuitionistic fuzzy points. Notes on IFS.
*1*, 79-84. - Kuratowski, K (1966). Topology (Volume I). New York, NY: Academic Press
- Vaidyanathaswamy, R (1944). The localisation theory in settopology. Proceedings of the Indian Academy of Sciences-Section A.
*20*, 51-61. https://doi.org/10.1007/BF03048958 - Jankovic, D, and Hamlett, TR (1990). New topologies from old via ideals. The American Mathematical Monthly.
*97*, 295-310. https://doi.org/10.1080/00029890.1990.11995593 - Mashhour, AS, Allam, AA, Mahmoud, FS, and Khedr, FH (1983). On supra topological spaces. Indian Journal of Pure and Applied Mathematics.
*14*, 502-510. - Turanl, N 1999. On intuitionistic fuzzy supra topological spaces., Proceedings of International Conference on Modeling and Simulation, Spain, pp.69-77.
- Kandil, A, Tantawy, OA, El-Sheikh, SA, and Hazza, SA (2015). New supra topologies from old via ideals. Journal of New Theory.
*2015*, 1-5. - Atanassov, KT (1984). Intuitionistic fuzzy sets. VII ITKR’s Session, Sofia (deposed in Central Sci-Technical Library of Bulg Acad of Sci, 1697/84). Sofia, Bulgaria: Bulgarian Academy of Sciences
- Riecan, B, and Atanassov, KT (2008). Some properties of two operations over intuitionistic fuzzy sets. Notes Intuit Fuzzy Sets.
*14*, 6-10.

International Journal of Fuzzy Logic and Intelligent Systems 2021;21:93~100 https://doi.org/10.5391/IJFIS.2021.21.1.93

© IJFIS

© The Korean Institute of Intelligent Systems. / Powered by INFOrang Co., Ltd