# Fuzzy Ideal Supra Topological Spaces

Authors: Fadhil Abbas
Journal: Advances in Fuzzy Systems
 (2020)

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## Abstract

In this paper, we introduce the notion of fuzzy ideals in fuzzy supra topological spaces. The concept of a fuzzys-local function is also introduced here by utilizing the s-neighbourhood structure for a fuzzy supra topological space. These concepts are discussed with a view to find new fuzzy supra topologies from the original one. The basic structure, especially a basis for such generated fuzzy supra topologies, and several relations between different fuzzy ideals and fuzzy supra topologies are also studied here. Moreover, we introduce a fuzzy set operator ΨS and study its properties. Finally, we introduce some sets of fuzzy ideal supra topological spaces (fuzzy ∗-supra dense-in-itself sets, fuzzy S∗-supra closedsets, fuzzy ∗-supra perfect sets, fuzzy regular-I-supra closedsets, fuzzy-I-supra opensets, fuzzy semi-I-supra opensets, fuzzy pre-I-supra opensets, fuzzy α-I-supra opensets, and fuzzy β-I-supra opensets) and study some characteristics of these sets, and then, we introduce some fuzzy ideal supra continuous functions.


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## Body

## 1. Introduction

The concept of fuzzy sets is an important concept in many fields. The concept was introduced by in 1965 [1]. The idea was welcomed because it addresses the uncertainty, something classical Cantor set theory could not address. Despite of some criticism expressed in the beginning by some specialists of mathematical logic, it has become an important subject in various fields and sciences. Zadeh writes in [1], “The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and potentially, may prove to have a much wider scope of applicability particularly in the fields of pattern classification and information processing.”Fuzzy set theory provides a natural way to deal with inaccuracy and a strict mathematical framework for the study of uncertain phenomena and concepts. It can also be considered as a modeling language, well suited for situations in which fuzzy relations criteria and phenomena exist. Despite the slow growth and progress of fuzzy set theory before the mid-1970s, the theory developed greatly afterward. This was caused by the first successful application of the theory to technological processes, in particular to systems based on ambiguous control rules called fuzzy control and boosted the interest in this area considerably.The concept of general topology is one of the most important mathematical topics and has wide applications in many applied sciences and mathematical subjects. The notion of fuzzy sets naturally plays a very significant role in the study of fuzzy topology introduced by Chang in 1968 [2]. Pu and Liu in 1980 [3] introduced the concept of quasi-coincidence and q-neighbourhoods by which the extensions of functions in fuzzy setting can very interestingly and effectively be carried out. The concept of an ideal in a topological space was first introduced by Kuratowski in 1966 [4] and Vaidyanathswamy in 1945 [5]. They also defined local functions in an ideal topological space. Furthermore, Jankovic and Hamlet in 1990 [6] studied the properties of ideal topological spaces and introduced another operator called the Ψ operator. They have also obtained a new topology from the original ideal topological space. Using the local function, they defined a Kuratowski closure operator in the new topological space.The concept of supra topology was introduced by Mashhour et al. in 1983 [7]. It is fundamental with respect to the investigation of general topological spaces. In 2016, Al-shami [8] discussed the concepts of compactness and separation axioms on supra topological spaces. Then, Al-Shami [9, 10] and Al-shami et al. [11] presented new types of supra compact spaces using supra α-open, supra semiopen, and supra preopensets. Later, the authors of [12–15] employed some generalizations of supra opensets to investigate several kinds of supra limit points of a set and supra Ti spaces i=0,1,2,3,4. In fact, they provided many interesting examples to show the validity of the obtained results. Recently, Al-shami [16] have defined supra paracompact spaces, and Assad et al. [17] have studied γ operation on supra topological spaces.Abd El-Monsef and Ramadan in 1987 [18] introduced the concept of fuzzy supra topological as a natural generalization of the notion of supra topology spaces. In addition to that, some properties of the concept of ideal supra topological spaces are obtained by Modak and Mistry in 2012 [19]. In 2015 [20], further properties of ideal supra topological spaces are investigated.In this paper, we introduce the notion of fuzzy ideals in fuzzy supra topological spaces.Section-wise description of the work carried out in this paper is given. Beginning with an introduction, necessary notation and preliminaries have been given.In Section3, the concept of a fuzzy s-local function is also introduced here by utilizing.InSection 4, we give the s-neighbourhood structure for a fuzzy supra topological space. These concepts are discussed with a view to find new fuzzy supra topologies from the original one. The basic structure, especially a basis for such generated fuzzy supra topologies, and several relations between different fuzzy ideals and fuzzy supra topologies are also studied here.InSection 5, we introduce a fuzzy set operator ΨS and study its properties.InSection 6, we introduce some sets of fuzzy ideal supra topological spaces (fuzzy ∗-supra dense-in-itself sets, fuzzy S∗-supra closedsets, fuzzy ∗-supra perfect sets, fuzzy regular-I-supra closedsets, fuzzy-I-supra opensets, fuzzy semi-I-supra opensets, fuzzy pre-I-supra opensets, fuzzy α-I-supra opensets, and fuzzy β-I-supra opensets) and study some characteristics of theses sets. Finally, in Section 7, we introduce some fuzzy ideal supra continuous functions.


## 2. Preliminaries

Definition 1.
(see [1]). Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function μA:X⟶0,1, and μAx is interpreted as the degree of membership of the element x in the fuzzy set A, for each x∈X. It is clear that A is completely determined by the set of tuples A=x,μAx:x∈X.Definition 2.
(see [1]). Let A and B be fuzzy sets of the formA=x,μAx:x∈X and B=x,μBx:x∈X. Then, we define(1)
A⊆B⇔μAx≤μBx for all x∈X(2)
A=B⇔A⊆B and B⊆A(3)
C=A∪B⇔μCx=maxμAx,μBx(4)
D=A∩B⇔μDx=minμAx,μBx(5)
E=Ac⇔μEx=1−μAx(6)
1x=X=x,1:x∈X and 0x=∅=x,0:x∈XDefinition 3.
(see [1]). For any family Ajj∈J of fuzzy sets in X, we define(1)
C=∪j∈JAj⇔μCx=supμAjx:j∈J(2)
D=∩j∈JAj⇔μDx=infμAjx:j∈JDefinition 4.
(see [21]). A fuzzy set in X is called a fuzzy point if and only if it takes the value 0 for all y∈X except one, say x∈X. If its value at x is α∈0,1, we denote this fuzzy point by xα, where the point x is called its support.Definition 5.
(see [18]). A subclass S⊆Px (P(X) is the collection of all fuzzy sets on X and is called a fuzzy supra topology on X if 0x,1x∈S, and S is closed under arbitrary union. The pair (X, S) is called a fuzzy supra topological space, and the members of S are called fuzzy supra opensets. A fuzzy set A is fuzzy supra closed if and only if its complement Ac is fuzzy supra open.Definition 6.
(see [18]). Let (X, S) be a fuzzy supra topological space and let A be a fuzzy set in X. Then, the fuzzy supra interior and the fuzzy supra closure of A in (X, S) are defined as(1)IntSA=∪U:U⊆A,U∈S,ClSA=∩F:A⊆F,Fc∈S,respectively.Corollary 1.
From Definition6, IntSA is a fuzzy supra openset and ClSA is a fuzzy supra closedset.Definition 7.
(see [22]). Let (X, S) be a fuzzy supra topological space. A fuzzy set A in X is said to be quasi-coincident with a fuzzy set B if there exists x∈X, such that Ax+Bx>1. In this case, we write AqB.Definition 8.
(see [22]). A fuzzy set A in a fuzzy supra topological space (X, S) is an s-neighbourhood of a fuzzy point xα if there is M∈S with xα∈M⊆A. The collection NSxα of all s-neighbourhoods of xα is called the s-neighbourhood system of xα.Definition 9.
(see [18]). Let S1 and S2 be two fuzzy supra topologies on a set X such that S1⊆S2. Then, we say that S2 is stronger(finer) than S1 or S1 is weaker(coarser) than S2.Definition 10.
(see [18]). Let (X, S) be a fuzzy supra topological space and β⊆S. Then, β is called a base for the fuzzy supra topology S if every fuzzy supra openset U∈S is a union of members of β. Equivalently, β is a fuzzy supra base for S if, for any fuzzy point xα∈U, there exists B∈β with xα∈B⊆U.Definition 11.
(see [18]). A mapping c:PX⟶PX is said to be a fuzzy supra closure operator if it satisfies the following axioms:(1)
c0x=0x(2)
A⊆cA for every fuzzy set A in X(3)
cA∪cB⊆cA∪B for every fuzzy sets A and B in X(4)
ccA=cA for every fuzzy set A in XTheorem 1.
(see [18]). Let X be a nonempty set, and let the mapping c:PX⟶PX be a fuzzy supra closure operator. Then, the collection S=A∈PX:cAc=Ac is fuzzy supra topology on X induced by the fuzzy supra closure operator c.Definition 12.
(see [23]). A nonempty collection of fuzzy sets I of a set X is called a fuzzy ideal on X if and only if(1)
A∈I and B⊆A⇒B∈I(heredity)(2)
A∈I and B∈I⇒A∪B∈I(finite additivity)


## 3. FuzzyS-Local Function

Definition 13.
A fuzzy supra topological space (X, S) with a fuzzy ideal I on X is called a fuzzy ideal supra topological space and denoted as (X, S, I).Definition 14.
Let (X, S, I) be a fuzzy ideal supra topology and let A be any fuzzy set in X. Then, the fuzzy s-local function A∗S(I, S) of A is the union of all fuzzy points xα such that if M∈NSxα and E∈I, then there is at least one y∈X for which My+Ay−1x>Ey.
In other words, we say that a fuzzy setA is fuzzy s-local in I at xα if there exists M∈NSxα, such that for every y∈X, My+Ay−1x≤Ey for some E∈I. We will occasionally write A∗S for A∗SI,S, and it will cause no ambiguity.Example 1.
The simplest fuzzy ideals onX are 0x and P(X). Then, I=0x⇔A∗S=ClSA, for any fuzzy set A in X and I=PX⇔A∗S=0x.Theorem 2.
Let (X, S, I) be a fuzzy ideal supra topological space, and let A and B be fuzzy sets in X. Then,(1)
0x∗S=0x(2)
IfA⊆B, then A∗S⊆B∗S(3)
IfI1⊆I2, then A∗SI2⊆A∗SI1(4)
A∗S=ClSA∗S⊆ClSA(5)
A∗S∗S⊆A∗S(6)
A∗S is a fuzzy supra closedset(7)
A∗S∪B∗S⊆A∪B∗S(8)
A∩B∗S⊆A∗S∩B∗S(9)
IfE∈I, then A∪E∗S=A∗S=A−E∗S(10)
IfU∈S, then U∩A∗S=U∩U∩A∗S⊆U∩A∗S(11)
IfE∈I, then E∗S=0x(12)
IfE∈I, then 1x−E∗S=1x∗SProof.
(1)
This is obvious from the definition of fuzzys-local function(2)
LetA⊆B. Then, Ax≤Bx for every x∈X. From the definition of fuzzy s-local function, if xα∈A∗S, then xα∈B∗S. Therefore, A∗S⊆B∗S.(3)
LetI1⊆I2 and let xα∈X be any fuzzy points, such that xα∉A∗SI1,S. Then, there is at least one M∈NSxα and for every y∈X, such that My+Ay−1x≤Ey for some E∈I1. But I1⊆I2, then E∈I2. This implies xα∉A∗SI2,S. Therefore, A∗SI2,S⊆A∗SI1,S.(4)
For any fuzzy ideal onX, 0x⊆I. Therefore by (3) and Example 1, for any fuzzy set A in X,A∗SI⊆A∗S0x=ClSA. Now, let xα∈ClSA∗S. Then, for every M∈NSxα, there is at least one y∈X, such that My+A∗Sy>1x. Hence, A∗Sy≠0x, and let β=A∗Sy. Clearly, yβ∈A∗S and β+My>1x, so that M∈NSyβ. Now, yβ∈A∗S implies there is at least one x1∈X, such that Vx1+Ax1−1x>Ex1, for each V∈NSyβ and E∈I. This is also true for M. So, there is at least one x2∈X such that Mx2+Ax2−1x>Ex2, for each E∈I. Since M∈NSxα; therefore, xα∈A∗S. Hence, A∗S=ClSA∗S⊆ClSA.(5)
From (4),A∗S∗S⊆ClSA∗S=A∗S(6)
Letxα∉A∗SI,S. Then, there is at least one M∈NSxα, such that for every y∈X,My+Ay−1x≤Ey for some E∈I, and xα∈M implies M≤1x−A∗SI, and we have 1x−A∗SI which is a fuzzy supra openset. Therefore, A∗S is fuzzy supra closed.(7)
We haveA⊆A∪B and B⊆A∪B. Then, from (2), A∗S⊆A∪B∗S and B∗S⊆A∪B∗S. Hence, A∗S∪B∗S⊆A∪B∗S.(8)
We haveA∩B⊆A and A∩B⊆B. Then, from (2), A∩B∗S⊆A∗S and A∩B∗S⊆B∗S. Hence, A∩B∗S⊆A∗S∩B∗S.(9)
We haveA⊆A∪E. Then, from (2), A∗S⊆A∪E∗S. Now, let xα∈A∪E∗S; then, for every M∈NSxα, there is at least one y∈X such that My+A∪Ey−1x>Ey, for each E∈I. If maxAy,Ey=Ey, then for each M∈NSxα and E∈I, My+Ey−1x>Ey, and this implies My>1x, and this is a contradiction. Then, for each M∈NSxα and E∈I,My+Ay−1x>Ey, and this implies xα∈A∗S. Therefore, A∪E∗S⊆A∗S. Hence, A∪E∗S=A∗S. Now, from the defined fuzzy s-local function, clear A∗S=A−E∗S; then, A∪E∗S=A∗S=A−E∗S.(10)
We haveU∩A⊆A. Then, from (2), U∩A∗S⊆A∗S. So, U∩U∩A∗S⊆U∩A∗S.(11)
This is obvious from the definition of fuzzys-local function(12)
This is obvious from (9)Theorem 3.
Let (X, S, I) be a fuzzy ideal supra topological space, and let A be any fuzzy set in X. Then, A∪A∗S∗S⊆A∗S.Proof.
Letxα∈A∪A∗S∗S. Then, for every M∈NSxα, there is at least one y∈X, such that My+A∪A∗Sy−1x>Ey, for each E∈I. Now, if maxAy,A∗Sy=Ay implies xα∈A∗S; therefore, the proof is done. If maxAy,A∗Sy=A∗Sy, then A∗Sy≠0x, and let β=A∗S. Clearly, yβ∈A∗S and My+β−1x>Ey. Now, yβ∈A∗S implies there is at least one x1∈X, such that Vx1+Ax1−1x>Ex1 for each V∈NSyβ and E∈I. This is also true for M. So, there is at least one x2∈X, such that Mx2+Ax2−1x>Ex2, for each E∈I. Since M∈NSxα, then xα∈A∗S. Therefore, A∪A∗S∗S⊆A∗S.Theorem 4.
Let (X, S, I) be a fuzzy ideal supra topological space. Then, the operator Cl∗S:PX⟶PX, defined by Cl∗SA=A∪A∗S for any fuzzy set A in X, is a fuzzy supra closure operator, and hence, it generates a fuzzy supra topology S∗I=A∈PX:Cl∗SAc=Ac which is finer than S.Proof.
(1)
By (1) in Theorem2, 0x∗S=0x, we have Cl∗S0x=0x(2)
Clear thatA⊆Cl∗SA for every fuzzy set A(3)
LetA and B be any two fuzzy sets. Then, Cl∗SA∪Cl∗SB=A∪A∗S∪B∪B∗S=A∪B∪A∗S∪B∗S⊆A∪B∪A∪B∗S=Cl∗SA∪B (by (7) in Theorem 2). Hence, Cl∗SA∪Cl∗SB⊆Cl∗SA∪B.(4)
LetA be any fuzzy set. Since, by (2), A⊆Cl∗SA; then, Cl∗SA⊆Cl∗SCl∗SA. On the other hand, Cl∗SCl∗SA=Cl∗SA∪A∗S=A∪A∗S∪A∪A∗S∗S⊆A∪A∗S∪A∗S=Cl∗SA (by Theorem 3); it follows that Cl∗SCl∗SA⊆Cl∗SA. Hence, Cl∗SCl∗SA=Cl∗SA. Consequently, Cl∗SA is a fuzzy supra closure operator. Also, it is easy to show that the collection S∗I=A∈PX:Cl∗SAc=Ac is a fuzzy supra topology on X which is called the fuzzy supra topology induced by the fuzzy supra closure operator.Proposition 1.
For any fuzzy ideal onX, if I=0x⇒Cl∗SA=A∪A∗S=A∪ClSA=ClSA for every a fuzzy set A in X. So, S∗0x=S, and if I=PX⇒Cl∗SA=A because A∗S=0x for every a fuzzy set A in X. So, S∗PX is a fuzzy discrete supra topology on X. Since 0x and P(X) are the tow extreme fuzzy ideal on X, for any fuzzy ideal I on X, we have 0x⊆I⊆PX. So, we can conclude by (2) in Theorem 2 that S∗0x⊆S∗I⊆S∗PX, i.e., S⊆S∗I, for any fuzzy ideal I on X. In particular, we have for any tow fuzzy ideals I1 and I2 on X, I1⊆I2⇒S∗I1⊆S∗I2.Theorem 5.
For any fuzzy ideal supra topological space (X, S, I), the class βI,S=U−E:U∈S,E∈I is the base for the fuzzy supra topology S∗I.Proof.
Since0x∈I, then S⊆β from which it follows X=∪β. Also, for every β1,β2∈β, we have, β1=U1−E1 and β2=U2−E2, where U1,U2∈S∗I, and E1,E2∈I. Then, β1∩β2=U1−E1∩U2−E2=U1∩E1c∩U1∩E2c=U1−U2∩E1−E2c=U1−U2−E1∪E2c∈β. Therefore, β is a base for S∗I.Example 2.
LetT be the fuzzy indiscrete supra topology on X, i.e., T=0x,1x. So, 1x is the only s-neighbourhoods of xα. Now, xα∈A∗S for a fuzzy set A if and only if for each E∈I, there is at least one y∈X, such that 1x+Ay−1x>Ey. This implies, for each E∈I,Ay>Ey for at least one y∈X. So, A∉I. Therefore, A∗S=1x if A∉I and A∗S=0x if A∈I. This implies that we have Cl∗SA=A∪A∗S=1x if A∉I and Cl∗SA=A if A∈I for any fuzzy set A of X. Hence, T∗=M:Mc∈I. Let S∪T∗I be the supremum fuzzy supra topology of S and T∗I, i.e., the smallest fuzzy supra topology is generated by S∪T∗I. Then, we have the following theorem.Theorem 6.
S∗I=S∪T∗I.Proof.
Follows from the fact thatβ forms a basis for S∗I.Theorem 7.
LetS1andS2 be tow fuzzy supra topologies on X. Then, for any fuzzy ideal I on X, S1⊆S2 implies(1)
A∗SS2,I⊆A∗SS1,I for every fuzzy set in X(2)
S1∗I⊆S2∗IProof.
(1)
Since everyS1s-neighbourhood of any fuzzy point xα is also a S2s-neighbourhood of xα. Therefore, A∗SS2,I⊆A∗SS1,I.(2)
Clearly,S1∗I⊆S2∗I as A∗SS2,I⊆A∗SS1,ITheorem 8.
Let (X, S, I) be a fuzzy supra topological space. Then,(1)
IfH∈I⇒Hc∈S∗I(2)
A∗S=Cl∗SA∗S for every fuzzy set A in X, i.e., A∗S is a fuzzy S∗-supra closedset.Proof.
(1)
For everyH∈I⇒H∗S=0x. Hence, Cl∗SH=H⇒Hc∈S∗I, i.e., H is a fuzzy S∗-supra closedset.(2)
From (5) in Theorem2, we have A∗S∗S⊆A∗S⇒A∗S=A∗S∪A∗S∗S=Cl∗SA. Hence, A∗S is a fuzzy S∗-supra closedset.


## 4.S-Compatible of Fuzzy Ideals with Fuzzy Supra Topology

Definition 15.
Let (X, S, I) be a fuzzy supra topological space. S is said to be fuzzy S-compatible with I, denoted by S∼I, if for every fuzzy set A in X; if for all fuzzy point xα∈A, there exists M∈NSxα such that My+Ay−1x≤Ey holds for every y∈X and some E∈I, and then, A∈I.Definition 16.
(see [20]). Let Bα,α∈Δ be any indexed family of fuzzy sets in X such that BαqA, for each α∈Δ, where A is a fuzzy set in X. Then, Bα,α∈Δ is said to be a quasicover of A if and only if Ay+∪α∈ΔBαy≥1x for every y∈X.Definition 17.
LetBα,α∈Δ be quasicover of A; if each Bα is a fuzzy supra openset, then this quasicover with be called a fuzzy quasi-supra open cover A in X. Therefore, in either case, Ac⊆∪α∈ΔBα.Theorem 9.
Let (X, S, I) be a fuzzy ideal supra topological space. Then, the following conditions are equivalent:(1)
S∼I(2)
If for every fuzzy setA in X has a fuzzy quasi-supra open coverBα,α∈Δ such that for each α,Ay+Bαy−1x≤Ey for some E∈I and for every y∈X, and then, A∈I(3)
For every fuzzy setA in X, A∩A∗S=0x implies A∈I(4)
For every fuzzy setA in X, A˜∈I, where A˜=∪xα, such that xα∈A but xα∉A∗SProof.
(1)⇒(2) Let Bα,α∈Δ be a fuzzy quasi-supra open cover of a fuzzy set A in X such that for each α∈Δ,Bαy+Ay−1x≤Ey for some E∈I and for every y∈X. Therefore, as Bα,α∈Δ is a fuzzy quasi-supra open cover of A, for each xα∈A, there exists at least one Bα0 such that xαqBα0 and for every y∈X,Bα0y+Ay−1x≤Ey for some E∈I. Obviously, Bα0∈NSxα. Therefore, as S∼I,A∈I.(2)⇒(1) Clear from the fact that a collection of fuzzy supra opensets which contain at least one open s-neighbourhood of each fuzzy point of A that constitutes a fuzzy quasi-supra open cover of A.(2)⇒(3) Let A∩A∗S=0x, i.e., minAy,A∗Sy=0x, for every y∈X. So, a fuzzy point xα∈A implies xα∉A∗S. That means, there is M∈NSxα such that for every y∈X,My+Ay−1x≤Ey for some E∈I. If xα∈A since M∈NSxα, there is a fuzzy supra openset V (in S) such that xαqV⊆M, and so, the collection of such V’s for each xα∈A constitutes a fuzzy quasi-supra open cover of A. Therefore, by condition (2), A∈I.(3)⇒(1) Let for every fuzzy point xα∈A, there is M∈NSxα such that for every y∈X,My+Ay−1x≤Ey for some E∈I. That means xα∉A∗S. Now, there are two cases: either A∗Sx=0x or A∗Sx≠0x, but α>A∗Sx≠0x. Let, if possible, xα∈A be such that α>A∗Sx≠0x. Let α1=A∗Sx. Then, the fuzzy point xα1∈A∗S and also xα1∈A. This implies for each V∈NSxα1, and for each E∈I, there is at least one y∈X such that Vy+Ay−1x>Ey. Since xα1∈A, this contradicts the assumption for every fuzzy point of A. So, A∗Sx=0x. That means, xα∈A implies xα∉A∗S. Now, this is true for every fuzzy set A in X. So, for every fuzzy set A in X, A∩A∗S=0x. Hence, by condition (3), we have A∈I, which implies S∼I.(3)⇒(4) Let the fuzzy point xα∈A˜. That means, xα∈A, but xα∉A∗S. So, there is M∈NSxα such that for every y∈X,My+Ay−1x≤Ey for some E∈I. Since A˜⊆A, so for every y∈X,My+A˜y−1x≤Ey for some E∈I. Therefore, xα∉A˜∗S, so that either A˜∗Sx=0x or A˜∗Sx≠0x, but α>A˜∗Sx≠0x. Let xα1, be a fuzzy point such that α1≤A˜∗Sx<α, i.e., xα1∈A˜∗S. So, for each V∈NSxα1 and for each E∈I, there is at least one y∈X such that Vy+A˜y−1x>Ey. Since A˜⊆A, for each V∈NSxα1 and for each E∈I, there is at least one y∈X such that Vy+Ay−1x>Ey. This implies xα1∈A∗S. But as α1<α,xα∈A˜, and therefore xα1∉A∗S. This is a contradiction. Hence, A˜∗Sx=0x, so that xα∈A˜ implies xα∉A˜∗S with A˜∗S=0x. Thus, we have A˜∩A˜∗S=0x for every fuzzy set A in X. Hence, by condition (3), A˜∈I.(4)⇒(1) Let for every fuzzy set A in X, A˜∈I. This means for every xα∈A˜ then xα∈A and xα∉A∗S then for every y∈X, there exist M∈NSxα such that My+Ay−1x≤Ey for some E∈I. This implies S∼I.Theorem 10.
Let (X, S, I) be a fuzzy ideal supra topological space. Then, the following are equivalent and implied by S∼I.(1)
For every fuzzy setA in X, A∩A∗S=0x implies A∗S=0x.(2)
For every fuzzy setA in X, A˜∗S=0x (Ã is defined as in (4) in Theorem 9).(3)
For every fuzzy setA in X, A∩A∗S∗S=A∗S.Proof.
Clear from Theorem9.Theorem 11.
Let (X, S, I) be a fuzzy ideal supra topological space, let S∼I. Then a fuzzy set A in X is supra closed with respect to S∗I if and only if it is the union of a fuzzy set which is supra closed with respect to S and a fuzzy set in I.Proof.
LetA be a fuzzy set in X such that it is fuzzy S∗-supra closedset. That means A∗S⊆A, and we have A=A˜∪A∗S. Since S∼I, A∈I. Also, A∗S is always fuzzy S-supra closedset by (3) in Theorem 2. Conversely, let A be any fuzzy set in X such that A=U∪V, where Cl∗SU=U⊆A. This means A∗S⊆U⊆A. So, we have Cl∗SA=A∪A∗S=A and implies A is a fuzzy S∗-supra closedset.Corollary 2.
The fuzzy supra topologyS is S-compatible with fuzzy ideal I on X implies βI,S, a basis for S∗I, and is itself a fuzzy topology and also β=S∗I.Proof.
Clear.


## 5. Fuzzy Set OperatorΨS

In this section, we introduce the fuzzy set operatorΨS, S-codense, and ΨS-C-fuzzy set, and we give new results.Definition 18.
Let (X, S, I) be a fuzzy ideal supra topological space. An operator ΨS:PX⟶S is defined as follows: for every fuzzy set A in X, ΨSA = {xα fuzzy point: there exists M∈NSxα such that M−A∈I}. We observe that ΨSA=1x−1x−A∗S. The behaviors of the operator ΨS have been discussed in the following theorem:Theorem 12.
Let (X, S, I) be a fuzzy ideal supra topological space. Let A and B be two fuzzy sets in X. Then,(1)
ΨSA is a fuzzy supra openset(2)
IntSA⊆ΨSA(3)
IfA⊆B, then ΨSA⊆ΨSB(4)
ΨSA∩B⊆ΨSA∩ΨSB(5)
ΨSA∪ΨSB⊆ΨSA∪B(6)
IfU∈S, then U⊆ΨSU(7)
ΨSA⊆ΨSΨSA(8)
ΨSA=ΨSΨSA if and only if 1x−A∗S=1x−A∗S∗S(9)
IfA−B∪B−A∈I, then ΨSA=ΨSB(10)
IfE∈I, then ΨSE=1x−1x∗S(11)
IfE∈I, then ΨSA−E=ΨSA(12)
IfE∈I, then ΨSA∪E=ΨSAProof.
(1)
Since1x−A∗S is a fuzzy supra closedset, 1x−1x−A∗S is a fuzzy supra openset. Hence, ΨSA is a fuzzy supra openset.(2)
From definition of theΨS operator, ΨSA=1x−1x−A∗S. Then, 1x−ClS1x−A⊆1x−1x−A∗S=ΨSA, from (4) in Theorem 2. Hence, IntSA⊆ΨSA.(3)
LetA⊆B. Then, 1x−B⊆1x−A. Then, from (2) in Theorem 2, 1x−B∗S⊆1x−A∗S. Therefore, ΨSA⊆ΨSB.(4)
We haveA∩B⊆A and A∩B⊆B. Then, from (3), ΨSA∩B⊆ΨSA∩ΨSB.(5)
We haveA⊆A∪B and B⊆A∪B. Then, from (3), ΨSA∪ΨSB⊆ΨSA∪B.(6)
LetU∈S. Then, 1x−U is a fuzzy supra closedset, and hence, ClS1x−U=1x−U. Then, 1x−U∗S⊆ClS1x−U=1x−U. Hence, U⊆1x−1x−U∗S, so U⊆ΨSU.(7)
From (2),ΨSA∈S, and from (6), ΨSA⊆ΨSΨSA.(8)
LetΨSA=ΨSΨSA. Then, 1x−1x−A∗S=ΨS1x−1x−A∗S=1x−1x−1x−1x−A∗S∗S=1x−1x−A∗S∗S. Therefore, 1x−A∗S=1x−A∗S∗S. Conversely suppose that 1x−A∗S=1x−A∗S∗S hold. Then, 1x−1x−A∗S=1x−1x−A∗S∗S and 1x−1x−A∗S=1x−1x−1x−1x−A∗S∗S=1x−1x−ΨSA∗S. Hence, ΨSA=ΨSΨSA.(9)
LetA−B∪B−A∈I, and let A−B=E1,B−A=E2. We observe that E1,E2∈I by heredity, and B=A−E1∪E2. Thus, ΨSA=ΨSA−E1=ΨSA−E1∪E2=ΨSB.(10)
By (9) in Theorem2, we obtain if E∈I, then ΨSE=1x−1x∗S(11)
This follows from (9) in Theorem2, and ΨSA−E=1x−1x−A−E∗S=1x−1x−A∪E∗S=1x−1x−A∗S=ΨSA(12)
This follows from (9) in Theorem2, and ΨSA∪E=1x−1x−A∪E∗S=1x−1x−A−E∗S=1x−1x−A∗S=ΨSATheorem 13.
Let (X, S, I) be a fuzzy ideal supra topological space. If η=A∈PX:A⊆ΨSA. Then, η is a fuzzy supra topology for X.Proof.
Letη=A∈PX:A⊆ΨSA. By (1) in Theorem 2, 0x∗S=0x and ΨS1x=1x−1x−1x∗S=1x−0x∗S=1x. Moreover, ΨS0x=1x−1x−0x∗S=1x−1x=0x. Therefore, we obtain that 0x⊆ΨS0x and 1x⊆ΨS1x=1x, and thus, 0x,1x∈η. Now, if Aα:α∈Δ⊆η, then Aα⊆ΨSAα⊆ΨS∪Aα for every α, and hence, ∪Aα⊆ΨS∪Aα. This shows that η is a fuzzy supra topology.Definition 19.
A fuzzy ideal I in a space (X, S, I) is called S-codense fuzzy ideal if S∩I=0x. The following theorem is related to S-codense fuzzy ideal.Theorem 14.
Let (X, S, I) be a fuzzy ideal supra topological space, and let I be S-codense with S. Then, 1x=1x∗S.Proof.
It is obvious that1x∗S⊆1x. For converse, suppose xα∈1x but xα∉1x∗S, there is at least one M∈NSxα and for every y∈X such that My+1x−1x≤Ey for some E∈I⇒My≤Ey. That means M∈I, a contradiction to the fact that S∩I=0x. Hence, 1x=1x∗S.Definition 20.
Let (X, S, I) be a fuzzy ideal supra topological space. A fuzzy set A in X is called the ΨS-C-fuzzy set if A⊆ClSΨSA. The collection of all ΨS-C-fuzzy sets in (X, S, I) is denoted by ΨSX,S.Theorem 15.
Let (X, S, I) be a fuzzy ideal supra topological space. If A∈S, then A∈ΨSX,S.Proof.
From (6) in Theorem12, it follows that S⊆ΨSX,S.Theorem 16.
LetAα:α∈Δ be a collection of nonempty ΨS-C-fuzzy sets in a fuzzy ideal supra topological space (X, S, I); then, ∪α∈ΔAα∈ΨSX,S.Proof.
For eachα∈Δ,Aα⊆ClSΨSAα⊆ClSΨS∪α∈ΔAα. This implies that ∪α∈ΔAα⊆ClSΨS∪α∈ΔAα. Thus, ∪α∈ΔAα∈ΨSX,S.


## 6. Some Sets of a Fuzzy Ideal Supra Topological Space

Definition 21.
Let (X, S, I) be a fuzzy ideal supra topological space and let A be any fuzzy set in X. Then, A is said to be(1)
Fuzzy∗-supra dense-in-itself set if A⊆A∗S(2)
FuzzyS∗-supra closedset if A∗S⊆A(3)
Fuzzy∗-supra prefect set if A=A∗S(4)
Fuzzy regular-I-supra closedset ifA=IntSA∗STheorem 17.
Let (X, S, I) be a fuzzy ideal supra topological space, and let A be any fuzzy set in X. Then, the following statements hold:(1)
Every fuzzy regular-I-supra closedset is a fuzzy∗-supra prefect set(2)
Every fuzzy∗-supra prefect set is a fuzzy S∗-supra closedset(3)
Every fuzzy∗-supra prefect set is a fuzzy ∗-supra dense-in-itself setProof.
(1)
LetA be a fuzzy regular-I-supra closedset. Then, we have A=IntSA∗S. Since IntSA⊆A by (2) in Theorem 2, then IntSA∗S⊆A∗S. We have A=IntSA∗S⊆A∗S. Since A=IntSA∗S, then A∗S=IntSA∗S∗S⊆IntSA∗S=A. Therefore, we obtain A=A∗S. This shows that A is a fuzzy ∗-supra prefect set.(2)
LetA be a fuzzy ∗-supra perfect set. Then, we have A=A∗S; therefore, we obtain A∗S⊆A. This shows that A is a fuzzy S∗-supra closedset.(3)
LetA be a fuzzy ∗-supra perfect set. Then, we have A=A∗S; therefore, we obtain A⊆A∗s. This shows that A is a fuzzy ∗-supra dense-in-itself set.Remark 1.
The converses of Theorem17 need not be true as the following examples show.Example 3.
LetX=a,b,c and let A and B be fuzzy sets in X defined as follows:(2)Aa=0.4,Ab=0.7,Ac=0.5,Ba=0.6,Bb=0.3,Bc=0.5.
We putS=0x,1x,A. If we take I=0x, then B is a fuzzy ∗-supra perfect set but not a fuzzy regular-I-supra closedset.Example 4.
LetX=a,b,c, and let A and B be fuzzy sets in X defined as follows:(3)Aa=0.1,Ab=0.3,Ac=0.5,Ba=0.4,Bb=0.6,Bc=0.7.
We putS=0x,1x,A. If we take I=PX, then B is a fuzzy S∗-supra closedset but not a fuzzy ∗-supra perfect set.Example 5.
In Example3, A is a fuzzy ∗-supra dense-in-itself set but not a fuzzy ∗-supra perfect set.Definition 22.
Let (X, S, I) be a fuzzy ideal supra topological space, and let A be any fuzzy set in X. Then, A is said to be(1)
Fuzzy-I-supra openset ifA⊆IntSA∗S(2)
Fuzzy semi-I-supra openset ifA⊆Cl∗SIntSA(3)
Fuzzy pre-I-supra openset ifA⊆IntSCl∗SA(4)
Fuzzyα-I-supra openset if A⊆IntSCl∗SIntSA(5)
Fuzzyβ-I-supra openset if A⊆ClSIntSCl∗SA
A fuzzy setA of a fuzzy ideal supra topological space (X, S, I) is said to be fuzzy-I-closedset (resp. fuzzy semi-I-supra closedset, fuzzy pre-I-supra closedset, fuzzy α-I-supra closedset, and fuzzy β-I-supra closedset) if its complement is a fuzzy-I-openset (resp. fuzzy semi-I-supra openset, fuzzy pre-I-supra openset, fuzzy α-I-supra openset, and fuzzy β-I-supra openset).Theorem 18.
Let (X, S, I) be a fuzzy ideal supra topological space. Then, the following statements hold:(1)
Every fuzzy supra openset is a fuzzyα-I-supra openset(2)
Every fuzzy-I-supra openset is a fuzzy pre-I-supra openset(3)
Every fuzzyα-I-supra openset is a fuzzy semi-I-supra openset(4)
Every fuzzyα-I-supra openset is a fuzzy pre-I-supra openset(5)
Every fuzzy semi-I-supra openset is a fuzzyβ-I-supra openset(6)
Every fuzzy pre-I-supra openset is a fuzzyβ-I-supra opensetProof.
(1)
LetA be a fuzzy supra openset. Then, we have A=IntSA. Since, A⊆IntSA⊆Cl∗SIntSA. But if A is a fuzzy supra openset, then A=IntSA⊆IntSCl∗SIntSA. This shows that A is a fuzzy α-I-supra openset.(2)
LetA be a fuzzy-I-supra openset. Then, we have A⊆IntSA∗S, but A∗S⊆Cl∗SA. Then, A⊆IntSCl∗SA. This shows that A is a fuzzy pre-I-supra openset.(3)
LetA be a fuzzy α-I-supra openset. Then, we have A⊆IntSCl∗SIntSA\\⊆Cl∗SIntSA. This shows that A is a fuzzy semi-I-supra openset.(4)
LetA be a fuzzy α-I-supra openset. Then, we have A⊆IntSCl∗SIntSA\\⊆IntSCl∗SA. This shows that A is a fuzzy pre-I-supra openset.(5)
LetA be a fuzzy semi-I-supra openset. Then, we have A⊆Cl∗SIntSA⊆ClSIntSCl∗SA. This shows that A is a fuzzy β-I-supra openset.(6)
LetA be a fuzzy pre-I-supra openset. Then, we have A⊆IntSCl∗SA⊆ClSIntSCl∗SA. This shows that A is a fuzzy β-I-supra openset.Remark 2.
The converses of Theorem18 need not be true as the following examples show.Example 6.
In Example4, B is a fuzzy α-I-supra openset, but not a fuzzy supra openset.Example 7.
In Example4, A is a fuzzy pre-I-supra openset, but not a fuzzy-I-supra openset.Example 8.
LetX=a,b,c and let A, B, and C be a fuzzy sets in X defined as follows:(4)Aa=0.7,Ab=0.4,Ac=0.5,Ba=0.2,Bb=0.3,Bc=0.4,Ca=0.8,Cb=0.7,Cc=0.6.
We putS=0x,1x,B. If we take I=0x, then A is a fuzzy semi-I-supra openset, but not a fuzzy α-I-supra openset.Example 9.
In Example8, if we put S=0x,1x,A and we take I=0x, then C is a fuzzy pre-I-supra openset, but not a fuzzy α-I-supra openset.Example 10.
In Example3, B is a fuzzy β-I-supra openset, but not a fuzzy semi-I-supra openset.Example 11.
LetX=a,b,c, and let A, B, and C be fuzzy sets in X defined as follows:(5)Aa=0.7,Ab=0.4,Ac=0.2,Ba=0.9,Bb=0.8,Bc=0.7,Ca=0.1,Cb=0.2,Cc=0.3.
We putS=0x,1x,A,C,A∪C. If we take I=0x, then B is a fuzzy β-I-supra openset, but not a fuzzy pre-I-supra openset.Theorem 19.
Let (X, S, I) be a fuzzy ideal supra topological space, and let A be any fuzzy set in X. If A is a fuzzy regular-I-supra closedset, then A is a fuzzy semi-I-supra openset.Proof.
LetA be a fuzzy regular-I-supra closedset. Then, we have A=IntSA∗S\\⊆IntSA∗S∪IntSA=Cl∗SIntSA. This shows that A is a fuzzy semi-I-supra openset.Remark 3.
The converses of Theorem19 need not be true as the following example shows.Example 12.
In Example11, A is a fuzzy semi-I-supra openset, but not a fuzzy regular-I-supra closedset.


## 7. Some Fuzzy Ideal Supra Continuous Functions

Definition 23.
A functionf:X,S,I⟶Y,φ is said to be fuzzy ∗-supra perfectly continuous (resp. fuzzy regular-I-supra closed continuous and fuzzy contra ∗-supra continuous) if for every V∈φ,f−1Vis a fuzzy ∗-supra perfect (resp. fuzzy regular-I-supra closed and fuzzy S∗-supra closed) set of (X, S, I).Theorem 20.
For a functionf:X,S,I⟶Y,φ, the following statements hold:(1)
Every fuzzy regular-I-supra closed continuous is fuzzy∗-supra perfectly continuous(2)
Every fuzzy∗-supra perfectly continuous is fuzzy contra ∗-supra continuousProof.
This follows from Theorem17 and Definition 23.Remark 4.
The converses of Theorem20 need not be true as shown in the following examples.Example 13.
LetX=a,b,candY=x,y,z, and let A and B be fuzzy supra subsets defined as follows:(6)Aa=0.7,Ab=0.4,Ac=0.8,Bx=0.3,By=0.6,Bz=0.2.
LetS=0x,1x,A,φ=0Y,1Y,B, and I=0x. Then, the function f:X,S,I⟶Y,φ defined by fa=x,fb=y, and fc=z. Then, f is fuzzy ∗-supra perfectly continuous but not fuzzy regular-I-supra closed continuous.Example 14.
LetX=a,b,candY=x,y,z, and let A and B be fuzzy supra subsets defined as follows:(7)Aa=0.8,Ab=0.2,Ac=0.4,Bx=0.9,By=0.4,Bz=0.7.
LetS=0x,1x,A,φ=0Y,1Y,B, and I=0x. Then, the function f:X,S,I⟶Y,φ is defined by fa=x,fb=y, and fc=z; then, f is fuzzy contra ∗-supra continuous but not fuzzy ∗-supra perfectly continuous.Definition 24.
A functionf:X,S,I⟶Y,φ is said to be fuzzy-I-supra continuous (resp. fuzzy semi-I-supra continuous, fuzzy pre-I-supra continuous, fuzzy α-I-supra continuous, and fuzzy β-I-supra continuous) if for every V∈φ,f−1V is a fuzzy-I-supra openset (resp. fuzzy semi-I-supra open, fuzzy pre-I-supra open, fuzzy α-I-supra open, and fuzzy β-I-supra open) of (X, S, I).Theorem 21.
For a functionf:X,S,I⟶Y,φ, the following statements hold:(1)
Every fuzzy supra continuous is fuzzyα-I-supra continuous(2)
Every fuzzy-I-supra continuous is fuzzy pre-I-supra continuous(3)
Every fuzzyα-I-supra continuous is fuzzy semi-I-supra continuous(4)
Every fuzzyα-I-supra continuous is fuzzy pre-I-supra continuous(5)
Every fuzzy semi-I-supra continuous is fuzzyβ-I-supra continuous(6)
Every fuzzy pre-I-supra continuous is fuzzyβ-I-supra continuousProof.
This follows from Theorem18 and Definition 24.Remark 5.
The converses of Theorem21 need not be true as shown in the following examples.Example 15.
In Example14, f is fuzzy α-I-supra continuous but not fuzzy supra continuous.Example 16.
In Example14, if we take I=PX, then f is fuzzy pre-I-supra continuous but not fuzzy-I-supra continuous.Example 17.
LetX=a,b,candY=x,y,z, and let A and B be fuzzy supra subsets defined as follows:(8)Aa=0.4,Ab=0.1,Ac=0.2,Bx=0.6,By=0.9,Bz=0.8.
LetS=0x,1x,A,φ=0Y,1Y,B, and I=0x. Then, the function f:X,S,I⟶Y,φ is defined by fa=x,fb=y, and fc=z. Then, f is fuzzy semi-I-supra continuous and fuzzy pre-I-supra continuous but not fuzzy α-I-supra continuous.Example 18.
In Example13, f is fuzzy β-I-supra continuous but not fuzzy semi-I-supra continuous.Example 19.
In Example17, if we take I=PX, then f is fuzzy β-I-supra continuous but not fuzzy pre-I-supra continuous.Theorem 22.
Let (X, S, I) be a fuzzy ideal supra topological space and let A be any fuzzy set in X. If A is fuzzy regular-I-supra closedset, then A is a fuzzy semi-I-supra openset.Proof.
This follows from Theorem19 and Definitions 23 and 24.Remark 6.
The converses of Theorem22 need not be true as the following example shows.Example 20.
In Example17, A is fuzzy semi-I-supra continuous but not fuzzy regular-I-supra closed continuous.


## 8. Conclusion

The present paper is focused on the notion of fuzzy ideals in fuzzy supra topological spaces. The concept of a fuzzys-local function is also introduced here by utilizing the s-neighbourhood structure for a fuzzy supra topological space. These concepts are discussed with a view to find new fuzzy supra topologies from the original one. The basic structure, especially a basis for such generated fuzzy supra topologies, and several relations between different fuzzy ideals and fuzzy supra topologies are also studied here.



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