# A New Notion of Fuzzy Local Function and Some Applications

Authors: I. M. Taha; S. E. Abbas
Journal: Advances in Fuzzy Systems
 (2022)

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## Abstract

In this paper, a new notion of fuzzy local function calledr-fuzzy local function is introduced, and some properties are given in a fuzzy ideal topological space X,τ,ℓ in Šostak sense. After that, the concepts of fuzzy upper (resp., lower) almost ℓ-continuous, weakly ℓ-continuous, and almost weakly ℓ-continuous multifunctions are introduced. Some properties and characterizations of these multifunctions along with their mutual relationships are discussed with the help of examples.


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## Body

## 1. Introduction

Zadeh [1] introduced the basic idea of a fuzzy set as an extension of classical set theory. The basic notions of fuzzy sets have been improved and applied in different directions. Along this direction, we can refer [2–4]. A fuzzy multifunction (multivalued mapping) is a fuzzy set valued function [5–8]. The difference between fuzzy multifunctions and fuzzy functions has to do with the definition of an inverse image. For a fuzzy function, there is one inverse but for a fuzzy multifunction there are two types of inverses. By these two definitions of inverse, we can define the continuity of fuzzy multifunction. Fuzzy multifunctions have many applications, for instance, decision theory and artificial intelligence. Taha [9–11] introduced and studied the concepts of r-fuzzy ℓ-open, r-generalized fuzzy ℓ-open, and r-fuzzy δ-ℓ-open sets in a fuzzy ideal topological space X,τ,ℓ in Šostak sense [12]. Moreover, Taha [10, 11, 13] introduced and studied the concepts of fuzzy upper and lower α-ℓ-continuous (resp., β-ℓ-continuous, δ-ℓ-continuous, and generalized ℓ-continuous) multifunctions via fuzzy ideals [14].We lay out the remainder of this article as follows. Section2 contains some basic definitions and results that help in understanding the obtained results. In Section 3, we introduce and study the notion of r-fuzzy local function by using fuzzy ideal topological space and the definition of fuzzy difference between two fuzzy sets. Additionally, from the definition of r-fuzzy local function, we introduce a stronger form of fuzzy upper (resp., lower) precontinuous multifunctions [15], namely, fuzzy upper (resp., lower) ℓ-continuous multifunctions. In Sections 4, 5 and 6 we introduce the concepts of fuzzy upper (resp., lower) almost ℓ-continuous, weakly ℓ-continuous, and almost weakly ℓ-continuous multifunctions. Several properties of these new multifunctions are established. Finally, Section 7 gives some conclusions and suggests some future works.


## 2. Preliminaries

In this section, we present the basic definitions which we need in the next sections. Throughout this paper,X refers to an initial universe. The family of all fuzzy sets in X is denoted by IX and for λ∈IX, λcx=1−λx for all x∈X (where I=0,1 and I°=0,1. For t∈I, t¯x=t for all x∈X. All other notations are standard notations of fuzzy set theory. Let us define the fuzzy difference between two fuzzy sets λ,μ∈IX as follows:(1)λ∧¯μ=0¯,ifλ≤μ,λ∧μc,otherwise.Recall that a fuzzy ideaℓ on X [14] is a map ℓ:IX⟶I that satisfies the following conditions: (i) ∀λ,μ∈IX and λ≤μ⇒ℓμ≤ℓλ; (ii) ∀λ,μ∈IX⇒ℓλ∨μ≥ℓλ∧ℓμ. Also, ℓ is called proper if ℓ1¯=0 and there exists μ∈IX such that ℓμ〉0. The simplest fuzzy ideals on X, ℓ0, and ℓ1 are defined as follows:(2)ℓ0λ=1,ifλ=0¯,0,otherwise,and ℓ1λ=1∀λ∈IX. If ℓ1 and ℓ2 are fuzzy ideals on X, we say that ℓ1 is finer than ℓ2 (ℓ2 is coarser than ℓ1), denoted by ℓ2≤ℓ1, iff ℓ2λ≤ℓ1λ∀λ∈IX. Let τ be a fuzzy topology on X in Šostak sense, the triple X,τ,ℓ is called fuzzy ideal topological space.A mappingF:X⊸Y is called a fuzzy multifunction [15, 16] iff Fx∈IY for each x∈X. The degree of membership of y in Fx is denoted by Fxy=GFx,y for any x,y∈X×Y. Also, F is normalized iff for each x∈X, there exists y0∈Y such that GFx,y0=1 and F is Crisp iff GFx,y=1 for each x∈X and y∈Y. The upper inverse Fuμ, the lower inverse Flμ of μ∈IY, and the image Fλ of λ∈IX are defined as follows: Fuμx=∧y∈YGFcx,y∨μy, Flμx=∨y∈YGFx,y∧μy, and Fλy=∨x∈XGFx,y∧λx. All definitions and properties of image, lower, and upper are found in [15].


## 3. Fuzzy Ideal andr-Fuzzy Local Function

Definition 1.
LetX,τ,ℓ be a fuzzy ideal topological space, λ∈IX and r∈I°. Then, the r-fuzzy local function λr∗ of λ is defined as follows:(3)λr∗=∧μ∈IX:ℓλ∧¯μ≥r,τμc≥r.Remark 1.
.(1)
If we takeℓ=ℓ0 for each λ∈IX, we have(4)λr∗=∧μ∈IX:λ≤μ,τμc≥r=Cτλ,r.(2)
If we takeℓ=ℓ1 (resp., ℓλ≥r) for each λ∈IX, we have λr∗=0¯.
We will occasionally writeλr∗ or λr∗ℓ for λr∗ℓ,τ.Theorem 1.
LetX,τ,ℓ be a fuzzy ideal topological space and ℓ1 and ℓ2 be two fuzzy ideals on X. Then, for any fuzzy sets λ,ν∈IX,(1)
0¯r∗=0¯(2)
Ifλ≤ν⇒λr∗≤νr∗(3)
Ifℓ2≤ℓ1⇒λr∗ℓ1≤λr∗ℓ2(4)
λr∗  =  Cτλr∗,r≤Cτλ,r(5)
λr∗r∗≤λr∗ and λr∗c≠λcr∗(6)
λ∨νr∗=λr∗∨νr∗ and λ∧νr∗≤λr∗∧νr∗(7)
Ifℓν≥r⇒λ∨νr∗=λr∗(8)
τ≤τ∗ℓ≤τ∗ℓ1Proof.
(1)
From Definition1, we have 0¯r∗=0¯.(2)
Suppose thatλr∗≰νr∗ if λ≤ν. By the definition of νr∗, there exists μ∈IX with νr∗≤μ, ℓν∧¯μ≥r, and τμc≥r such that λr∗≰μ.Hence,λ≤ν implies λ∧¯μ≤ν∧¯μ and ℓλ∧¯μ≥ℓν∧¯μ≥r. Hence, λr∗≤μ; it is a contradiction. Then, λr∗≤νr∗.(3)
Suppose thatλr∗ℓ1≰λr∗ℓ2 if ℓ2≤ℓ1. By the definition of λr∗ℓ2, there exists μ∈IX with λr∗ℓ2≤μ, ℓ2λ∧¯μ≥r, and τμc≥r such that λr∗ℓ1≰μ. Hence, ℓ2≤ℓ1 implies ℓ1λ∧¯μ≥ℓ2λ∧¯μ≥r. Hence, λr∗ℓ1≤μ; it is a contradiction. Then, λr∗ℓ1≤λr∗ℓ2.(4)
From Definition2, we have λr∗  =  Cτλr∗,r. Since ℓ0≤ℓ for any fuzzy ideal ℓ, λr∗ℓ≤λr∗ℓ0=Cτλ,r. Thus, λr∗  =  Cτλr∗,r≤Cτλ,r.(5)
By (4), we haveλr∗r∗=Cτλr∗r∗,r≤Cτλr∗,r=λr∗. In general, the converse is not true, as shown by Example 1.(6)
Hence,λ and ν≤λ∨ν implies λr∗≤λ∨νr∗ and νr∗≤λ∨νr∗. Thus, λr∗∨νr∗≤λ∨νr∗. Now, we show that λ∨νr∗≤λr∗∨νr∗.Supposeλ∨νr∗≰λr∗∨νr∗ implies λ∨νr∗≰λr∗ and λ∨νr∗≰νr∗.Case 1. Since λ∨νr∗≰λr∗, by the definition of λr∗, there exists μ1∈IX with λr∗≤μ1, ℓλ∧¯μ1≥r, and τμ1c≥r such that λ∨νr∗≰μ1. Hence, λ∨ν∧¯μ1=λ∧¯μ1∨ν∧¯μ1 implies ℓλ∨ν∧¯μ1≥r. Hence, λ∨νr∗≤μ1, which is a contradiction. Thus, λ∨νr∗≤λr∗.Case 2. Since λ∨νr∗≰νr∗, by the definition of νr∗, there exists μ2∈IX with νr∗≤μ2, ℓν∧¯μ2≥r, and τμ2c≥r such that λ∨νr∗≰μ2. Hence, λ∨ν∧¯μ2=λ∧¯μ2∨ν∧¯μ2 implies ℓλ∨ν∧¯μ2≥r. Hence, λ∨νr∗≤μ2, which is a contradiction. Thus, λ∨νr∗≤νr∗. Then, λ∨νr∗≤λr∗∨νr∗.Also,λ∧ν≤λ and λ∧ν≤ν imply λ∧νr∗≤λr∗ and λ∧νr∗≤νr∗. Thus, λ∧νr∗≤λr∗∧νr∗.(7)
Hence,ℓν≥r implies νr∗=0¯. Thus, λ∨νr∗=λr∗∨νr∗=λr∗.(8)
By (3),τ∗ℓ0≤τ∗ℓ≤τ∗ℓ1, i.e., τ≤τ∗ℓ≤τ∗ℓ1.Example 1.
Defineτ,ℓ:IX⟶I as follows:(5)τλ=1,ifλ∈0¯,1¯,12,ifλ=0.5¯,34,ifλ=0.8¯,0,otherwise,ℓν=1,ifν=0¯,12,ifν=0.5¯,23,if0¯<ν<0.5¯,0,otherwise..
InX,τ,ℓ, 0¯=0.6¯1/2∗1/2∗≠0.6¯1/2∗=0.5¯ and 1¯=0.5¯1/2∗c≠0.5¯c1/2∗=0¯.Lemma 1.
LetX,τ,ℓ be a fuzzy ideal topological space, and λj:j∈J⊂IX. Then, ∨j∈Jλjr∗≤∨j∈Jλjr∗ (resp., ∧j∈Jλjr∗≤∧j∈Jλjr∗).Proof.
Sinceλj≤∨j∈Jλj for each j∈J, by Theorem 1 (2), we have λjr∗≤∨j∈Jλjr∗ for each j∈J. This implies ∨j∈Jλjr∗≤∨j∈Jλjr∗. Other case is similarly proved.Definition 2.
LetX,τ,ℓ be a fuzzy ideal topological space. Then, for each λ∈IX and r∈I°, we define an operator Cl∗:IX×I°⟶IX as follows:(6)Cl∗λ,r=λ∨λr∗.
The fuzzy topology generated byCl∗ is denoted by τ∗, i.e., τ∗λ=∨r:Cl∗λc,r=λc and τ≤τ∗. Now, if ℓ=ℓ0, then Cl∗λ,r=λ∨λr∗=λ∨Cτλ,r=Cτλ,r for each λ∈IX. So, τ∗=τ. Again if ℓ=ℓ1 (resp., ℓλ≥r), then Cl∗λ,r=λ for each λ∈IX.Theorem 2.
LetX,τ,ℓ be a fuzzy ideal topological space. Then, for any fuzzy sets λ,ν∈IX and r∈I°, the operator Cl∗:IX×I°⟶IX satisfies the following properties:(1)
Cl∗0¯,r=0¯(2)
λ≤Cl∗λ,r≤Cτλ,r(3)
Ifλ≤ν, then Cl∗λ,r≤Cl∗ν,r(4)
Cl∗λ∨ν,r=Cl∗λ,r∨Cl∗ν,r(5)
Cl∗λ∧ν,r≤Cl∗λ,r∧Cl∗ν,r(6)
Cl∗Cl∗λ,r,r=Cl∗λ,rProof.
(1)
Hence,Cl∗0¯,r=0¯∨0¯r∗ and 0¯r∗=0¯ implies Cl∗0¯,r=0¯.(2)
Hence,Cl∗λ,r=λ∨λr∗ implies λ≤Cl∗λ,r. Since λ≤Cτλ,r and from Theorem 1(4), we have λr∗≤Cτλ,r implying Cl∗λ,r≤Cτλ,r. Thus, λ≤Cl∗λ,r≤Cτλ,r.(3)
Fromλ≤ν and Theorem 1 (2), we have λ∨λr∗≤ν∨νr∗, i.e., Cl∗λ,r≤Cl∗ν,r.(4)
By the definition ofCl∗ and from Theorem 1 (6), we have Cl∗λ∨ν,r=λ∨ν∨λ∨νr∗=λ∨ν∨λr∗∨νr∗=λ∨λr∗∨ν∨νr∗=Cl∗λ,r∨Cl∗ν,r.(5)
Hence,λ∧ν≤λ and λ∧ν≤ν imply Cl∗λ∧ν,r≤Cl∗λ,r and Cl∗λ∧ν,r≤Cl∗ν,r. Thus, Cl∗λ∧ν,r≤Cl∗λ,r∧Cl∗ν,r.(6)
From (2) and (3), we haveCl∗λ,r≤Cl∗Cl∗λ,r,r. Now, we show that Cl∗λ,r≥Cl∗Cl∗λ,r,r. By (4) and the definition of Cl∗, we have Cl∗Cl∗λ,r,r=Cl∗λ∨λr∗,r=Cl∗λ,r∨Cl∗λr∗,r=Cl∗λ,r∨λr∗∨λr∗r∗=Cl∗λ,r∨λr∗r∗≤Cl∗λ,r∨λr∗=Cl∗λ,r.
The following theorem is similarly proved, as in Theorem2.Theorem 3.
LetX,τ,ℓ be a fuzzy ideal topological space. Then, for each λ∈IX and r∈I°, we define an operator int∗:IX×I°⟶IX as follows:(7)int∗λ,r=λ∧λcr∗c.For eachλ,ν∈IX, the operator int∗ satisfies the following properties:(1)
int∗1¯,r=1¯(2)
Iτλ,r≤int∗λ,r≤λ(3)
Ifλ≤ν, then int∗λ,r≤int∗ν,r(4)
int∗int∗λ,r,r=int∗λ,r(5)
int∗λ∧ν,r=int∗λ,r∧int∗ν,r(6)
int∗λ,r=Iτλ,r, if ℓ=ℓ0(7)
int∗λc,r=Cl∗λ,rcDefinition 3.
A fuzzy multifunctionF:X,τ,ℓ⊸Y,η is called(1)
Fuzzy upper (resp., lower)ℓ-continuous at a fuzzy point xt∈domF iff xt∈Fuμ (resp., xt∈Flμ) for each μ∈IY, ημ≥r, there exists λ∈IX, τλ≥r, and xt∈λ such that λ∧domF≤Fuμr∗ (resp., λ≤Flμr∗).(2)
Fuzzy upperℓ-continuous (resp., fuzzy lower ℓ-continuous) iff it is fuzzy upper ℓ-continuous (resp., fuzzy lower ℓ-continuous) at every xt∈domF.Remark 2.
IfF is normalized, then F is fuzzy upper ℓ-continuous at a fuzzy point xt∈domF iff xt∈Fuμ for each μ∈IY, ημ≥r there exists λ∈IX, τλ≥r and xt∈λ such that λ≤Fuμr∗.Theorem 4.
LetF:X,τ,ℓ⊸Y,η be a fuzzy multifunction (resp., normalized fuzzy multifunction). Then, F is fuzzy lower (resp., upper) ℓ-continuous iff Flμ≤IτFlμr∗,r (resp., Fuμ≤IτFuμr∗,r) for each μ∈IY with ημ≥r and r∈Io.Proof..
⇒ Let xt∈domF, μ∈IY with ημ≥r and xt∈Flμ. Then, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤Flμr∗. Thus, xt∈λ≤IτFlμr∗,r and hence Flμ≤IτFlμr∗,r.
⇐ Let xt∈domF, μ∈IY with ημ≥r and xt∈Flμ. Then, Flμ≤IτFlμr∗,r and hence xt∈IτFlμr∗,r≤Flμr∗. Thus, F is fuzzy lower ℓ-continuous. Other case is similarly proved.Remark 3.
.(1)
Every fuzzy lower (resp., upper)ℓ-continuous multifunction is fuzzy lower (resp., upper) precontinuous [15](2)
If we takeℓ=ℓ0, we have F as fuzzy lower (resp., upper) ℓ-continuous iff it is fuzzy lower (resp., upper) precontinuous(3)
Fuzzy upper (resp., lower)ℓ-continuity and fuzzy upper (resp., lower) semicontinuity [15] are independent notions, as shown by example 2.Example 2.
LetX=x1,x2, Y=y1,y2,y3, and F:X⊸Y be a fuzzy multifunction defined by GFx1,y1=0.1,GFx1,y2=1.0,GFx1,y3=0.3,GFx2,y1=0.5,GFx2,y2=0.1, and GFx2,y3=1.0. Define τ1,τ2,ℓ1,ℓ2:IX⟶I and η:IY⟶I as follows:(8)τ1λ=1,ifλ∈0¯,1¯,12,ifλ=0.2¯,34,ifλ=0.3¯,0,otherwise,τ2λ=1,ifλ∈0¯,1¯,12,ifλ=0.3¯,0,otherwise,ℓ1ν=1,ifν=0¯,23,if0¯<ν<0.3¯0,otherwise,,ℓ2ν=1,ifν=0¯,12,if0¯<ν<0.2¯0,otherwise,,ημ=1,ifμ∈0¯,1¯,12,ifμ=0.2¯,0,otherwise..
Then,(1)
F:X,τ1,ℓ1⊸Y,η is fuzzy upper (resp., lower) semicontinuous, but it is not fuzzy upper (resp., lower) ℓ-continuous because Fu0.2¯=0.2¯ (resp., Fl0.2¯=0.2¯) and 0.2¯≤Iτ10.2¯,1/2=0.2¯, but 0.2¯≰Iτ10.2¯1/2∗,1/2=0¯(2)
F:X,τ2,ℓ2⊸Y,η is fuzzy upper (resp., lower) ℓ-continuous, but it is not fuzzy upper (resp., lower) semicontinuous because Fu0.2¯=0.2¯ (resp., Fl0.2¯=0.2¯) and 0.2¯≤Iτ20.2¯1/2∗,1/2=0.3¯, but 0.2¯≰Iτ20.2¯,1/2=0¯Corollary 1.
LetF:X,τ,ℓ⊸Y,η and H:Y,η⊸Z,γ be two fuzzy multifunctions (resp., normalized fuzzy multifunctions). Then, H°F is fuzzy lower (resp., upper) ℓ-continuous if F is fuzzy lower (resp., upper) ℓ-continuous and H is fuzzy lower (resp., upper) semicontinuous.


## 4. Fuzzy Upper and Lower Almostℓ-Continuity

Definition 4.
A fuzzy multifunctionF:X,τ⊸Y,η,ℓ is called(1)
Fuzzy upper almostℓ-continuous at a fuzzy point xt∈domF iff xt∈Fuμ for each μ∈IY and ημ≥r, there exists λ∈IX with τλ≥r and xt∈λ such that λ∧domF≤FuIηCl∗μ,r,r.(2)
Fuzzy lower almostℓ-continuous at a fuzzy point xt∈domF iff xt∈Flμ for each μ∈IY and ημ≥r, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤FlIηCl∗μ,r,r.(3)
Fuzzy upper almostℓ-continuous (resp., fuzzy lower almost ℓ-continuous) iff it is fuzzy upper almost ℓ-continuous (resp., fuzzy lower almost ℓ-continuous) at every xt∈domF.Remark 4.
.(1)
IfF is normalized, F is fuzzy upper almost ℓ-continuous at a fuzzy point xt∈domF iff xt∈Fuμ for each μ∈IY and ημ≥r, there exists λ∈IX, τλ≥r and xt∈λ such that λ≤FuIηCl∗μ,r,r.(2)
Fuzzy upper (lower) semicontinuity⇒ fuzzy upper (lower) almost ℓ-continuity ⇒ fuzzy upper (lower) almost continuity [17].(3)
Fuzzy upper (lower) almostℓ0-continuity ⇔ fuzzy upper (lower) almost continuity.Theorem 5.
For a fuzzy multifunctionF:X,τ⊸Y,η,ℓ, μ∈IY, and r∈Io, the following statements are equivalent:(1)
F is fuzzy lower almost ℓ-continuous(2)
Flμ≤IτFlIηCl∗μ,r,r,r, if ημ≥r(3)
CτFuCηint∗μ,r,r,r≤Fuμ, if ημc≥rProof.
(1)
⇒ (2) Let xt∈domF, μ∈IY with ημ≥r and xt∈Flμ. Then, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤FlIηCl∗μ,r,r. Thus, xt∈λ≤IτFlIηCl∗μ,r,r,r, i.e., Flμ≤IτFlIηCl∗μ,r,r,r.(2)
⇒ (3) Let μ∈IY with ημc≥r. Then, by (2), Fuμc=Flμc≤IτFlIηCl∗μc,r,r,r=CτFuCηint∗μ,r,r,rc. Thus,(9)CτFuCηint∗μ,r,r,r≤Fuμ.(3)
⇒ (1) Let xt∈domF, μ∈IY with ημ≥r and xt∈Flμ. Then, by (3), IτFlIηCl∗μ,r,r,rc=CτFuCηint∗μc,r,r,r≤Fuμc=Flμc, i.e., Flμ≤IτFlIηCl∗μ,r,r,r. Thus, xt∈IτFlIηCl∗μ,r,r,r≤FlIηCl∗μ,r,r. Hence, F is fuzzy lower almost ℓ-continuous.
The following theorem is similarly proved as in Theorem5.Theorem 6.
For a normalized fuzzy multifunctionF:X,τ⊸Y,η,ℓ, μ∈IY and r∈Io, the following statements are equivalent:(1)
F is fuzzy upper almost ℓ-continuous(2)
Fuμ≤IτFuIηCl∗μ,r,r,r, if ημ≥r(3)
CτFlCηint∗μ,r,r,r≤Flμ, if ημc≥rExample 3.
Define a fuzzy multifunctionF:X⊸Y as in example 2,τ1,τ2:IX⟶I and η1,η2,ℓ1,ℓ2:IY⟶I as follows:(10)τ1λ=1,ifλ∈0¯,1¯,34,ifλ=0.3¯,0,otherwise,τ2λ=1,ifλ∈0¯,1¯,12,ifλ=0.5¯0,otherwise,,η1μ=1,ifμ∈0¯,1¯,12,ifμ=0.2¯,23,ifμ=0.3¯,0,otherwise,η2μ=1,ifμ∈0¯,1¯,12,ifμ=0.5¯,34,ifμ=0.4¯,0,otherwise,ℓ1ν=1,ifν=0¯,23,if0¯〈ν〈0.2¯,0,otherwise,ℓ2ν=1,ifν=0¯,23,if0¯<ν≤0.4¯0,otherwise.,
Then,(1)
F:X,τ1⊸Y,η1,ℓ1 is fuzzy upper (resp., lower) almost ℓ-continuous, but it is not fuzzy upper (resp., lower) semicontinuous because 0.2¯=Fu0.2¯≤Iτ1FuIη1Cl∗0.2¯,1/2,1/2,1/2=0.3¯ and 0.3¯=Fu0.3¯≤Iτ1FuIη1Cl∗0.3¯,1/2,1/2,1/2=0.3¯, but 0.2¯≰Iτ10.2¯,1/2=0¯,(2)
F:X,τ2⊸Y,η2,ℓ2 is fuzzy upper (resp., lower) almost continuous, but it is not fuzzy upper (resp., lower) almost ℓ-continuous because 0.4¯=Fu0.4¯≤Iτ2FuIη2Cη20.4¯,1/2,1/2,1/2=0.5¯ and Fu0.5¯≤Iτ2FuIη2Cη20.5¯,1/2,1/2,1/2=0.5¯, but 0.4¯=Fu0.4¯≰Iτ2FuIη2Cl∗0.4¯,1/2,1/2,1/2=0¯.Theorem 7.
For a fuzzy multifunctionF:X,τ⊸Y,η,ℓ, μ∈IY, and r∈Io, the following statements are equivalent:(1)
F is fuzzy lower almost ℓ-continuous(2)
τFlμ≥r, if μ=IηCl∗μ,r,r(3)
τFlIηCl∗μ,r,r≥r, if ημ≥rProof.
(1)
⇒ (2) If μ=IηCl∗μ,r,r, then ημ≥r. By Theorem 5 (2), Flμ≤IτFlIηCl∗μ,r,r,r=IτFlμ,r. Thus, τFlμ≥r.(2)
⇔ (3) is obvious.(3)
⇒ (1) Let xt∈domF, μ∈IY with ημ≥r and xt∈Flμ. Then, by (3) and μ≤IηCl∗μ,r,r, τFlIηCl∗μ,r,r≥r, and xt∈Flμ≤FlIηCl∗μ,r,r. Thus, F is fuzzy lower almost ℓ-continuous.
The following theorems are similarly proved as in Theorem7.Theorem 8.
For a fuzzy multifunctionF:X,τ⊸Y,η,ℓ, μ∈IY, and r∈Io, the following statements are equivalent:(1)
F is fuzzy lower almost ℓ-continuous(2)
τFuμc≥r if μ=Cηint∗μ,r,r(3)
τFuCηint∗μ,r,rc≥r if ημc≥rTheorem 9.
For a normalized fuzzy multifunctionF:X,τ⊸Y,η,ℓμ∈IY and r∈Io, the following statements are equivalent:(1)
F is fuzzy upper almost ℓ-continuous(2)
τFuμ≥r if μ=IηCl∗μ,r,r(3)
τFuIηCl∗μ,r,r≥r if ημ≥rTheorem 10.
For a normalized fuzzy multifunctionF:X,τ⊸Y,η,ℓ, μ∈IY, and r∈Io, the following statements are equivalent:(1)
F is fuzzy upper almost ℓ-continuous(2)
τFlμc≥r if μ=Cηint∗μ,r,r(3)
τFlCηint∗μ,r,rc≥r if ημc≥rTheorem 11.
LetF:X,τ⊸Y,η,ℓ be a fuzzy multifunction. Then, F is fuzzy lower almost ℓ-continuous iff CτFuμ,r≤FuCημ,r for any μ∈IY with μ≤Cηint∗μ,r,r and r∈I°.Proof.
⇒ Let F be a fuzzy lower almost ℓ-continuous. Then, for any μ∈IY with μ≤Cηint∗μ,r,r=ν (say), where ν=Cηint∗ν,r,r. By Theorem 8, τFuνc≥r and thus CτFuμ,r≤CτFuν,r=FuCηint∗ν,r,r≤FuCημ,r.
⇐ Let μ∈IY with μ=Cηint∗μ,r,r. Then, μ≤Cηint∗μ,r,r and CτFuμ,r≤FuCημ,r=Fuμ. Therefore, we obtain τFuμc≥r. Thus, by Theorem 8, F is fuzzy lower almost ℓ-continuous.
11. The following theorem is similarly proved as in Theorem.Theorem 12.
LetF:X,τ⊸Y,η,ℓ be a normalized fuzzy multifunction. Then, F is fuzzy upper almost ℓ-continuous iff CτFlμ,r≤FlCημ,r for any μ∈IY with μ≤Cηint∗μ,r,r and r∈I°.Theorem 13.
LetFi:X,τ⊸Y,η,ℓi∈Γ be a family of fuzzy lower almost ℓ-continuous. Then, ∪i∈ΓFi is fuzzy lower almost ℓ-continuous.Proof.
Letμ∈IY. Then, ∪i∈ΓFilμ=∨i∈ΓFilμ. Since Fii∈Γ is a family of fuzzy lower almost ℓ-continuous, τFilμ≥r for any μ=IηCl∗μ,r,r and i∈Γ. Then, τ∪i∈ΓFilμ=τ∨i∈ΓFilμ≥∧i∈ΓτFilμ≥r. Hence, ∪i∈ΓFi is fuzzy lower almost ℓ-continuous.Theorem 14.
LetF1 and F2:X,τ⊸Y,η,ℓ be two normalized fuzzy upper almost ℓ-continuous. Then, F1∪F2 is fuzzy upper almost ℓ-continuous.Proof.
Letμ∈IY. Then, F1∪F2uμ=F1uμ∧F2uμ. Since F1 and F2 are two normalized fuzzy upper almost ℓ-continuous, τFiuμ≥r for any μ=IηCl∗μ,r,r and i∈1,2. Then, τF1∪F2uμ=τF1uμ∧F2uμ≥τF1uμ∧τF2uμ≥r. Hence, F1∪F2 is fuzzy upper almost ℓ-continuous.Corollary 2.
LetF:X,τ⊸Y,η,ℓ and H:Y,η⊸Z,γ be two fuzzy multifunctions (resp., normalized fuzzy multifunctions). Then, H°F is fuzzy lower (resp., upper) almost ℓ-continuous if F is fuzzy lower (resp., upper) almost ℓ-continuous and H is fuzzy lower (resp., upper) semicontinuous.


## 5. Fuzzy Upper and Lower Weaklyℓ-Continuity

Definition 5.
A fuzzy multifunctionF:X,τ⊸Y,η,ℓ is called(1)
Fuzzy upper weaklyℓ-continuous at a fuzzy point xt∈domF iff xt∈Fuμ for each μ∈IY and ημ≥r, there exists λ∈IX with τλ≥r and xt∈λ such that λ∧domF≤FuCl∗μ,r.(2)
Fuzzy lower weaklyℓ-continuous at a fuzzy point xt∈domF iff xt∈Flμ for each μ∈IY and ημ≥r, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤FlCl∗μ,r.(3)
Fuzzy upper weaklyℓ-continuous (resp., fuzzy lower weakly ℓ-continuous) iff it is fuzzy upper weakly ℓ-continuous (resp., fuzzy lower weakly ℓ-continuous) at every xt∈domF.Remark 5.
.(1)
IfF is normalized, F is fuzzy upper weakly ℓ-continuous at a fuzzy point xt∈domF iff xt∈Fuμ for each μ∈IY and ημ≥r, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤FuCl∗μ,r.(2)
Fuzzy upper (lower) almostℓ-continuity ⇒ fuzzy upper (lower) weakly ℓ-continuity ⇒ fuzzy upper (lower) weakly continuity [17].(3)
Fuzzy upper (lower) weaklyℓ0-continuity ⇔ fuzzy upper (lower) weakly continuity.Theorem 15.
A fuzzy multifunctionF:X,τ⊸Y,η,ℓ is fuzzy lower weakly ℓ-continuous iff Flμ≤IτFlCl∗μ,r,r for each μ∈IY, ημ≥r, and r∈Io.Proof.
⇒ Let xt∈domF, μ∈IY with ημ≥r and xt∈Flμ. Then, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤FlCl∗μ,r. Thus, xt∈λ≤IτFlCl∗μ,r,r and hence Flμ≤IτFlCl∗μ,r,r.
⇐ Let xt∈domF, μ∈IY with ημ≥r and xt∈Flμ. Then, xt∈Flμ≤IτFlCl∗μ,r,r. Thus, xt∈IτFlCl∗μ,r,r≤FlCl∗μ,r. Hence, F is fuzzy lower weakly ℓ-continuous.
The following theorem is similarly proved as in Theorem15.Theorem 16.
A normalized fuzzy multifunctionF:X,τ⊸Y,η,ℓ is fuzzy upper weakly ℓ-continuous iff Fuμ≤IτFuCl∗μ,r,r for each μ∈IY with ημ≥r and r∈Io.Example 4.
LetX=x1,x2,x3 , Y=y1,y2,y3 and F:X⊸Y be a fuzzy multifunction defined by GFx1,y1=0.8,GFx1,y2=0.3,GFx1,y3=0.3,GFx2,y1=0.1,GFx2,y2=1.0, GFx2,y3=0.1,GFx3,y1=0.1,GFx3,y2=0.2,GFx3,y3=1.0. Define μ1∈IX and μ2∈IY as follows: μ1=x1/0.4,x2/0.1,x3/0.2 and μ2=y1/0.4,y2/0.1,y3/0.2. Define τ:IX⟶I and η,ℓ1,ℓ2:IY⟶I as follows:(11)τλ=1,ifλ∈0¯,1¯,12,ifλ=0.5¯,0,otherwise,ημ=1,ifμ∈0¯,1¯,34,ifμ=μ2,0,otherwise,ℓ1ν=1,ifν=0¯,12,if0¯〈ν≤0.4¯0,otherwise,,ℓ2ν=1,ifν=0¯,12,if0¯<ν≤0.2¯,0,otherwise..
Then,(1)
F:X,τ⊸Y,η,ℓ1 is fuzzy upper (resp., lower) weakly continuous, but it is not fuzzy upper (resp., lower) weakly ℓ-continuous because μ1=Fuμ2≤IτFuCημ2,1/2,1/2=0.5¯, but μ1=Fuμ2≰IτFuCl∗μ2,1/2,1/2=0¯.(2)
F:X,τ⊸Y,η,ℓ2 is fuzzy upper (resp., lower) weakly ℓ-continuous, but it is not fuzzy upper (resp. lower) almost ℓ-continuous because μ1=Fuμ2≤IτFuCl∗μ2,1/2,1/2=0.5¯, but μ1=Fuμ2≰IτFuIηCl∗μ2,1/2,1/2,1/2=0¯.Theorem 17.
A fuzzy multifunctionF:X,τ⊸Y,η,ℓ is fuzzy lower weakly ℓ-continuous iff CτFuint∗μ,r,r≤Fuμ for each μ∈IY with ημc≥r.Proof..
⇒ Let μ∈IY with ημc≥r. Then, by Theorem 15, Fuμc=Flμc≤IτFlCl∗μc,r,r=CτFuint∗μ,r,rc. Thus, CτFuint∗μ,r,r≤Fuμ.
⇐ Let xt∈domF, μ∈IY with ημ≥r and xt∈Flμ. Then, IτFlCl∗μ,r,rc=CτFuint∗μc,r,r≤Fuμc=Flμc. Hence, Flμ≤IτFlCl∗μ,r,r, i.e., xt∈IτFlCl∗μ,r,r≤FlCl∗μ,r. Thus, F is fuzzy lower weakly ℓ-continuous.
The following theorem is similarly proved as in Theorem17.Theorem 18.
A fuzzy multifunctionF:X,τ⊸Y,η,ℓ is fuzzy upper weakly ℓ-continuous iff CτFlint∗μ,r,r≤Flμ for each μ∈IY with ημc≥r.Theorem 19.
IfF:X,τ⊸Y,η,ℓ is a normalized fuzzy upper weakly ℓ-continuous multifunction and Fλ≤IηCl∗Fλ,r,r for each λ∈IX, then F is fuzzy upper almost ℓ-continuous.Proof.
Letxt∈domF, μ∈IY, ημ≥r, and xt∈Fuμ. Then, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤FuCl∗μ,r, Fλ≤FFuCl∗μ,r≤Cl∗μ,r. Since Fλ≤IηCl∗Fλ,r,r≤IηCl∗μ,r,r, λ≤FuFλ≤FuIηCl∗μ,r,r. Then, F is fuzzy upper almost ℓ-continuous.Corollary 3.
LetF:X,τ⊸Y,η,ℓ and H:Y,η⊸Z,γ be two fuzzy multifunctions (resp., normalized fuzzy multifunctions). Then, H°F is fuzzy lower (resp., upper) weakly ℓ-continuous if F is fuzzy lower (resp., upper) weakly ℓ-continuous and H is fuzzy lower (resp., upper) semicontinuous.Theorem 20.
LetF:X,τ⊸Y,η,ℓ be a fuzzy lower weakly ℓ-continuous multifunction. Then, Flμ≤IτFlCl∗μ,r,r for any μ∈IY with μ≤IηCl∗μ,r,r and r∈I°.Proof.
LetF be a fuzzy lower weakly ℓ-continuous and μ≤IηCl∗μ,r,r. Then, if xt∈Flμ≤FlIηCl∗μ,r,r, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤FlCl∗IηCl∗μ,r,r,r≤FlCl∗μ,r. Thus, λ≤IτFlCl∗μ,r,r and Flμ≤IτFlCl∗μ,r,r.
The following theorem is similarly proved as in Theorem20.Theorem 21.
LetF:X,τ⊸Y,η,ℓ be a normalized fuzzy upper weakly ℓ-continuous multifunction. Then, Fuμ≤IτFuCl∗μ,r,r for any μ∈IY with μ≤IηCl∗μ,r,r and r∈I°.


## 6. Fuzzy Upper and Lower Almost Weaklyℓ-Continuity

Definition 6.
A fuzzy multifunctionF:X,τ⊸Y,η,ℓ is called(1)
Fuzzy upper almost weaklyℓ-continuous at a fuzzy point xt∈domF iff xt∈Fuμ for each μ∈IY and ημ≥r, there exists λ∈IX with τλ≥r and xt∈λ such that λ∧domF≤CτFuCl∗μ,r,r.(2)
Fuzzy lower almost weaklyℓ-continuous at a fuzzy point xt∈domF iff xt∈Flμ for each μ∈IY and ημ≥r, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤CτFlCl∗μ,r,r.(3)
Fuzzy upper almost weaklyℓ-continuous (resp., fuzzy lower almost weakly ℓ-continuous) iff it is fuzzy upper almost weakly ℓ-continuous (resp., fuzzy lower almost weakly ℓ-continuous) at every xt∈domF.Remark 6.
.(1)
IfF is normalized, then F is fuzzy upper almost weakly ℓ-continuous at a fuzzy point xt∈domF iff xt∈Fuμ for each μ∈IY and ημ≥r, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤CτFuCl∗μ,r,r(2)
Fuzzy upper (lower) weaklyℓ-continuity ⇒ fuzzy upper (lower) almost weakly ℓ-continuity ⇒ fuzzy upper (lower) almost weakly continuity [17](3)
Fuzzy upper (lower) almost weaklyℓ0-continuity ⇔ fuzzy upper (lower) almost weakly continuityTheorem 22.
For a fuzzy multifunctionF:X,τ⊸Y,η,ℓ, μ∈IY, and r∈Io, the following statements are equivalent:(1)
F is fuzzy lower almost weakly ℓ-continuous(2)
Flμ≤IτCτFlCl∗μ,r,r,r if ημ≥r(3)
CτIτFuint∗μ,r,r,r≤Fuμ if ημc≥rProof.
(1)
⇒ (2) Let xt∈domF, μ∈IY with ημ≥r and xt∈Flμ. Then, there exists λ∈IX with τλ≥r and xt∈λ such that λ≤CτFlCl∗μ,r,r. Thus, xt∈IτCτFlCl∗μ,r,r,r, i.e., Flμ≤IτCτFlCl∗μ,r,r,r.(2)
⇒ (3) Let μ∈IY with ημc≥r. Then, by (2), Fuμc=Flμc≤IτCτFlCl∗μc,r,r,r=CτIτFuint∗μ,r,r,rc. Thus,(12)CτIτFuint∗μ,r,r,r≤Fuμ.(3)
⇒ (1) Let xt∈domF, μ∈IY with ημ≥r and xt∈Flμ. Then, by (3), IτCτFlCl∗μ,r,r,rc=CτIτFuint∗μc,r,r,r≤Fuμc=Flμc, i.e., Flμ≤IτCτFlCl∗μ,r,r,r. Therefore, xt∈IτCτFlCl∗μ,r,r,r≤CτFlCl∗μ,r,r. Thus, F is fuzzy lower almost weakly ℓ-continuous.
The following theorem is similarly proved as in Theorem22.Theorem 23.
For a normalized fuzzy multifunctionF:X,τ⊸Y,η,ℓ, μ∈IY and r∈Io, the following statements are equivalent:(1)
F is fuzzy upper almost weakly ℓ-continuous(2)
Fuμ≤IτCτFuCl∗μ,r,r,r if ημ≥r(3)
CτIτFlint∗μ,r,r,r≤Flμ if ημc≥rExample 5.
LetX=x1,x2,x3, Y=y1,y2,y3, and F:X⊸Y be a fuzzy multifunction defined by GFx1,y1=0.8,GFx1,y2=0.3,GFx1,y3=0.3,GFx2,y1=0.1,GFx2,y2=1.0, GFx2,y3=0.1,GFx3,y1=0.1,GFx3,y2=0.2,GFx3,y3=1.0. Define μ1∈IX and μ2∈IY as follows: μ1=x1/0.4,x2/0.1,x3/0.2 and μ2=y1/0.4,y2/0.1,y3/0.2. Define τ1,τ2:IX⟶I and η,ℓ:IY⟶I as follows:(13)τ1λ=1,ifλ∈0¯,1¯,12,ifλ=0.5¯,0,otherwise,τ2λ=1,ifλ∈0¯,1¯,12,ifλ=0.3¯,0,otherwise,ημ=1,ifμ∈0¯,1¯,34,ifμ=μ2,0,otherwise,,ℓν=1,ifν=0¯,12,if0¯〈ν≤0.4¯,0,otherwise..
Then,(1)
F:X,τ2⊸Y,η,ℓ is fuzzy upper (resp., lower) almost weakly continuous, but it is not fuzzy upper (resp., lower) almost weakly ℓ-continuous(2)
F:X,τ1⊸Y,η,ℓ is fuzzy upper (resp., lower) almost weakly ℓ-continuous, but it is not fuzzy upper (resp., lower) weakly ℓ-continuousTheorem 24.
LetF:X,τ⊸Y,η,ℓ be a normalized fuzzy multifunction and F be a fuzzy upper almost weakly ℓ-continuous and fuzzy lower almost ℓ-continuous. Then, F is fuzzy upper weakly ℓ-continuous.Proof..
Letμ∈IY with ημ≥r and F be a fuzzy upper almost weakly ℓ-continuous. Then, by Theorem 23 (2), Fuμ≤IτCτFuCl∗μ,r,r,r. Since Cημ,r=Cηint∗Cημ,r,r,r, it follows from Theorem 8 (2) that τFuCημ,rc≥r, τFuCl∗μ,rc≥r, and Fuμ≤IτFuCl∗μ,r,r. Thus, by Theorem 16, F is fuzzy upper weakly ℓ-continuous.
24. The following theorem is similarly proved as in Theorem.Theorem 25.
LetF:X,τ⊸Y,η,ℓ be a normalized fuzzy multifunction and F be a fuzzy lower almost weakly ℓ-continuous and fuzzy upper almost ℓ-continuous. Then, F is fuzzy lower weakly ℓ-continuous.Corollary 4.
LetF:X,τ⊸Y,η,ℓ and H:Y,η⊸Z,γ be two fuzzy multifunctions (resp., normalized fuzzy multifunctions). Then, H°F is fuzzy lower (resp., upper) almost weakly ℓ-continuous if F is fuzzy lower (resp., upper) almost weakly ℓ-continuous and H is fuzzy lower (resp., upper) semicontinuous.


## 7. Conclusion and Future Work

In the present paper, based on fuzzy operatorsα, β, idX:IX×I°⟶IX, and θ, θ∗, ∂, idY:IY×I°⟶IY, we can give a generalized form of fuzzy (resp., normalized fuzzy) multifunction as F:X,τ,ℓ1⊸Y,η,ℓ2 is fuzzy lower (resp., upper) α,β,θ,∂,ℓ1,ℓ2-continuous multifunction if for every μ∈IY with ημ≥r and r∈Io, ℓ1αFl∂μ,r,r∧¯βFlθμ,r,r≥ημ (resp., ℓ1αFu∂μ,r,r∧¯βFuθμ,r,r≥ημ). It is clear that(1)
Fuzzy lower (resp., upper)idX,Iτ,IηCl∗,idY,ℓ01,ℓ2-continuous is fuzzy lower (resp., upper) almost ℓ2-continuous multifunction(2)
Fuzzy lower (resp., upper)idX,Iτ,Cl∗,idY,ℓ01,ℓ2-continuous is fuzzy lower (resp., upper) weakly ℓ2-continuous multifunction(3)
Fuzzy lower (resp., upper)idX,IτCτ,Cl∗,idY,ℓ01,ℓ2-continuous is fuzzy lower (resp., upper) almost weakly ℓ2-continuous multifunctionIn the upcoming work, we will define some new separation axioms on fuzzy ideal topological spaces. Also, we shall discuss the concepts given here in the frames of fuzzy softr-minimal spaces [3]. We hope that this work will contribute fuzzy ideal topological structure studies.



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