# Fuzzy Ideals and Fuzzy Filters of Pseudocomplemented Semilattices

Authors: Berhanu Assaye Alaba; Wondwosen Zemene Norahun
Journal: Advances in Fuzzy Systems
 (2019)

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## Abstract

In this paper, we introduce the concept of kernel fuzzy ideals and ⁎-fuzzy filters of a pseudocomplemented semilattice and investigate some of their properties. We observe that every fuzzy ideal cannot be a kernel of a ⁎-fuzzy congruence and we give necessary and sufficient conditions for a fuzzy ideal to be a kernel of a ⁎-fuzzy congruence. On the other hand, we show that every fuzzy filter is the cokernel of a ⁎-fuzzy congruence. Finally, we prove that the class of ⁎-fuzzy filters forms a complete lattice that is isomorphic to the lattice of kernel fuzzy ideals.


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## Body

## 1. Introduction

The theory of pseudocomplementation was introduced and extensively studied in semilattices and particularly in distributive lattices by O. Frink [1] and G. Birkhoff [2]. Later, pseudocomplement in Stone algebra has been studied by several authors like R. Balbes [3], G. Gra¨tzer [4], etc. In 1973, W. H. Cornish [5] studied congruence on pseudocomplemented distributive lattices and identified those ideals and filters that are congruence kernels and cokernels, respectively. Later, T. S. Blyth [6] studied ideals and filters of pseudocomplemented semilattices.On the other hand, the concept of fuzzy sets was firstly introduced by Zadeh [7]. Rosenfeld has developed the concept of fuzzy subgroups [8]. Since then, several authors have developed interesting results on fuzzy theory; see [8–19].In this paper, we introduce the concept of kernel fuzzy ideals and∗-fuzzy filters of a pseudocomplemented semilattice. We studied a ∗-fuzzy congruence on a pseudocomplemented semilattice. We observe that every fuzzy ideal cannot be a kernel of a ∗-fuzzy congruence. We give necessary and sufficient conditions for a fuzzy ideal to be a kernel of a ∗-fuzzy congruence. We also show that the class of kernel fuzzy ideals can be made a complete distributive lattice. Moreover, we study the image and preimage of kernel fuzzy ideals under a ∗-epimorphism mapping. In Section 4 we turn our attention to fuzzy filters. Here we show that every fuzzy filter is the cokernel of a ∗-fuzzy congruence and investigate a certain type of fuzzy filter called a ∗-fuzzy filter. We prove that these filters form a complete lattice that is isomorphic to the lattice of kernel fuzzy ideals.


## 2. Preliminaries

In this section, we recall some definitions and basic results on lattices, meet semilattices, and fuzzy theory.Definition 1 (see [2]).
An algebra(L,∧,∨) is said to be a lattice if it satisfies the following conditions: (1)
x∧x=x and x∨x=x,(2)
x∧y=y∧x and x∨y=y∨x,(3)
x∧(y∧z)=(x∧y)∧z and x∨(y∨z)=(x∨y)∨z,(4)
x∧(x∨y)=x and x∨(x∧y)=y.Definition 2 (see [2]).
A lattice(L,∧,∨) is a distributive lattice if and only if it satisfies the following identity: (1)x∧y∨z=x∧y∨x∧z.A latticeL is said to be bounded if there exist 0 and 1 in L such that 0∧x=0 and 1∨x=1 for all x∈L.IfL and M are lattices, then f:L→M is said to be a lattice morphism if for all x,y∈L.(1)
f(x∧y)=f(x)∧f(y)(2)
f(x∨y)=f(x)∨f(y).Iff is onto, then f is an epimorphism.Definition 3 (see [20]).
An algebra(L,∧,∨,∗,0,1) is a pseudocomplemented lattice if the following conditions hold: (1)
(L,∧,∨,0,1) is a bounded lattice,(2)
for allx,y∈L,x∧y=0⇔x∧y∗=x.
If(L,∧,∨,0,1) is a bounded distributive lattice, then (L,∧,∨,∗,0,1) is a pseudocomplemented distributive lattice.The lattice(L,∧,∨,0,1) is said to be a complemented lattice if satisfies the following conditions for all x∈L there exists x′∈L such that x∧x′=0 and x∨x′=1.Definition 4 (see [3]).
A pseudocomplemented distributive latticeL is called a Stone algebra if, for all x∈L, it satisfies the property: (2)x∗∨x∗∗=1.Definition 5 (see [1]).
An algebra(S,∧,∗,0) is a pseudocomplemented semilattice if for all x,y,z∈S the following conditions are satisfied: (1)
x∧x=x,(2)
x∧y=y∧x,(3)
x∧(y∧z)=(x∧y)∧z,(4)
x∧y=0⇔x∧y∗=x.It is well known thatS is a partially ordered set relative to the order relation defined by x≤y⇔x∧y=x and that relative to this order relation x∧y is the greatest lower bound of x and y. Hence the condition of Definition 5(4) is equivalent to(3)x∧y=0⇔x≤y∗.A nonempty subsetI of a semilattice S is called an ideal of S if, y∈I,x∈S⇒y∧x∈I.A nonempty subsetF of a semilattice S is called a filter of S if it satisfies the following:(1)
x,y∈F⇒x∧y∈F,(2)
y∈F,x∈S,y≤x⇒x∈F.Theorem 6 (see [1]).
For any two elementsx,y of a pseudocomplemented semilattice S, we have the following: (1)
0∗∗=0,(2)
x∧(x∧y)∗=x∧y∗,(3)
x∧x∗=0,(4)
x≤y⇒y∗≤x∗,(5)
x≤x∗∗,(6)
x∗∗∗=x∗,(7)
x∗∗∗∗=x∗∗,(8)
(x∧y)∗∗=x∗∗∧y∗∗,(9)
(x∧y)∗=(x∗∗∧y∗∗)∗.An elementx of a pseudocomplemented semilattice is called closed if x=x∗∗.Definition 7 (see [7]).
LetX be any nonempty set. A mapping μ:X→[0,1] is called a fuzzy subset of X.The unit interval[0,1] together the operations min and max form a complete distributive lattice. We often write ∧ for minimum or infimum and ∨ for maximum or supremum. That is, for all α,β∈[0,1] we have α∧β=min{α,β} and α∨β=max{α,β}.Definition 8 (see [8]).
Letμ and θ be fuzzy subsets of a set A. Define the fuzzy subsets μ∪θ and μ∩θ of A as follows: for each x∈A, (4)μ∪θx=μx∨θx,μ∩θx=μx∧θx.Then μ∪θ and μ∩θ are called the union and intersection of μ and θ, respectively.For any collection,{μj:j∈J} of fuzzy subsets of X, where J is a nonempty index set, the least upper bound ⋃j∈J‍μj, and the greatest lower bound ⋂j∈J‍μj of the μj’s are given by for each x∈X, (5)⋃j∈Jμjx=⋁j∈Jμjx,⋂j∈Jμjx=⋀j∈Jμjx, respectively.For eacht∈0,1 the set(6)μt=x∈A:μx≥t is called the level subset of μ at t [7].The characteristics function of any setA is defined as (7)χAx=1,ifx∈A0,ifx∉ADefinition 9 (see [8]).
Letf be a function from X into Y; μ be a fuzzy subset of X; and θ be a fuzzy subset of Y. The image of μ under f, denoted by f(μ), is a fuzzy subset of Y such that (8)fμy=supμx:x∈f-1y,iff-1y≠ϕ0,otherwise,wherey∈Y.The preimage of θ under f, symbolized by f-1(θ), is a fuzzy subset of X and (9)f-1θx=θfxforallx∈X.Definition 10 (see [18]).
A fuzzy subsetμ of a bounded lattice L is called a fuzzy ideal of L if for all x,y∈L the following conditions are satisfied: (1)
μ(0)=1,(2)
μ(x∨y)≥μ(x)∧μ(y),(3)
μ(x∧y)≥μ(x)∨μ(y).Definition 11 (see [18]).
A fuzzy subsetμ of a bounded lattice L is called a fuzzy filter of L if for all x,y∈L the following conditions are satisfied: (1)
μ(1)=1,(2)
μ(x∨y)≥μ(x)∨μ(y),(3)
μ(x∧y)≥μ(x)∧μ(y).Theorem 12 (see [21]).
LetL be a lattice, x∈L, and α∈[0,1]. Define a fuzzy subset αx of L as (10)αxy=1,ify≤xα,ify≰x is a fuzzy ideal of L.Remark 13 (see [21]).
α x is called the α-level principal fuzzy ideal corresponding to x.
Similarly, a fuzzy subsetαx of L defined as (11)αxy=1,ifx≤yα,ifx≰yis the α-level principal fuzzy filter corresponding to x.Definition 14 (see [18]).
A fuzzy subsetθ of L×L is said to be a fuzzy congruence on L if and only if, for any x,y,z∈L, the following hold: (1)
θ(x,x)=1,(2)
θ(x,y)=θ(y,x),(3)
θ(x,y)∧θ(y,z)≤θ(x,z),(4)
θ(x,y)≤θ(x∨z,y∨z)∧θ(x∧z,y∧z).


## 3. Kernel Fuzzy Ideals and∗-Fuzzy Ideals

In this section, we introduce the concept of kernel fuzzy ideals and∗-fuzzy ideals of a pseudocomplemented semilattice. We give necessary and sufficient conditions for fuzzy ideals to be a kernel of a ∗-fuzzy congruence.Throughout the rest of this paper,S stands for a pseudocomplemented semilattice (S,∧,∗,0,1) and a partially ordered set unless it is specified.Now we define a fuzzy ideal of a semilattice.Definition 15.
A fuzzy subsetμ of S is called a fuzzy ideal of S if for all x,y∈S the following conditions are satisfied: (1)
μ(0)=1,(2)
μ(x∧y)≥μ(x)∨μ(y).Theorem 16.
A fuzzy subsetμ of S is a fuzzy ideal if and only if each level subset of μ is an ideal of S. (In particular, a nonempty subset I of S is an ideal of S if and only if χI is a fuzzy ideal.)Proof.
Supposeμ is a fuzzy ideal of S. Let x∈μt and y∈S,t∈[0,1]. Then 0∈μt and μ(x)∨μ(y)≥t. Since μ is a fuzzy ideal, we get that μ(x∧y)≥t. Thus x∧y∈μt. So μt is an ideal of S.
Conversely, suppose that every level subset ofμ is a fuzzy ideal of S. Then μ1 is a fuzzy ideal of S. Thus μ(0)=1. Now we proceed to show that μ(x∧y)≥μ(x)∨μ(y); let x,y∈S such that μ(x)=α and μ(y)=β. Then either α≤β or β≤α. Without loss of generality, α≤β. Since μβ is an ideal of S and y∈μβ, then x∧y∈μβ. Thus μ(x∧y)≥μ(x)∨μ(y). So μ is a fuzzy ideal of S.Definition 17.
A fuzzy subsetθ of S×S is said to be a fuzzy congruence on S if and only if, for any x,y,z,w∈S, the following hold: (1)
θ(x,x)=1,(2)
θ(x,y)=θ(y,x),(3)
θ(x,y)∧θ(y,z)≤θ(x,z),(4)
θ(x,y)∧θ(z,w)≤θ(x∧z,y∧w).Definition 18.
A fuzzy congruence relationθ on S is called a ∗-fuzzy congruence if θ(x,y)≤θ(x∗,y∗) for all x,y∈S.It is clear that a fuzzy congruenceθ is a ∗-fuzzy congruence on S if and only if each level subset θt={(x,y)∈S×S:θ(x,y)≥t,t∈[0,1]} of θ is a ∗-congruence on S. Also, let λ be a congruence on S. Then λ is a ∗-congruence on S if and only if its characteristic function χλ is a ∗-fuzzy congruence on S.Theorem 19.
A fuzzy congruenceθ on S is a ∗-fuzzy congruence if and only if (12)θx∗,1≥θx,0.Proof.
Ifθ is a ∗-fuzzy congruence on S, then by Definition 18θ(x∗,1)≥θ(x,0). Suppose conversely that the condition holds. Let x,y∈S. Since θ is a fuzzy congruence on S, θ(x,y)≤θ(x∗∧x,x∗∧y)=θ(0,x∗∧y). By the assumption, we have θ(0,x∗∧y)≤θ(1,(x∗∧y)∗). Thus θ(x,y)≤θ(1,(x∗∧y)∗)≤θ(x∗,x∗∧(x∗∧y)∗)=θ(x∗,x∗∧y∗). Similarly, θ(x,y)≤θ(x∗∧y∗,y∗). Since θ is a fuzzy congruence, then θ(x∗,y∗)≥θ(x∗,x∗∧y∗)∧θ(x∗∧y∗,y∗). Thus θ(x∗,y∗)≥θ(x,y). So θ is a ∗-fuzzy congruence on S.Ifθ is a fuzzy congruence on S and x∈S, then the fuzzy subset θx of S defined by (13)θxy=θx,yforally∈S is called a fuzzy congruence class of S determined by θ and x. We thus have the following theorem.Theorem 20.
Ifθ is a fuzzy congruence on S, then the fuzzy congruence class θ0 of S determined by θ and 0 is a fuzzy ideal of S.Proof.
Supposeθ is a fuzzy congruence on S. Then θ(0,0)=1. Thus θ0(0)=1. Again for any x,y∈S, θ0(x)=θ(x,0)∧θ(y,y). Since θ is a fuzzy congruence on S, we have θ0(x)≤θ(x∧y,0)=θ0(x∧y). Similarly, θ0(y)≤θ0(x∧y). Thus θ0(x∧y)≥θ0(x)∨θ0(y). So θ0 is a fuzzy ideal of S.Now we define a kernel fuzzy ideal of a semilattice.Definition 21.
A fuzzy idealμ of S is called a kernel fuzzy ideal if μ=θ0, where θ is a ∗-fuzzy congruence on S.Example 22.
Consider the semilatticeS whose Hasse diagram is given in Figure 1.
ThenS is pseudocomplemented; we have 0∗=1,a∗=b,p∗=q∗=b∗=a,di∗=0 for i≥0. The fuzzy ideal μ of S defined as μ(0)=1,μ(a)=μ(p)=μ(q)=μ(b)=0.5, and μ(x)=0 for x≠0,a,b,p,q is not the kernel fuzzy ideal of a ∗-fuzzy congruence. For, suppose that θ were a ∗-fuzzy congruence with kernel μ; then θ0(1)≥θ(1,a)∧θ(a,0). Since θ(0,b)≤θ(0∗,b∗)=θ(1,a), we have θ0(1)≥θ(0,b)∧θ(a,0)=μ(b)∧μ(a)=0.5. This is a contradiction.Figure 1In Example22 we observe that every fuzzy ideal of a pseudocomplemented semilattice S is not a kernel fuzzy ideal. In the following theorem we give the necessary and sufficient conditions for a fuzzy ideal of a pseudocomplemented semilattice to be a kernel of a ∗-fuzzy congruence.Theorem 23.
A fuzzy idealμ of S is a kernel fuzzy ideal if and only if μ((x∗∧y∗)∗)≥μ(x)∧μ(y) for all x,y∈S.Proof.
Letμ be a kernel fuzzy ideal of a ∗-fuzzy congruence θ on S. Then μ=θ0. Suppose x,y∈S. Then by the assumption θ(x∗,1)≥θ(x,0)=μ(x) and θ(y∗,1)≥μ(y). Thus μ(x)∧μ(y)≤θ(x∗,1)∧θ(y∗,1). Since θ is a fuzzy congruence on S, we have (14)θx∗,1∧θy∗,1≤θx∗∧y∗,1≤θx∗∧y∗∗,0=μx∗∧y∗∗ Thus μ((x∗∧y∗)∗)≥μ(x)∧μ(y).
Conversely, suppose that the condition holds. Consider the fuzzy relationθμ defined on S by (15)θμx,y=supμz:x∧z∗=y∧z∗,z∈S. Clearly, θμ is both reflexive and symmetric. It is also transitive; if x,y∈S, then (16)θμx,y∧θμy,z=supμw:x∧w∗=y∧w∗∧supμu:y∧u∗=z∧u∗=supμw∧μu:x∧w∗=y∧w∗,y∧u∗=z∧u∗ If x∧w∗=y∧w∗,y∧u∗=z∧u∗, then x∧(w∗∧u∗)∗∗=z∧(w∗∧u∗)∗∗. By the assumption, we have (17)θμx,y∧θμy,z≤supμw∗∧u∗∗:x∧w∗∧u∗∗∗=z∧w∗∧u∗∗∗≤supμk:x∧k∗=z∧k∗,k∈L=θμx,z
So thatθμ is a fuzzy equivalence relation on S. By the similar procedure it can be easily verified that θμ is a fuzzy congruence on S.
To showθμ is a ∗-fuzzy congruence on S, let x∈S. Then θμ(x∗,1)=sup{μ(z):x∗∧z∗=z∗}. Since S is a pseudocomplemented semilattice, x∗∧z∗=z∗⇔x∧z∗=0⇔x∧z∗∗=x. Thus θμ(x∗,1)=sup{μ(z):x≤z∗∗}. So that(18)θμx,0=supμz:x∧z∗=0=supμz:x≤z∗∗≤θμx∗,1.It follows by Theorem 19 that θμ is a ∗-fuzzy congruence on S.
Now takingx=y in the condition we obtain μ(x)=μ(x∗∗). Now we proceed to show that the kernel of θμ is μ. For any x∈S,(19)θμ0x=supμz:x∧z∗=0=supμz:x≤z∗∗≥μxLet y∈S satisfying x≤y∗∗. Then μ(x)≥μ(y). This implies that μ(x) is an upper bound of {μ(z):x≤z∗∗}. This shows that μ(x)≥(θμ)0(x). Thus μ=(θμ)0. So μ is a kernel of θμ.Corollary 24.
μ is a kernel fuzzy ideal if and only if (1)
μ(x)=μ(x∗∗),(2)
μ(k)=sup{μ(x)∧μ(y):x∗∧y∗=k∗}.Proof.
Letμ be a kernel fuzzy ideal of S. Then μ((x∗∧y∗)∗)≥μ(x)∧μ(y) for all x,y∈S.
(1) Sincex≤x∗∗ and μ is a fuzzy ideal, we get μ(x)=μ(x∧x∗∗)≥μ(x∗∗). Again if x=y, then μ(x)≤μ(x∗∗). Thus μ(x∗∗)=μ(x) for all x∈S.
(2) Letx,y,k∈S such that x∗∧y∗=k∗. Then (x∗∧y∗)∗=k∗∗. By (1) and by the assumption, we have μ(k)≥μ(x)∧μ(y). This implies μ(k)≥sup{μ(x)∧μ(y):x∗∧y∗=k∗,x,y,k∈S}. On the other hand, since k∗=k∗∧k∗ and μ(k)≤μ(k), we have μ(k)≤sup{μ(x)∧μ(y):x∗∧y∗=k∗,x,y,k∈S}. Thus μ(k)=sup{μ(x)∧μ(y):x∗∧y∗=k∗,x,y,k∈S}.
Conversely, if (1) and (2) hold, for anyx,y∈S there exists k∈S such that x∗∧y∗=k∗, then μ(k)≥μ(x)∧μ(y). Thus by (1), we get that μ((x∗∧y∗)∗)=μ(k∗∗)≥μ(x)∧μ(y). So μ is a kernel fuzzy ideal of S.The setS(S)={x∗∗:x∈S} is called the skeleton of S. The elements of S(S) are called skeletal.Corollary 25.
Letα∈[0,1]. αx is a kernel fuzzy ideal if and only if x is a skeletal element of S.Proof.
Letαx be a kernel fuzzy ideal. Then by Corollary 24(1) αx(x∗∗)=αx(x), which implies x∗∗≤x. Since x≤x∗∗ for all x∈S, we get that x=x∗∗. Thus x is a skeletal element.
Conversely, Suppose thatx is a skeletal element of S. To show αx is a kernel fuzzy ideal, let y,z∈S such that y≤x and z≤x. Then x∗≤y∗∧z∗ and αx(y)∧αx(z)=1. Thus (y∗∧z∗)∗≤x∗∗=x. So αx((y∗∧z∗)∗)=1. This shows that αx((y∗∧z∗)∗)≥αx(y)∧αx(z). If y≰x and z≰x, then trivially holds. Hence αx is a kernel fuzzy ideal.Corollary 26.
The following conditions onS are equivalent: (1)
Every fuzzy ideal ofS is a kernel fuzzy ideal.(2)
Every level principal fuzzy ideal is a kernel fuzzy ideal.(3)
S is a Boolean algebra.Theorem 27.
A fuzzy subsetμ of S is a kernel fuzzy ideal if and only if each level subset of μ is kernel ideal of S.Proof.
Letμ be a kernel fuzzy ideal of S. Then by Theorem 16μt is an ideal of S,∀t∈[0,1]. To show μt is a kernel ideal, let x,y∈μt. Then by the assumption, μ((x∗∧y∗)∗)≥μ(x)∧μ(y)≥t. Thus (x∗∧y∗)∗∈μt.
Conversely, suppose that every level subset ofμ is a kernel ideal of S. Then by Theorem 16μ is a fuzzy ideal of S. To show μ is a kernel fuzzy ideal, let x,y∈S such that μ(x)=α and μ(y)=β. Then either α≤β or β≤α. Without loss of generality, α≤β. Then μβ⊆μα and by the assumption, (x∗∧y∗)∗∈μα. This shows that μ((x∗∧y∗)∗)≥α=α∧β. Thus μ((x∗∧y∗)∗)≥μ(x)∧μ(y). So μ is a kernel fuzzy ideal of S.Corollary 28.
A nonempty subsetI of S is a kernel ideal of S if and only if χI is a kernel fuzzy ideal.Definition 29.
A fuzzy idealμ of S is called a ∗-fuzzy ideal if μ(x)=μ(x∗∗) for all x∈S.Corollary 30.
Every kernel fuzzy ideal is a∗-fuzzy ideal.Corollary 31.
IfL is a pseudocomplemented distributive lattice, then a fuzzy ideal μ of L is a kernel fuzzy ideal if and only if it is a ∗-fuzzy ideal.Proof.
Supposeμ is a kernel fuzzy ideal of L. Then by Corollary 24(1) μ is a ∗-fuzzy ideal of L.
Conversely, supposeμ is a ∗-fuzzy ideal. Since L is a pseudocomplemented distributive lattice, for any x,y∈L we have (x∨y)∗=x∗∧y∗. Now μ((x∗∧y∗)∗)=μ((x∨y)∗∗). Since μ is a ∗-fuzzy ideal, we get μ((x∗∧y∗)∗)=μ(x∨y)≥μ(x)∧μ(y). Thus μ is a kernel fuzzy ideal of L.Theorem 32.
A∗-fuzzy ideal μ is a kernel fuzzy ideal if and only if μ(supS(S){x∗∗,y∗∗})≥μ(x)∧μ(y) for all x,y∈S.Proof.
Letx,y∈S. Define supS(S){x∗∗,y∗∗}=(x∗∧y∗)∗. Since z∗∗∗=z∗ for all z∈S, we have (x∗∧y∗)∗=(x∗∧y∗)∗∗∗. Thus (x∗∧y∗)∗∈S(S). Since μ is a kernel fuzzy ideal, μ(supS(S){x∗∗,y∗∗})=μ((x∗∧y∗)∗)≥μ(x)∧μ(y).Theorem 33.
A fuzzy subsetμ of S is a ∗-fuzzy ideal if and only if each level subset of μ is a ∗-ideal of S.Proof.
Letμ be a ∗-fuzzy ideal of S. Then μ(x)=μ(x∗∗) for each x∈S and by Theorem 16μt is an ideal of S,t∈[0,1]. To show μt is a ∗-ideal, let x∈μt. Then μ(x∗∗)=μ(x)≥t and x∗∗∈μt. Thus each level subset of μ is a ∗-fuzzy ideal.
Conversely, suppose every level fuzzy subset ofμ is a ∗-ideal of S. Then by Theorem 16μ is a fuzzy ideal of S. Since x≤x∗∗ and μ is a fuzzy ideal, we have μ(x∗∗)≤μ(x). Let μ(x)=t. Then x∗∗∈μt. Thus μ(x∗∗)≥μ(x). So μ is a ∗-fuzzy ideal of S.Corollary 34.
A nonempty subsetI of S is a ∗-ideal of S if and only if χI is a ∗-ideal.Theorem 35.
Letμ be a kernel fuzzy ideal of S. Then the smallest ∗-fuzzy congruence on S with kernel μ is given by(20)θμx,y=supμz:x∧z∗=y∧z∗,z∈S.Proof.
It is shown in the proof of Theorem23 that θμ is a ∗-fuzzy congruence with kernel μ. Now we proceed to show that θμ is the smallest ∗-fuzzy congruence with kernel μ.
Letη be a ∗-fuzzy congruence with kernel μ. For x,y∈S,(21)θμx,y=supμz:x∧z∗=y∧z∗,z∈S.Since η is a ∗-fuzzy congruence, we have η(x∧z∗,x)≥η(x,x)∧η(z∗,1)=η(z∗,1)≥η(z,0)=μ(z). Similarly, η(y∧z∗,y)≥μ(z). If x∧z∗=y∧z∗, then (22)ηx,y≥ηx,x∧z∗∧ηx∧z∗,y=ηx,x∧z∗∧ηy∧z∗,y≥μz This shows that η(x,y) is an upper bound of {μ(z):x∧z∗=y∧z∗,z∈S}. Thus θμ⊆η. So θμ is the smallest ∗-fuzzy congruence with kernel μ.Lemma 36.
Ifμ and θ are ∗-fuzzy ideals of S, then so is μ∪θ.Proof.
Supposeμ and θ are ∗-fuzzy ideals of S. Clearly (μ∪θ)(0)=1. Let x,y∈S. Then (μ∪θ)(x∧y)=μ(x∧y)∨θ(x∧y). Since μ and θ are fuzzy ideals, we have (μ∪θ)(x∧y)≥(μ(x)∨θ(x))∨(μ(y)∨θ(y))=(μ∪θ)(x)∨(μ∪θ)(y). Thus μ∪θ is a fuzzy ideal of S. Now (μ∪θ)(x)=μ(x)∨θ(x). Since μ and θ are ∗-fuzzy ideals, we get that (μ∪θ)(x)=μ(x∗∗)∨θ(x∗∗)=(μ∪θ)(x∗∗) for all x∈S. Thus μ∪θ is a ∗-fuzzy ideal of S.The class of all∗-fuzzy ideals of S is denoted by FI∗(S). It is clear that the set FI∗(S) of ∗-fuzzy ideals of S, ordered by fuzzy set inclusion, is a complete distributive lattice in which the lattice operations are fuzzy set-theoretic.The class of all kernel fuzzy ideals ofS is denoted by FKI(S). We now prove that FKI(S) is a complete distributive lattice.Theorem 37.
Ifμ,θ∈FKI(S), the supremum of μ and θ is given by (23)μ∨_θx=supμz∧θw:x≤z∗∧w∗∗,z,w∈S.Proof.
Letη=μ∨_θ. First, we need to show that η is a kernel fuzzy ideal of S. Since 0≤(0∗∧0∗)∗, we have η(0)=1. For any x,y∈S, (24)ηx=supμz∧θw:x≤z∗∧w∗∗,z,w∈S≤supμz∧θw:x∧y≤z∗∧w∗∗,z,w∈S=ηx∧y This shows that η(x∧y)≥η(x)∨η(y). Thus η is a fuzzy ideal of S.
We now show thatη is a kernel fuzzy ideal. For any x,y∈S, (25)ηx∧ηy=supμz1∧θw1:x≤z1∗∧w1∗∗,z1,w1∈S∧supμz2∧θw2:y≤z2∗∧w2∗∗,z2,w2∈S=supμz1∧μz2∧θw1∧θw2:x≤z1∗∧w1∗∗,y≤z2∗∧w2∗∗ If x≤(z1∗∧w1∗)∗ and y≤(z2∗∧w2∗)∗, then z1∗∧w1∗≤x∗ and z2∗∧w2∗≤y∗. Thus (x∗∧y∗)∗≤((z1∗∧z2∗)∗∗∧(w1∗∧w2∗)∗∗)∗. Since μ and θ are kernel fuzzy ideals, we get that μ(z1)∧μ(z2)≤μ((z1∗∧z2∗)∗) and θ(w1)∧θ(w2)≤θ((w1∗∧w2∗)∗). Based on this we have (26)ηx∧ηy≤supμz1∗∧z2∗∗∧θw1∗∧w2∗∗:x∗∧y∗∗≤z1∗∧z2∗∗∗∧w1∗∧w2∗∗∗∗≤supμu∧θv:x∗∧y∗∗≤u∗∧v∗∗,u,v∈S=ηx∗∧y∗∗ Thus η is a kernel fuzzy ideal of S.
To showη is the smallest kernel fuzzy ideal of S, let λ be a kernel fuzzy ideal of S containing μ and θ. Then for any x∈S, (27)ηx=supμz∧θw:x≤z∗∧w∗∗≤supλz∧λw:x≤z∗∧w∗∗ If x≤(z∗∧w∗)∗, then λ(x)≥λ(z)∧λ(w). This implies λ(x) is an upper bound of {λ(z)∧λ(w):x≤(z∗∧w∗)∗}. Thus η⊆λ. Hence η is the smallest kernel fuzzy ideal containing μ and θ.Theorem 38.
The setFKI(S) forms a complete distributive lattice with respect to inclusion ordering of fuzzy sets.Proof.
Clearly(FKI(S),⊆) is a partially ordered set. For μ,θ∈FKI(S), clearly μ∧θ,μ∨_θ∈(FKI(S)). So (FKI(S),∧,∨_) is a lattice.
Forμ,θ,η∈FKI(S), clearly (μ∩θ)∨_(μ∩η)⊆μ∩(θ∨_η). Now for any x∈L, (28)μ∩θ∨_ηx=supμx∧θy∧ηz:x≤y∗∧z∗∗,y,z∈S=supμx∧θy∧μx∧ηz:x≤y∗∧z∗∗=supμx∧θy∗∗∧μx∧ηz∗∗:x≤y∗∧z∗∗≤supμx∧y∗∗∧θx∧y∗∗∧μx∧z∗∗∧ηx∧z∗∗:x≤y∗∧z∗∗=supμ∩θx∧y∗∗∧μ∩ηx∧z∗∗:x≤y∗∧z∗∗ If x≤(y∗∧z∗)∗, then x∧y∗∧z∗=0. Since S is a pseudocomplemented semilattice, we have y∧z∗=y∧(y∧z)∗ for all y,z∈S. Now we proceed to show that x≤((x∧y∗∗)∗∧(x∧z∗∗)∗)∗. (29)x∧x∧y∗∗∗∧x∧z∗∗∗∗=x∧x∧x∧y∗∗∗∧x∧z∗∗∗∗=x∧x∧x∧y∗∗∗∧x∧x∧z∗∗∗∗=x∧x∧y∗∧x∧z∗∗=x∧x∧y∗∧z∗∗=x∧0∗=x Thus x≤((x∧y∗∗)∗∧(x∧z∗∗)∗)∗. Based on this fact we have (30)μ∩θ∨_ηx≤supμ∩θx∧y∗∗∧μ∩ηx∧z∗∗:x≤x∧y∗∗∗∧x∧z∗∗∗∗≤supμ∩θw∧μ∩ηu:x≤w∗∧u∗∗,w,u∈S≤μ∩θ∨_μ∩ηx Thus μ∩(θ∨_η)=(μ∩θ)∨_(μ∩η). So FKI(S) is distributive.
Now we show the completeness. Since(FKI(S),⊆) is a poset and χS is greatest element of FKI(S), it is enough to show that every subfamily of FKI(S) has infimum. Let {μα:α∈J} be a subfamily of FKI(S). Then ⋂α∈J‍μα is a fuzzy ideal of S. For any x,y∈S, (31)⋂α∈Jμαx∧⋂α∈Jμαy=inf⁡μαx:α∈J∧inf⁡μβy:β∈J≤inf⁡μαx∧μαy:α∈J≤inf⁡μαx∗∧y∗∗:α∈J=⋂α∈Jμαx∗∧y∗∗ Thus ⋂α∈I‍μα is a kernel fuzzy ideal of L. So FKI(S) is a complete distributive lattice.Corollary 39.
IfL is a pseudocomplemented lattice, then FI∗(L) forms a distributive lattice with respect to inclusion ordering of fuzzy sets.Proof.
Clearly(FI∗(L),⊆) is a partially ordered set. For μ,θ∈FI∗(L), define (32)μ∧θ=μ∩θ,μ∨_θx=supμz∧θw:x≤z∗∧w∗∗,z,w∈L. Obviously μ∧θ,μ∨_θ∈(FI∗(L)). Let η=μ∨_θ and x,y∈L. Then (33)ηx∧ηy=supμz1∧θw1:x≤z1∗∧w1∗∗,z1,w1∈L∧supμz2∧θw2:y≤z2∗∧w2∗∗,z2,w2∈L=supμz1∧μz2∧θw1∧θw2:x≤z1∗∧w1∗∗,y≤z2∗∧w2∗∗=supμz1∨z2∧θw1∨w2:x≤z1∗∧w1∗∗,y≤z2∗∧w2∗∗ If x≤(z1∗∧w1∗)∗ and y≤(z2∗∧w2∗)∗, then x∨y≤(z1∗∧w1∗)∗∨(z2∗∧w2∗)∗, which implies x∨y≤((z1∗∧w1∗)∗∨(z2∗∧w2∗)∗)∗∗=((z1∗∧w1∗)∧(z2∗∧w2∗))∗=((z1∨z2)∗∧(w1∨w2)∗))∗. Using this fact we have (34)ηx∧ηy≤supμz1∨z2∧θw1∨w2:x∨y≤z1∨z2∗∧w1∨w2∗∗≤supμu1∧θu2:x∨y≤u1∗∧u2∗∗=ηx∨y Thus η(x∨y)≥η(x)∧η(y). So FI∗(L) is a distributive lattice.Theorem 40.
IfL is a pseudocomplemented distributive lattice, then the following statements are equivalent: (1)
ifμ,θ are kernel fuzzy ideals of L, then so is μ∨θ,(2)
L is a Stone lattice.Proof.
(1)⇒(2): Let x∈L. Then x∗ and x∗∗ are skeletal elements of L. Thus by Corollary 25, αx∗ and αx∗∗ are kernel fuzzy ideals. By the assumption, αx∗∨αx∗∗=αx∗∨x∗∗ is a kernel fuzzy ideal. To prove our claim, it suffices to show that x∗∨x∗∗=1 for all x∈L. Since αx∗∨x∗∗ is a kernel fuzzy ideal, then by Corollary 25, x∗∨x∗∗∈S(L). This implies x∗∨x∗∗=(x∗∨x∗∗)∗∗=(x∗∗∧x∗)∗=1 for all x∈L. Thus L is a Stone lattice.
(2)⇒(1): If L is a Stone lattice, then (x∧y)∗=x∗∨y∗ for all x,y∈L. Let μ and θ be kernel fuzzy ideals. Then μ∨θ is a fuzzy ideal of L. To show μ∨θ is a kernel fuzzy ideal, it suffices to show that (μ∨θ)(x)=(μ∨θ)(x∗∗). Since x≤x∗∗ and μ∨θ is a fuzzy ideal, we have (μ∨θ)(x∗∗)≤(μ∨θ)(x).
Now,(μ∨θ)(x)=sup{μ(y)∧θ(z):x=y∨z}. Since L is a Stone lattice, x=y∨z⇒x∗∗=y∗∗∨z∗∗. Based on this we have (35)μ∨θx≤supμy∧θz:x∗∗=y∗∗∨z∗∗=supμy∗∗∧θz∗∗:x∗∗=y∗∗∨z∗∗=supμw∧θu:x∗∗=w∨u=μ∨θx∗∗ Thus μ∨θ is a ∗-fuzzy ideal of L. So by Corollary 31, μ∨θ is a kernel fuzzy ideal of LIfS and M are pseudocomplemented semilattices, then a semilattice morphism f:S→M will be called a ∗-morphism if f(x∗)=(f(x))∗ for all x∈S.Lemma 41.
Letf:S→M be an epimorphism. Then (1)
Ifμ is a fuzzy ideal of S, then f(μ) is a fuzzy ideal of M.(2)
Ifθ is a fuzzy ideal of M, then f-1(θ) is a fuzzy ideal of S.Proof.
(1) Ifμ is a fuzzy ideal of S, then f(μ)(0M)=sup{μ(a):a∈f-1(0M)}≥μ(0S)=1. Thus f(μ)(0M)=1. For any x,y∈M, (36)fμx=supμz:z∈f-1x≤supμz∨μw:z∈f-1x,w∈f-1y Since z∈f-1(x) and w∈f-1(y), we get that z∧w∈f-1(x∧y). Based on this we have (37)fμx≤supμz∧w:z∧w∈f-1x∧y≤supμu:u∈f-1x∧y=fμx∧y Similarly, f(μ)(y)≤f(μ)(x∧y). Thus f(μ)(x∧y)≥f(μ)(x)∨f(μ)(y). So f(μ) is a fuzzy ideal of M.
(2) Ifθ is a fuzzy ideal of M, then f-1(θ)(0S)=θ(f(0S))=θ(0M)=1. For any x,y∈S, we have f-1(θ)(x∧y)=(θ)(f(x)∧f(y))≥(θ)(f(x))∨(θ)(f(y))=f-1(θ)(x)∨f-1(θ)(y). Thus f-1(θ) is a fuzzy ideal of S.Theorem 42.
Letf:S→M be a ∗-epimorphism. Then (1)
Ifμ is a kernel fuzzy ideal of S, then f(μ) is a kernel fuzzy ideal of M.(2)
Ifθ is a kernel fuzzy ideal of M, then f-1(θ) is a kernel fuzzy ideal of S.Proof.
(1) Letμ be a kernel fuzzy ideal of S. Then by Lemma 41(1) f(μ) is a fuzzy ideal of M. To show f(μ) is a kernel fuzzy ideal, let x,y∈M. Then (38)fμx∧fμy=supμz:z∈f-1x,z∈S∧supμw:w∈f-1y,w∈S=supμz∧μw:z∈f-1x,w∈f-1y If z∈f-1(x) and w∈f-1(y), then (z∗∧w∗)∗∈f-1((x∗∧y∗)∗). Since μ is a kernel fuzzy ideal of S, we have that μ(z)∧μ(w)≤μ((z∗∧w∗)∗). Based on this fact we have (39)fμx∧fμy≤supμz∗∧w∗∗:z∗∧w∗∗∈f-1x∗∧y∗∗≤supμu:u∈f-1x∗∧y∗∗,u∈S=fμx∗∧y∗∗ This shows that f(μ)((x∗∧y∗)∗)≥f(μ)(x)∧f(μ)(y) for any x,y∈M. Hence f(μ) is a kernel fuzzy ideal of M.
(2) Letθ be a kernel fuzzy ideal of M. Then by Lemma 41(2) f-1(θ) is a fuzzy ideal of S. To show f-1(θ) is a kernel fuzzy ideal, let x,y∈S. Then(40)f-1θx∗∧y∗∗=θfx∗∧y∗∗=θfx∗∧fy∗∗≥f-1θx∧f-1θy. Thus f-1(θ) is a kernel fuzzy ideal of S.Theorem 43.
Letf:S→M be a ∗-epimorphism. Then the map g:FKI(S)→FKI(M) defined by μ↦f(μ) is a lattice epimorphism.Proof.
Clearlyχ{0S},χS∈FKI(S). Since Kerf={0} and f is onto, we get f(χ{0S})=χ{0M} and f(χS)=χM. This implies g(χ{0S})=χ{0M} and g(χS)=χM.
Letμ,θ∈FKI(S). Then μ∩θ and μ∨_θ are kernel fuzzy ideals of S. Thus f(μ∩θ) and f(μ∨_θ) are kernel fuzzy ideals of M. Since μ∩θ⊆μ and μ∩θ⊆θ, we have f(μ∩θ)⊆f(θ)∩f(μ). For any y∈M,(41)fμ∩fθy=supμz:z∈f-1y,z∈S∧supθw:w∈f-1y,w∈S. Since f is a homomorphism and f(z)=y,f(w)=y, we get f(z∧w)=y. Using this fact, we have (42)fμ∩fθy≤supμz∧w:z∧w∈f-1y∧supθz∧w:z∧w∈f-1y=supμz∧w∧θz∧w:z∧w∈f-1y=supμ∩θz∧w:z∧w∈f-1y≤supμ∩θu:u∈f-1y=fμ∩θy. So f(μ)∩f(θ)=f(μ∩θ). Again clearly, f(μ)∨_f(θ)⊆f(μ∨_θ). For any x∈M, (43)fμ∨_fθx=supfμx1∧fθx2:x≤x1∗∧x2∗∗=supsupμw1:w1∈f-1x1∧supμw2:w2∈f-1x2:x≤x1∗∧x2∗∗ and (44)fμ∨_θx=supμ∨_θz:z∈f-1x,z∈S=supsupμz1∧θz2:z≤z1∗∧z2∗∗:z∈f-1x. If f(z)=x and z≤(z1∗∧z2∗)∗, then x=f(z)≤(f(z1)∗∧f(z2)∗)∗. Put f(z1)=y1 and f(z2)=y2. Then z1∈f-1(y1),z2∈f-1(y2) and x≤(y1∗∧y2∗)∗. Based on this fact we have (45)fμ∨_θx≤supsupμz1:z1∈f-1y1∧supμz2:z2∈f-1y2:x≤y1∗∧y2∗∗=supfμy1∧fμy2:x≤y1∗∧y2∗∗=fμ∨_fθx Thus f(μ∨_θ)=f(μ)∨_f(θ). So g is a homomorphism. To show g is an epimorphism, let μ∈FKI(M). Then f-1(μ)∈FKI(M). Since f is onto, we have f(f-1(μ))=μ. Thus g is onto. So g is a lattice epimorphism.


## 4.∗-Fuzzy Filters

Turning our attention to fuzzy filters, we first introduce the concept of∗-fuzzy filter of a pseudocomplemented semilattice. We prove that, in contrast to the situation concerning fuzzy ideals, every fuzzy filter of a pseudocomplemented semilattice is the cokernel of a ∗-fuzzy congruence. Moreover, we have shown that there is an isomorphism between the class of kernel fuzzy ideals and the class of ∗-fuzzy filters of a pseudocomplemented semilattice.Now we define a fuzzy filter of a semilattice.Definition 44.
A fuzzy subsetμ of S is called a fuzzy filter of S if, for all x,y∈S, the following conditions are satisfied: (1)
μ(1)=1,(2)
μ(x∧y)≥μ(x)∧μ(y),(3)
μ(x)≤μ(y) whenever x≤y.Theorem 45.
A fuzzy subsetμ of S is a fuzzy filter of S if and only if (46)μ1=1,μx∧y=μx∧μyforallx,y∈S.Proof.
Letμ be a fuzzy filter of S and x,y∈S. Since x∧y≤x and x∧y≤y, by the assumption we get that μ(x∧y)≤μ(x)∧μ(y). Thus μ(1)=1 and μ(x∧y)=μ(x)∧μ(y) for all x,y∈S.
Conversely, suppose the condition holds. Then conditions (1) and (2) of Definition44 are satisfied. Let x,y∈S such that x≤y. Then μ(x)=μ(x∧y)=μ(x)∧μ(y). Thus μ(x)≤μ(y). So μ is a fuzzy filter of S.Theorem 46.
A fuzzy subsetμ of S is a fuzzy filter of S if and only if each level subset of μ is a filter of S. (In particular, a nonempty subset F of S is a filter of S if and only if χF is a fuzzy filter of S.)Proof.
Supposeμ is a fuzzy filter of S. Clearly 1∈μt for all t∈[0,1]. To show μt is a filter of S, let x,y∈μt. Then μ(x)∧μ(y)≥t. Since μ is a fuzzy filter of S, we get μ(x∧y)≥μ(x)∧μ(y)≥t. Thus x∧y∈μt. Again, if x∈μt and z∈S such that x≤z, then by Theorem 45 we have μ(x)=μ(x∧z)=μ(x)∧μ(z). Thus μ(z)≥μ(x)≥t. This implies z∈μt. Hence μt is a filter of S for all t∈[0,1].
Suppose conversely that the condition holds. Then clearlyμ(1)=1. Let x,y∈S such that μ(x)=α and μ(y)=β. Then x∈μα and y∈μβ. Since α,β∈[0,1], either α≤β or β≤α. Without loss of generality α≤β, then μβ⊆μα. Which implies x,y∈μα. Since μα is a filter, we get that x∧y∈μα. Thus μ(x∧y)≥μ(x)∧μ(y). Finally, let x,y∈S such that x≤y. If μ(x)=α, then x∈μα and x≤y. By the assumption, we have y∈μα. Thus μ(x)≤μ(y). So μ is a fuzzy filter of S.Theorem 47.
Ifθ is a fuzzy congruence on S, then the fuzzy congruence class θ1 of S determined by θ and 1 is a fuzzy filter of S.Proof.
Supposeθ be a fuzzy congruence on S. Then θ(1,1)=1. Thus θ1(1)=1. Let x,y∈S. Then θ1(x)∧θ1(y)=θ(x,1)∧(y,1). Since θ is a fuzzy congruence, we have θ1(x)∧θ1(y)≤θ(x∧y,1)=θ1(x∧y). Thus θ1(x∧y)≥θ1(x)∧θ1(y). Let x≤y. Then θ1(x)=θ(x,1)∧θ(y,y)≤θ(x∧y,y)=θ(x,y). Since θ is a fuzzy congruence, we have θ(y,1)≥θ(y,x)∧θ(x,1)=θ(x,1). Thus θ1(x)≤θ1(y). So θ1 is a fuzzy filter of S.Definition 48.
A fuzzy filterμ of S is called a cokernel fuzzy filter if μ=θ1, where θ is a ∗-fuzzy congruence on S.Theorem 49.
Ifη is a fuzzy filter of S, then the fuzzy relation θη defined by(47)θηx,y=supηz:x∧z=y∧z,z∈Sis a ∗-fuzzy congruence with cokernel fuzzy filter η; moreover, θη is the smallest ∗-fuzzy congruence.Proof.
It is clear thatθη is a fuzzy congruence on S. Now we proceed to show that the fuzzy cokernel of θη is η. For any x∈S,(48)θμx,1=supηz:x∧z=z=supηz:z≤x≥ηx Let w∈S satisfying w≤x. Then η(x)≥η(w). This implies that η(x) is an upper bound of {η(z):w≤x}. Thus η=(θμ)1. So η is a fuzzy cokernel of θη.
To showθη is a ∗-fuzzy congruence on S, let x∈S. Then(49)θηx∗,1=supηz:x∗∧z=z=supηz:z≤x∗,θηx,0=supηw:x∧w=w=supηw:w≤x∗≤θηx∗,1.Thus θη is a ∗-fuzzy congruence on S. Finally, let λ be a ∗-fuzzy congruence with cokernel η and x,y,z∈S. Then η(z)=λ(1,z)=λ(1,z)∧λ(x,x)≤λ(x,x∧z). Similarly, λ(y∧z,y)≥η(z). Let x∧z=y∧z. Since λ is a fuzzy congruence, we have that (50)λx,y≥λx,x∧z∧λx∧z,y=λx,x∧z∧λy∧z,y≥ηz. This implies that λ(x,y) is an upper bound of {η(z):x∧z=y∧z}. Thus λ(x,y)≥θη(x,y). So θη is the smallest ∗-fuzzy congruence with cokernel η.We now observe that the condition for fuzzy filters that is dual to that given in Theorem23, namely, η((x∗∧y∗)∗)≤η(x)∧η(y), is of no interest; for if this held, then we would have η(1)=η((x∗∧x∗∗)∗)≤η(x)∧η(x∗)=η(0). Thus η=χS. However, the condition that is dual to condition (1) of Corollary 24 is of considerable interest.Definition 50.
A fuzzy filterμ of S is called a ∗-fuzzy filter if μ(x)=μ(x∗∗) for all x∈L.Theorem 51.
A fuzzy subsetμ of S is a ∗-fuzzy filter if and only if each level subset of μ is a ∗-filter of S.Proof.
Suppose a fuzzy subsetμ of S is a ∗-fuzzy filter. Then μ(x)=μ(x∗∗) for all x∈S and by Theorem 46 every level subset of μ is a filter of S. To show μα is a ∗-filter, let x∗∗∈μα,α∈[0,1]. Then by the assumption x∈μα. Thus μα is a ∗-filter of S.
Conversely, suppose thatμα is a ∗-filter for all α∈[0,1]. Then x∗∗∈μα⇒x∈μα and by Theorem 46, μ is a fuzzy filter of S. Since x≤x∗∗ and μ is a fuzzy filter, then μ(x)≤μ(x∗∗) for each x∈S. If μ(x∗∗)=α, then by the assumption x∈μα. This shows that μ(x)≥μ(x∗∗). Thus μ(x)=μ(x∗∗) for all x∈S. So μ is a ∗-fuzzy filter of S.Corollary 52.
A nonempty subsetF of S is a ∗-filter of S if and only if χF is a ∗-fuzzy filter.The class of fuzzy filters and∗-fuzzy filters of S are denoted by FF(S) and FF∗(S) respectively.Lemma 53.
Letμ be a fuzzy filter of S. Define α(μ)(x)=μ(x∗). Then α(μ) is a kernel fuzzy ideal of S.Proof.
To showα(μ) is a fuzzy ideal of S, it is enough to show that α(μ)(0)=1 and α(μ)(x)≥α(μ)(y) whenever x≤y. Clearly α(μ)(0)=1. If x,y∈S such that x≤y, then y∗≤x∗. Since μ is a fuzzy filter, we get that μ(y∗)≤μ(x∗). This implies α(μ)(x)≥α(μ)(y). Thus α(μ) is a fuzzy ideal of S. To show α(μ) is a kernel fuzzy ideal, let x,y∈S. Then α(μ)((x∗∧y∗)∗)=μ(x∗∧y∗)≥μ(x∗)∧μ(y∗)=α(μ)(x)∧α(μ)(y). Thus α(μ) is a kernel fuzzy ideal of S. Now we can define a mapping α:FF(S)→FKI(S) by μ↦α(μ).Lemma 54.
Letθ be a kernel fuzzy ideal of S. Define β(θ)(x)=θ(x∗). Then β(θ) is a ∗-fuzzy filter of S.Proof.
Clearlyβ(θ)(1)=1. If x,y∈S and x≤y, then y∗≤x∗ and β(θ)(x)≤β(θ)(y). We now show that β(θ)(x∧y)≥β(θ)(x)∧β(θ)(y). Let x,y∈S. Then (x∧y)∗=(x∗∗∧y∗∗)∗. Now β(θ)(x∧y)=μ((x∗∗∧y∗∗)∗)≥μ(x∗)∧μ(y∗)=β(θ)(x)∧β(θ)(y). Thus β(θ) is a fuzzy ideal of S. Finally, for each x∈S, β(θ)(x)=θ(x∗)=β(θ)(x∗∗). Hence β(θ) is a ∗-fuzzy ideal of S. Now we can define a mapping β:FKI(S)→FF(S) by θ↦β(θ).Theorem 55.
(1) For any fuzzy filterμ of S, we have μ⊆β(α(μ)).
(2) For everyμ∈FKI(S), we have μ=α(β(μ)).Proof.
(1) For anyx∈S, we have β(α(μ))(x)=μ(x∗∗). Since x≤x∗∗ for each x∈S and μ is a fuzzy filter, we have μ(x)≤μ(x∗∗). Thus μ(x)≤β(α(μ))(x) for all x∈S. So μ⊆β(α(μ)).
(2) Letμ be a kernel fuzzy ideal of S. Then by Corollary 24(1) we have μ(x)=μ(x∗∗) for all x∈S. Thus μ=α(β(μ)).Corollary 56.
β ( α ( μ ) ) = μ if and only if μ is a ∗-fuzzy filter.Lemma 57.
For any fuzzy filterμ of S, the map μ→β(α(μ)) is a closure operator on FF(S). That is, (1)
μ⊆β(α(μ)),(2)
βα(β(α(μ)))=β(α(μ)),(3)
μ⊆θ⇒β(α(μ))⊆β(α(θ)).Proof.
(1) Sinceμ is a fuzzy filter of S, by Theorem 55(1) we have μ⊆β(α(μ)).
(2) Sinceμ is a fuzzy filter, by Lemma 53α(μ) is a kernel fuzzy ideal of S. Again by Lemma 54, β(α(μ)) is a ∗-fuzzy filter. Thus by Corollary 56, βα(β(α(μ)))=β(α(μ)).
(3) For anyx∈S, we have β(α(μ))(x)=μ(x∗∗) and β(α(θ))(x)=θ(x∗∗). Since μ⊆θ, we get that μ(x∗∗)≤θ(x∗∗). This shows that β(α(μ))(x)≤β(α(θ))(x) for all x∈S. Thus β(α(μ))⊆β(α(θ)).The class of all∗-fuzzy filters of S is denoted by FF∗(S). We now prove that FF∗(S) is a complete distributive lattice.Theorem 58.
Ifμ,θ∈FF∗(S), the supremum of μ and θ is given by (51)μ∨_θx=supμz∧θw:x∗≤z∧w∗,z,w∈S.Proof.
Letη=μ∨_θ. We need to show that η is a ∗-fuzzy filter of S. Since 1∗≤(1∧1)∗, we have η(1)=1. For any x,y∈S, (52)ηx∧ηy=supμz1∧θw1:x∗≤z1∧w1∗,z1,w1∈S∧supμz2∧θw2:y∗≤z2∧w2∗,z2,w2∈S=supμz1∧μz2∧θw1∧θw2:x∗≤z1∧w1∗,y∗≤z2∧w2∗ If x∗≤(z1∧w1)∗ and y∗≤(z2∧w2)∗, then (x∧y)∗∗=x∗∗∧y∗∗≥(z1∧w1)∗∗∧(z2∧w2)∗∗=((z1∧z2)∧(w1∧w2))∗∗. Thus (x∧y)∗≤((z1∧z2)∧(w1∧w2))∗. Using this fact we have (53)ηx∧ηy≤supμz1∧z2∧θw1∧w2:x∧y∗≤z1∧z2∧w1∧w2∗≤supμu1∧θu2:x∧y∗≤u1∧u2∗=ηx∧y Thus η(x∧y)≥η(x)∧η(y).
On the other hand, ifx≤y, then y∗≤x∗ and η(x)≤η(y). Now we proceed to show that η is a ∗-fuzzy filter of S. For any x∈S, η(x∗∗)=sup{μ(z)∧θ(w):x∗≤(z∧w)∗}=η(x). Thus η is a ∗-fuzzy filter of S.
To showη is the smallest ∗-fuzzy filter containing μ and θ, let λ be a ∗-fuzzy filter containing μ and θ. Then for any x∈S, (54)ηx=supμz∧θw:x∗≤z∧w∗≤supλz∧λw:x∗≤z∧w∗ If x∗≤(z∧w)∗, then (z∧w)∗∗≤x∗∗. Since λ is a ∗-fuzzy filter, λ(z∧w)≤λ(x). Thus λ(z)∧λ(w)≤λ(x). So λ(x) is an upper bound of {λ(z)∧λ(w):x∗≤(z∧w)∗}. Thus η(x)≤λ(x). So η is the smallest ∗-fuzzy filter containing μ and θ.Theorem 59.
The setFF∗(S) forms a complete distributive lattice with respect to inclusion ordering of fuzzy sets.Proof.
Clearly(FF∗(S),⊆) is a partially ordered set. For μ,θ∈FF∗(S), clearly μ∧θ,μ∨_θ∈FF∗(S). Thus (FF∗(S),∧,∨_) is a lattice.
Forμ,θ,η∈FF∗(S), clearly (μ∩θ)∨_(μ∩η)⊆μ∩(θ∨_η). For any x∈S, (55)μ∩θ∨_ηx=supμx∧θy∧ηz:x∗≤y∧z∗,y,z∈S=supμx∧θy∧μx∧ηz:x∗≤y∧z∗ If x∗≤(y∧z)∗, then x∗∧(y∧z)∗∗=0. Now we proceed to show that x∗≤((x∗∧y∗)∗∧(x∗∧z∗)∗)∗.(56)x∗∗∧x∗∧y∗∗∧x∗∧z∗∗=x∗∧y∗∗∧x∗∧z∗∗∧x∗∧y∗∗∧x∗∧z∗∗∧x∗∗asy∧z∗=y∧y∧z∗=x∗∧y∗∗∧x∗∧z∗∗∧x∗∧x∗∧y∗∗∧x∗∧x∗∧z∗∗∗=x∗∧y∗∗∧x∗∧z∗∗∧x∗∧y∗∗∧x∗∧z∗∗∗=x∗∧y∗∗∧x∗∧z∗∗∧x∗∧y∧z∗∗∗=x∗∧y∗∗∧x∗∧z∗∗∧0∗=x∗∧y∗∗∧x∗∧z∗∗ Thus x∗≤((x∗∧y∗)∗∧(x∗∧z∗)∗)∗. Since x,y≤(x∗∧y∗)∗ and x,z≤(x∗∧z∗)∗, we have that, (57)μ∩θ∨_ηx≤supμ∩θx∗∧y∗∗∧μ∩ηx∗∧z∗∗:x∗≤x∗∧y∗∗∧x∗∧z∗∗∗≤supμ∩θw∧μ∩ηu:x∗≤w∧u∗,u,w∈S≤μ∩θ∨_μ∩ηx Thus μ∩(θ∨_η)=(μ∩θ)∨_(μ∩η). So FF∗(S) is distributive.
Let{μα:α∈J} be a subfamily of FF∗(S). Then ⋂α∈J‍μα is a ∗-fuzzy filter of S. Thus FF∗(S) is a complete distributive lattice.Corollary 60.
IfL is a pseudocomplemented lattice, then FF∗(L) forms a distributive lattice with respect to inclusion ordering of fuzzy sets.Theorem 61.
F F ∗ ( S ) ≅ F K I ( S ).Proof.
Defineβ^:FKI(S)→FF∗(S) by β^(μ)=β(μ)∀μ∈FKI(S). Then by Lemma 54, β(μ)∈FF∗(S). If we denote by α^:FF∗(S)→FKI(S) the restriction of α to FF∗(S), by Theorem 55, the compositions β^∘α^ and α^∘β^ are identity mappings. Thus β^ and α^ are mutually inverse isomorphisms.


## 5. Conclusion

In this work, we introduce the concept of kernel fuzzy ideals and∗-fuzzy filters of a pseudocomplemented semilattice and investigate some of their properties. We give a necessary and sufficient condition for a fuzzy congruence to be a ∗-fuzzy congruence. Furthermore, we identified those fuzzy ideals which can be kernel fuzzy ideals. We also proved that the class of kernel fuzzy ideals form a complete distributive lattice. Moreover, we prove that every fuzzy filter of a pseudocomplemented semilattice is the cokernel of a ∗-fuzzy congruence. Finally, we have shown that there is an isomorphism between the class of ∗-fuzzy filters and the class of kernel fuzzy ideals. Our future work will focus on fuzzy semiprime ideals in general lattices.



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*Source: 4263923-2019-07-08.xml*