# μ-Fuzzy Filters in Distributive Lattices

Authors: Wondwosen Zemene Norahun
Journal: Advances in Fuzzy Systems
 (2020)

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## Abstract

In this paper, we introduce the concept ofμ-fuzzy filters in distributive lattices. We study the special class of fuzzy filters called μ-fuzzy filters, which is isomorphic to the set of all fuzzy ideals of the lattice of coannihilators. We observe that every μ-fuzzy filter is the intersection of all prime μ-fuzzy filters containing it. We also topologize the set of all prime μ-fuzzy filters of a distributive lattice. Properties of the space are also studied. We show that there is a one-to-one correspondence between the class of μ-fuzzy filters and the lattice of all open sets in Xμ. It is proved that the space Xμ is a T0 space.


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## Body

## 1. Introduction

In 1970, Mandelker [1] introduced the concept of relative annihilators as a natural generalization of relative pseudo-complement and he characterized distributive lattices with the help of these annihilators. The concept of coannihilators and μ-filters in a distributive lattice with greatest element “1” was introduced by Rao and Badawy [2] and they characterized μ-filters in terms of coannihilators. For a filter F in L, μF=x++:x∈F is an ideal in the set A+L of all coannihilators, and conversely μ←I=x∈L:x++∈I is a filter in L when I is any ideal in A+L. A filter F of L is called a μ-filter if μ←μF=F.In 1965, Zadeh [3] mathematically formulated the fuzzy subset concept. He defined fuzzy subset of a nonempty set as a collection of objects with grade of membership in a continuum, with each object being assigned a value between 0 and 1 by a membership function. Fuzzy set theory was guided by the assumption that classical sets were not natural, appropriate, or useful notions in describing the real-life problems, because every object encountered in this real physical world carries some degree of fuzziness. A lot of work on fuzzy sets has come into being with many applications to various fields such as computer science, artificial intelligence, expert systems, control systems, decision-making, medical diagnosis, management science, operations research, pattern recognition, neural network, and others (see [4–7]).In 1971, Rosenfeld used the notion of a fuzzy subset of a set to introduce the concept of a fuzzy subgroup of a group [8]. Rosenfeld’s paper inspired the development of fuzzy abstract algebra. Since then, several authors have developed interesting results on fuzzy subgroups (see [9–17]), fuzzy ideals of rings (see [16, 18–21]), and fuzzy ideals of lattices (see [22–28]).Alaba and Norahun [29] studied the concept of α-fuzzy ideals of a distributive lattice in terms of annulates. They also studied the space of prime α-fuzzy ideals of a distributive lattice. In this paper, we introduce the dual of the concept of α-fuzzy ideals which is called μ-fuzzy filters in a distributive lattice with greatest element “1.” We study the special class of fuzzy filters called μ-fuzzy filters. We prove that the set of all μ-fuzzy filters of a distributive lattice forms a complete distributive lattice isomorphic to the set of all fuzzy ideals of A+L. We also show that there is a one-to-one correspondence between the class of prime μ-fuzzy filters of L and the set of all prime ideals of A+L. We prove that every μ-fuzzy filter is the intersection of all prime μ-fuzzy filters containing it. Moreover, we study the space of all prime μ-fuzzy filters in a distributive lattice. The set of prime μ-fuzzy filters of L is denoted by Xμ. For a μ-fuzzy filter θ of L, open subset of Xμ is of the form Xθ=η∈Xμ:θ⊈η and Vθ=η∈Xμ:θ⊆η is a closed set. We also show that the set of all open sets of the form Xxβ=η∈Xμ:xβ⊈η,x∈L,β∈0,1 forms a basis for the open sets of Xμ. The set of all μ-fuzzy filters of L is isomorphic with the set of all open sets in Xμ.


## 2. Preliminaries

We refer to Birkhoff [30] for the elementary properties of lattices.Definition 1 (see [2]).
For any setS of a lattice L, define S+ as follows:(1)S+=x∈L:s∨x=1,for alls∈S.
Here,S+ is called the coannihilator of S. If S=x, we write x+ instead of x+. Then, clearly L+=1 and 1+=L. For any subset S of a lattice L, it is clear that S+ is a filter in L.Lemma 1.
(see [2]). For any x,y∈L, the following conditions hold.(1)
x≤y⟹x+⊆y+(2)
x∧y+=x+∩y+(3)
x∨y++=x++∩y++(4)
x+=L if and only if x=1
The set of all coannihilator denotesA+L. Each coannihilator is a coannihilator filter, and hence, for two coannihilators x+ and y+, their supremum and infimum in A+L are(2)a∨b++=a++∩b++,a++∨¯b++=a∧b++a,respectively.In a distributive latticeL with 1, the set of all coannihilators A+L of L is a lattice A+L,∩,∨¯ and a sublattice of the Boolean algebra of coannihilator filters of L.For a filterF in L,(3)μF=x++:x∈Fis an ideal in A+L and the set(4)μ←I=x∈L:x++∈Iis a filter of L when I is any ideal in A+L. A filter F of L is called a μ-filter if μ←μF=F.Definition 2 (see [3]).
LetX be any nonempty set. A mapping μ:X⟶0,1 is called a fuzzy subset of X.
The unit interval0,1 together with the operations min and max form a complete lattice satisfying the infinite meet distributive law. We often write ∧ for minimum or infimum and ∨ for maximum or supremum. That is, for all α,β∈0,1, we have, α∧β=minα,β and α∨β=maxα,β.
The characteristic function of any setA is defined as(5)χAx=1,ifx∈A,0,ifx∉A.Definition 3 (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,(6)μ∪θx=μx∨θxandμ∩θx=μx∧θx.
Then,μ∪θ and μ∩θ are called the union and intersection of μ and θ, respectively.
For any collection,μi:i∈I of fuzzy subsets of X, where I is a nonempty index set, and the least upper bound ∪i∈Iμi and the greatest lower bound ∩i∈Iμi of the μi’s are given for each x∈X,(7)∪i∈Iμix=∨i∈Iμixand∩i∈Iμix=∧i∈Iμix,respectively.
For eacht∈0,1, the set(8)μt=x∈A:μx≥tis called the level subset of μ at t [3].Definition 4 (see [27]).
A fuzzy subsetμ of a lattice L is called a fuzzy ideal of L if; for all x,y∈L, the following conditions are satisfied:(1)
μ0=1(2)
μx∨y≥μx∧μy(3)
μx∧y≥μx∨μyDefinition 5 (see [27]).
A fuzzy subsetμ of a lattice L is called a fuzzy filter of L if, for all x,y∈L, the following condition is satisfied:(1)
μ1=1(2)
μx∨y≥μx∨μy(3)
μx∧y≥μx∧μy
In [27], Swamy and Raju observed the following:(1)
A fuzzy subsetμ of a lattice L is a fuzzy ideal of L if and only if(9)μ0=1andμx∨y=μx∧μy,for allx,y∈L.(2)
A fuzzy subsetμ of a lattice L is a fuzzy filter of L if and only if(10)μ1=1andμx∧y=μx∧μy,for allx,y∈L.
Letμ be a fuzzy subset of a lattice L. The smallest fuzzy filter of L containing μ is called a fuzzy filter of L induced by μ and denoted by μ and(11)μ=∩θ∈FFL:μ⊆θ.Lemma 2 (see [23]).
For any two fuzzy subsetsμ and θ of a distributive lattice L, we have(12)μ·θ=μ∧θ.The above result works dually.For any two fuzzy subsetsμ and θ of a distributive lattice L, we have(13)μ+θ=μ∧θ.The binary operations “+” and “·” on the set of all fuzzy subsets of a distributive lattice L are as follows:(14)μ+θx=Supμy∧θz:y,z∈L,y∨z=xandμ·θx=Supμy∧θz:y,z∈L,y∧z=x.Ifμ and θ are fuzzy ideals of L, then μ·θ=μ∧θ=μ∩θ and μ+θ=μ∨θIfμ and θ are fuzzy filters of L, then μ+θ=μ∧θ and μ·θ=μ∨θThe set of all fuzzy filters ofL is denoted by FFL.


## 3.μ-Fuzzy Filters

In this section, we introduce the concept ofμ-fuzzy filters in a distributive lattice with greatest element “1.” We study some basic properties of the class of μ-fuzzy filters. We prove that the class of μ-fuzzy filters forms a complete distributive lattice isomorphic to the class of fuzzy filters of A+L. We also show that there is a one-to-one correspondence between the set of all prime μ-fuzzy filters of L and prime fuzzy ideals of A+L. Finally, we observe that every μ-fuzzy filter is the intersection of all prime μ-fuzzy filters containing it.Throughout the rest of this paper,L stands for the distributive lattice with greatest element “1” unless otherwise mentioned.Theorem 1.
Letθ be a fuzzy filter of L. Then, the fuzzy subset μθ of A+L defined by(15)μθx++=Supθy:y++=x++,y∈Lis a fuzzy ideal of A+L.Proof.
Letθ be a fuzzy filter of L. Clearly, μθ1++=1. For any x++,y++∈A+L,(16)μθx++∧μθy++=Supθa:a++=x++,a∈L∧Supθb:b++=y++,b∈L=Supθa∧θb:a++=x++,b++=y++≤Supθa∧θb:a++∨¯b++=x++∨¯y++=Supθa∧b:a∧b++=x∧y++≤Supθc:c++=x∧y++=μθx++∨¯y++.
Thus,μθx++∨¯y++≥μθx++∧μθy++.
On the other hand,(17)μθx++=Supθa:a++=x++≤Supθa∨y:a++∧y++=x++∧y++≤Supθc:c++=x∨y++=μθx++∧y++.
Similarly,μθy++≤μθx++∧y++. So,(18)μθx++∧y++≥μθx++∨μθy++.
Hence,μθ is a fuzzy ideal of A+L.Lemma 3.
Letη be a fuzzy ideal of A+L. Then, the fuzzy subset μ←η of L defined as μ←ηx=ηx++ is a fuzzy filter of L.Proof.
Letη be a fuzzy ideal of A+L. Since 1++ is the smallest element of A+L, we get μ←η1=1. For any x,y∈L,(19)μ←ηx∨y=ηx++∧ηy++=μ←ηx∧μ←ηy.
Thus,μ←η is a fuzzy filter of L.Lemma 4.
Ifθ and η are fuzzy filters of L, then θ⊆η implies μθ⊆μη.Lemma 5.
Ifθ,η are fuzzy ideals of A+L, then θ⊆η implies μ←θ⊆μ←η.Theorem 2.
The setFIA+L of all fuzzy ideals of A+L forms a complete distributive lattice, where the infimum and supremum of any family θj:j∈J of fuzzy ideals are given by(20)∧ηj=∩ηjand∨ηj=∪ηj.Theorem 3.
The mappingμ is a homomorphism of FFL into FIA+L.Proof.
Letηandθ be two fuzzy filters of L. Then, by Lemma 4, we have μη∩θ⊆μη∩μθ and μη∨¯μθ⊆μη∨θ. For any x++∈A+L,(21)μηx++∧μθy++=Supηa:a++=x++∧Supθb:b++=y++.
Sincea++=x++ and b++=y++, we get a∧b++=x++. Based on this, we have(22)μηx++∧μθx++≤Supηa∧b:a∧b++=x++∧Supθa∧b:a∧b++=x++=Supηa∧b∧θa∧b:a∧b++=x++=Supη∩θa∧b:a∧b++=x++≤Supη∩θc:c++=x++=μη∩θx++.
Thus,μη∩μθ⊆μη∩θ. So, μη∩μθ=μη∩θ.
On the other hand,(23)μη∨θx++=Supη∨θa:a++=x++=SupSupηy∧θz:a=y∧z,y∧z++=x++≤SupSupηb1∧θb2:b1++=y++,b2++=z++,y∧z++=x++=SupSupηb1:b1++=y++∧Supθb2:b2++=z++,y∧z++=x++=Supμηy++∧μθz++:y∧z++=x++=Supμηy++∧μθz++:y++∨¯z++=x++=μη∨¯μθx++.
Then,μη∨θ⊆μη∨¯μθ. Thus, μη∨θ=μη∨¯μθ. So, μ is a homomorphism.Corollary 1.
For any two fuzzy filtersθ and η of L, we have(24)μ←μθ∩η=μ←μθ∩μ←μη.Proof.
For anyx∈L, μ←μη∩θx=μη∩θx++. Since μ is a homomorphism, we have μ←μη∩θ=μ←μη∩μ←μθ.Lemma 6.
For any fuzzy idealθ of A+L,μμ←θ=θ.Proof.
Sinceθ is a fuzzy ideal of A+L, by Lemma 3, μ←θ is a fuzzy filter of L and μμ←θ is a fuzzy ideal of A+L. Now, we proceed to show μμ←θ=θ.(25)μμ←θx++=Supμ←θa:a++=x++=Supθa++:a++=x++=θx++.
Thus,μμ←θ=θ.Lemma 7.
For any fuzzy filterθ of L, the map θ⟶μ←μθ is a closure operator on FFL. That is,(1)
θ⊆μ←μθ(2)
μ←μμ←μθ=μ←μθ(3)
θ⊆η⟹μ←μθ⊆μ←μη, for any two fuzzy filters θ,η of L
Now, we defineμ-fuzzy filter.Definition 6.
A fuzzy filterθ of L is called a μ-fuzzy filter of L if θ=μ←μθ.
Thus,μ-fuzzy filters are simply the closed elements with respect to the closure operator of Lemma 7, and μ←μθ is the smallest μ-fuzzy filter containing θ, for any fuzzy filter θ of L.Theorem 4.
For a nonempty fuzzy subsetθ of L, θ is a μ-fuzzy filter if and only if each level subset of θ is a μ-filter of L.Proof.
Letθ be a μ-fuzzy filter of L. Then, θt=μ←μθt. Now, we proceed to show μ←μθt=μ←μθt for all t∈0,1. We know that μ←μθt⊆μ←μθt. To show the other inclusion, let x∈μ←μθt. Then, x++∈μθt, and there is y∈θt such that x++=y++. Thus, μ←μθx≥t. So, θt=μ←μθt. Therefore, each level subset of θ is a μ-filter of L.
Conversely, assume that each level subset ofθ is a μ-filter. Then, θ is a fuzzy filter and θ⊆μ←μθ. To prove our claim, let t=μ←μθx. Then, for each ϵ>0, there is a∈L such that a++=x++ and θa>t−ϵ, which implies a∈θt−ϵ,a++=x++ and x∈μ←μθt−ϵ=θt−ϵ. This shows that x∈∩ϵ>0θt−ϵ=θt. Thus, μ⟶μθ⊆θ. So, θ is a μ-fuzzy filter of L.Corollary 2.
For a nonempty subsetF of L, F is a μ-filter if and only if χF is a μ-fuzzy filter of L.Theorem 5.
Letθ be a fuzzy filter of L. Then, θ is a μ-fuzzy filter if and only if, for each x,y∈L,x+=y+ implies θx=θy.Proof.
Letθ be a μ-fuzzy filter of L and x,y∈L such that x++=y++. Then,(26)θx=Supθa:a++=x++,a∈L=Supθa:a++=y++,a∈L=μ←μθy=θy.
Conversely, suppose that for eachx,y∈L,x++=y++ implies θx=θy. For any x∈L,(27)μ←μθx=Supθa:a++=x++,a∈L=θx.
Thus,θ is a μ-fuzzy filter of L.Theorem 6.
A fuzzy filterθ of L is a μ-fuzzy filter if and only if(28)∧a∈x++θa≥θx,for allx∈L.Proof.
Suppose a fuzzy filterθ of L is a μ-fuzzy filter. Then, by Theorem 4, every level subset is a μ-filter of L. Let x∈L such that θx=t. Since θt is a μ-filter of L, then x++⊆θt, which implies a∈θt for all a∈x++. Thus, θa≥θx for each a∈x++. So,(29)∧a∈x++θa≥θx,for allx∈L.
Suppose conversely that the condition holds. To proveθ is a μ-fuzzy filter, it suffices to show that μ←μθ⊆θ. For any x∈L, μ←μθx=Supθb:b++=x++. If b++=x++, then x∈b++. By the assumption, θx≥θb for each b++=x++. This shows that θx is an upper bound of θb:b++=x++. Thus, μ←μθx≤θx for all x∈L. So, μ←μθ⊆θ. Hence, θ is a μ-fuzzy filter of L.
Let us denote the set of allμ-fuzzy filters of L by FFμL.Theorem 7.
The classFFμL,∧,∨¯ of all μ-fuzzy filters of L forms a complete distributive lattice with respect to set inclusion.Proof.
Clearly,FFμL,⊆ is a partially ordered set. For η,θ∈FFμL, define(30)η∧θ=η∩θ,η∨¯θ=μ←μη∨θ.
Then, clearlyη∧θ,η∨¯θ∈FFμL. We need to show η∨¯θ is the least upper bound of η,θ. Since θ,η⊆η∨θ⊆η∨¯θ, η∨¯θ is an upper bound of μ,θ. Let λ be any upper bound for η,θ in FFμL. Then, η∨θ⊆λ, which implies that μ←η∨θ⊆μ←μλ=λ. Therefore, μ←μη∨θ is the supremum of both η,θ in FFμL. Hence, FFμL,∧,∨¯ is a lattice.
We now prove the distributivity. Letη,θ,λ∈FFμL. Then,(31)η∨¯θ∩λ=μ←μη∨θ∩λ=μ←μη∨θ∩η∨λ=μ←μη∨θ∩μ←μη∨λ=η∨¯θ∩η∨¯λ.
Thus,FFμL is a distributive lattice.
Now, we proceed to show the completeness. Since1 and L are μ-filters, χ1 and χL are least and greatest elements of FFμL, respectively. Let θi:i∈I⊆FFμL. Then, ∩i∈Iθi is a fuzzy filter of L and ∩i∈Iθi⊆μ←μ∩i∈Iθi.(32)∩i∈Iθi⊆θi,∀i∈I⟹μ←μ∩i∈Iθi⊆θi,∀i∈I⟹μ←μ∩i∈Iθi⊆∩i∈Iθi.
Thus,μ←μ∩i∈Iθi=∩i∈Iθi. So, FFμL,∧,∨¯ is a complete distributive lattice.Theorem 8.
The setFFμL is isomorphic to the lattice of fuzzy ideals of A+L.Proof.
Define(33)f:FFμL⟶FIA+L,fη=μη,∀θ∈FFμL.
Letη,θ∈FFμL and fη=fθ. Then, μη=μθ. Thus, μ←μη=μ←μθ. So, η=θ. Hence, f is one to one.
Letλ∈FIA+L. Then, by Lemma 3, μ←λ is a fuzzy filter of L. Now, we proceed to show that μ←λ is a μ-fuzzy filter of L. Let x∈L. Then, μ←μμ←λx=μμ←λx++. Thus, by Lemma 6, we get that μμ←λx++=μ←ηx. So, μ←λ=μ←μμ←λ. Thus, for each λ∈FIA+L,fμ←λ=λ. Therefore, f is onto.
Now, for anyη,θ∈FIμL, fη∨¯θ=fμ←μη∨θ=μμ←μη∨θ=μη∨θ=μη∨μθ=fη∨fθ. Similarly, fη∩θ=fη∩fθ. Therefore, f is an isomorphism of FFμL onto the lattice of fuzzy filters of A+L.Theorem 9.
The following are equivalent for each non-constantμ-fuzzy filter λ of L.(1)
For allθ,η∈FFL,(34)θ∩η⊆λ⇒θ⊆λorη⊆λ.(2)
For any fuzzy pointsxγ and yβ of L,(35)xγ+yβ⊆λ⇒xγ⊆λoryβ⊆λ.(3)
For allθ,η∈FFμL,(36)θ∩η⊆λ⇒θ⊆λorη⊆λ.Proof.
1⟹2. Let x,y∈L such that xγ+yβ⊆λ. Then, xγ+yβ⊆λ. Since L is a distributive lattice, by dual of Lemma 2, we have xγ∩yβ⊆λ. Since xγ and yβ are fuzzy filters of L, by the assumption, xγ⊆λ or yβ⊆λ. This shows that xγ⊆λ or yβ⊆λ.2⇒3. Let θ,η∈FFμL such that θ∩η⊆λ. Now, we need to show θ⊆λ or η⊆λ. Suppose not. Then, θ⊈λ and η⊈λ, which implies there exist x,y∈L such that θx>λx and ηy>λy. Put θx=γ and ηy=β. Then, xγ⊈λ and yβ⊈λ. Since xγ+yβ⊆θ∩η, we have xγ+yβ⊆λ. By the assumption, we get that xγ⊆λ or yβ⊆λ, which is a contradiction. Thus, θ⊆λ or η⊆λ.3⇒1. Suppose θ,η∈FFL such that θ∩η⊆λ. Then, by Corollary 1 we have μ←μθ∩μ←μη⊆λ. Since μ←μθ and μ←μη are μ-fuzzy filters, by the assumption, we get that μ←μθ⊆λ or μ←μη⊆λ, which implies θ⊆λ or η⊆λ.Definition 7.
By a primeμ-fuzzy filter, we mean a nonconstant μ-fuzzy filter of L satisfying (1) and hence all of the conditions of Theorem 9.
We have proved in Theorem8 that there is an order isomorphism between the class of μ-fuzzy filters and the set of fuzzy ideals of A+L. Now, we show that there is an isomorphism between the prime μ-fuzzy filters and the prime fuzzy ideals of the lattice of coannihilators.Theorem 10.
There is an isomorphism between the primeμ-fuzzy filters and the prime fuzzy ideals of the lattice of coannihilator.Proof.
By Theorem8, the map f is an isomorphism from FFμL into FIA+L. Let σ be a prime μ-fuzzy filter of L. Then, μσ∈FIA+L. Now, we prove μσ is a prime fuzzy ideal of FIA+L. Let θ,η∈FIA+L such that θ∩η⊆μσ. Since f is onto, there exist λ,γ∈FFμL such that fλ=θ and fγ=η. Thus, μλ∩γ⊆μσ. Since μ⟵ is an isotone, we have μ←μλ∩γ⊆μ←μσ. Thus, λ∩γ⊆σ. Since σ is a prime fuzzy filter, either λ⊆σ or γ⊆σ. This shows that either μλ⊆μσ or μγ⊆μσ. Thus, θ⊆μσ or η⊆μσ. Hence, μσ is a prime fuzzy ideal of A+L.
Conversely, suppose thatθ is a prime fuzzy ideal in A+L. Since f is onto, there exists a μ-fuzzy filter σ in FFμL such that θ=μσ. Let η,λ∈FFL such that η∩λ⊆σ. Since μ is an isotone, we get μη∩λ⊆μσ=θ. Thus, μη∩μλ⊆μσ. Since μσ is a prime fuzzy ideal of A+L, either μη⊆μσ or μλ⊆μσ. This implies η⊆μ⟵μσ or λ⊆μ←μσ. Since σ is a μ-fuzzy filter, we get η⊆σ or λ⊆σ. Thus, σ is prime fuzzy filter in FFμL. So, the prime μ-fuzzy filters correspond to prime fuzzy ideals of A+L.Theorem 11.
Letθ be a μ-fuzzy filter of L and η be a fuzzy ideal of L such that θ∩η≤α, α∈0,1. Then, there exists a prime μ-fuzzy filter λ of L such that θ⊆λ and λ∩η≤α.Proof.
PutP=σ∈FFμL:σ⊆ηandη∩σ≤α. Since θ∈P, P is nonempty, and it forms a poset together with the inclusion ordering of fuzzy sets. Let A=θii∈I be any chain in P. Then, clearly ∪i∈Iθi is a μ-fuzzy filter. Since θi∩η≤α for each i∈I, we get that ∪i∈Iθi∩η≤α. Thus, ∪i∈Iθi∈A. By applying Zorn’s lemma, we get a maximal element, say σ∈P; that is, σ is a μ-fuzzy filter of L such that θ⊆σ and σ∩η≤α.
Now, we proceed to showσ is a prime fuzzy filter. Assume that σ is not prime fuzzy filter. Let γ1∩γ2⊆σ such that γ1⊈σ and γ2⊈σ,γ1,γ2∈FFL. If we put σ1=μ←μγ1∨σ and σ2=μ←μγ2∨σ, then both σ1 and σ2 are μ-fuzzy filters of L properly containing σ. Since σ is maximal in P, we get σ1,σ2∉P. Thus, σ1∩η≰α and σ2∩η≰α. This implies there exist x,y∈L such that σ1∩ηx>α and σ2∩ηy>α, which implies σ1∩σ2∩ηx∧y>α⇒μ←μσ∨γ1∧γ2x∧y∧ηx∧y>α. This shows that θ∩ηx∧y>α. This is a contradiction. Thus, σ is prime μ-fuzzy filter of L.Corollary 3.
Letθ be a μ-fuzzy filter of L, a∈L and α∈0,1. If θa≤α, then there exists a prime μ-fuzzy filter η of L such that θ⊆η and ηa≤α.Proof.
PutP=σ∈FFμL:σ⊆ηandη∩σ≤α. Since θ∈P, P is nonempty, and it forms a poset together with the inclusion ordering of fuzzy sets. Let A=θii∈I be any chain in P. Clearly, ∪i∈Iθi is a μ-fuzzy filter. Since θia≤α for each i∈I, α is an upper bound of θia:i∈I. Thus, ∪i∈Iθia≤α. So, ∪i∈Iθi is a μ-fuzzy filter containing θ and ∪i∈Iθia≤α. Hence, ∪i∈Iθi∈P. By applying Zorn’s lemma, we get a maximal element, say σ∈P; that is, σ is a μ-fuzzy filter of L such that θ⊆σ and σa≤α.
Now, we proceed to showσ is a prime fuzzy filter. Assume that σ is not prime fuzzy filter. Let γ1∩γ2⊆σ and γ1⊈σ and γ2⊈σ,γ1,γ2∈FFL. If we put σ1=μ←μγ1∨σ and σ2=μ←μγ2∨σ, then both σ1 and σ2 are μ-fuzzy filters of L properly containing σ. Since σ is maximal in P, we get σ1,σ2∉P. Thus, σ1a>α and σ2a>α. Now, sigma1∩σ2a=μ←μγ1∨σ∩γ2∨σa=μ←μγ1∩γ2∨σa=σa>α. This is a contradiction. Hence, σ is prime μ-fuzzy filter.Corollary 4.
Anyμ-fuzzy filter of L is the intersection of all prime μ-fuzzy filters containing it.Proof.
Letθ be a proper μ-fuzzy filter of L. Consider the following.(37)λ=∩η:ηis a primeμ−fuzzy filter andθ⊆η.
Clearly,θ⊆λ. Assume that λ⊈θ. Then, there is a∈L such that λa>θa. Let θa=α. Consider the set(38)P=η∈FIμL:θ⊆ηandηa≤α.
By the above corollary, we can find a primeμ-fuzzy filter γ of L such that θ⊆γ and γa≤α. This implies λ⊆γ. This shows that λ≤α, which is a contradiction. Thus, λ⊆θ. So, λ=θ.


## 4. The Space of Primeμ-Fuzzy Filters

In this section, we study the space of primeμ-fuzzy filters of a distributive lattice and some properties of the space also.LetXμ be the set of all prime μ-fuzzy filters of a distributive lattice. Let Vθ=η∈Xμ:θ⊆η, where θ is a fuzzy subset of L and Xθ=η∈Xμ:θ⊈η=Xμ−Vθ. We let μ∗=μ1, i.e., μ∗=x∈L:μx=1.Lemma 8.
For any fuzzy filtersλ and θ of L, we have(1)
λ⊆θ⇒Xλ⊆Xθ(2)
Xλ∨θ=Xλ∪Xθ(3)
Xλ∩θ=Xλ∩XθProof.
(1)
Letλ⊆θ and η∈Xλ. Then, λ⊈η and θ⊈η. Thus, η∈Xθ.(2)
Sinceλ,θ⊆λ∨θ, by (1), we have Xλ∪Xθ⊆Xλ∨θ. Now, we proceed to show the other inclusion; let η∈Xλ∨θ. Then, λ∨θ⊈η. This shows that either λ⊈η or θ⊈η. So, η∈Xλ∪Xθ. Hence, Xλ∨θ=Xλ∪Xθ.(3)
By (1), we have Xλ∩θ⊆Xλ∩Xθ. On the other hand, let η∈Xλ∩Xθ. Then, λ⊈η and θ⊈η. Since η is a prime fuzzy filter, we get that λ∩θ⊈η. This shows that η∈Xλ∩θ. Thus, Xλ∩θ=Xλ∩Xθ.Lemma 9.
Letθ be a fuzzy subset of L. Then, Xθ=Xθ.Proof.
To prove our claim, it suffices to showXθ⊆Xθ. Let η∈Xθ. Then, θ⊈η. We need to show θ⊈η. Suppose not. Then, θ⊆η, which implies that θ⊆η, which is a contradiction. Thus, θ⊈η. So, Xθ=Xθ.Theorem 12.
Letx,y∈L and β∈0,1. Then,(1)
Xx∧yβ=Xxβ∪Xyβ(2)
Xx∨yβ=Xxβ∩Xyβ(3)
∪x∈L,β∈0,1Xxβ=XμProof.
(1)
Ifλ∈Xxβ∪Xyβ, then either xβ⊈λ or yβ⊈λ. This shows that β>λx or β>λy. Thus, β>λx∧λy=λx∧y. So, x∧yβ⊈λ. Hence, λ∈Xx∧yβ.On the other hand, letλ∈Xx∧yβ. Then, β>λx∧y=λx∧λy. This implies either xβ⊈λ or yβ⊈λ. Thus, λ∈Xxβ∪Xyβ.(2)
Ifλ∈Xxβ∩Xyβ, then xβ⊈λ and yβ⊈λ. This implies β>λx and β>λy. This shows that x,y∉λ∗. Since λ is prime fuzzy filter, card Im λ=2 and λ∗ is prime. Thus, x∨y∉λ∗, which implies β>λx∨y. Thus, x∨yβ⊈λ and hence Xxβ∩Xyβ⊆Xx∧yβ.Conversely, letλ∈Xx∨yβ. Then, x∨yβ⊈λ, which implies β>λx∨y≥λx∨λy. Thus, β>λx and β>λy. This shows that xβ⊈λ and yβ⊈λ. Thus, λ∈Xxβ∩Xyβ. So, Xx∨yβ⊆Xxβ∩Xyβ. Therefore, Xx∨yβ=Xxβ∩Xyβ.(3)
Clearly,∪x∈L,β∈0,1Xxβ⊆Xμ. Let λ∈Xμ. Then, Imλ=1,γ,γ∈0,1. This implies there is x∈L such that λx=γ. If we take some β∈0,1 such that β>γ, then xβ⊈λ. Thus, Xμ⊆∪x∈L,β∈0,1Xxβ. So, Xμ=∪x∈L,β∈0,1Xxβ.Lemma 10.
Letβ1,β1∈0,1; β=minβ1,β2; and x,y∈L. Then,(39)Xxβ1∩Xyβ2=Xx∨yβ.Proof.
Letλ∈Xxβ1∩Xyβ2. Then, xβ1⊈λ and xβ2⊈λ. This implies that β1>λx and β2>λy. Since λ∗ is prime filter and x,y∉λ∗, we have x∨y∉λ∗ and λx=λy=λx∨y. This shows that β>λx∨y. Thus, x∨yβ⊈λ. So, λ∈Xx∨yβ. To show the other inclusion, let λ∈Xx∨yβ. Then, β>λx∨y≥λx∨λy. This implies β1>λx and β2>λy. Thus, xβ1⊈λ and yβ2⊈λ. So, λ∈Xxβ1∩Xyβ2. Hence, Xxβ1∩Xyβ2=Xx∨yβ.Lemma 11.
Letθi:i∈I be any family of fuzzy filters of L. Then,(40)∩i∈IVθi=V∪i∈Iθi.Proof.
Sinceθi⊆∪i∈Iθi for each i∈I, we have V∪i∈Iθi⊆Vθi for each i∈I. Thus, V∪i∈Iθi⊆∪i∈IVθi.
Conversely, letλ∈∪i∈IVθi. Then, λ∈Vθi for each i∈I. This implies θi⊆μ. Thus, for any x∈L, μx is an upper bound of θix:i∈I. This implies that Supθix:i∈I≤λx. This shows that ∪i∈Iθi⊆λ and ∪i∈Iθi⊆λ. So, μ∈V∪i∈Iθi. Thus, ∩i∈IVθi⊆V∪i∈Iθi. Hence, ∩i∈IVθi=V∪i∈Iθi.Theorem 13.
The collectionT=Xθ:θis a fuzzy filter of Lis a topology onXμ.Proof.
First, we define two fuzzy subsetsη1,η2 of L as follows: η1x=0 and η2x=1 for all x∈L. Then, η1 and η2 are fuzzy filters of L. Since η1⊆λ, for all λ∈Xμ, we get that Vη1=Xμ. This shows that Xη1=ϕ. Since each λ∈Xμ is nonconstant, η2⊈λ for all λ∈Xμ. So Xη2=Xμ. Hence, ϕ,Xμ∈T.
Next, letXθ1,Xθ2∈T. Then, by Lemma 8, we get that Xθ1∩Xθ2=Xθ1∩θ2. This shows that T is closed under finite intersection.
Now, we proceed to show thatT is closed under arbitrary union. Let θi:i∈I be any family of fuzzy filters of L. Then, by Lemma 11 we have(41)∩i∈IVθi=V∪i∈Iθi,
which implies∪i∈IXθi=X∪i∈Iθi. Thus, by Lemma 9, we get that(42)X∪i∈Iθi=X∪i∈Iθi.
So,T is closed under arbitrary union. Therefore, T is a topology on Xμ. The space Xμ,T will be called the space of prime μ-fuzzy filters in L.
In the above theorem, we proved that the family ofXθ is a topology on Xμ. In the following result, we show that the set of all open sets of the form Xxβ is a basis for the topology on Xμ.Theorem 14.
The collectionℬ=Xxβ:x∈L,β∈0,1 forms base for some topology Xμ.Proof.
LetXθ be any open set in Xμ and λ∈Xθ. Then, θ⊈λ and there is x∈L such that θx>λx. Put θx=β; then xβ⊆θ and λ∈Xxβ. To show Xxβ⊆Xθ, let η∈Xxβ. Then, xβ⊈η and θx>ηx. This shows that η∈Xθ. Thus, λ∈Xxβ⊆Xθ. Hence, for any open set Xθ in Xμ we can find Xxβ in ℬ such that Xxβ⊆Xθ. Therefore, ℬ is a base for T.Theorem 15.
The spaceXμ is a T0-space.Proof.
Take any two different elementsη and θ in Xμ. Then, either η⊈θ or θ⊈η. Without loss of generality, we can assume that η⊈θ. Then, θ∈Xη and η∉Xη. Thus, Xμ is a T0-space.Theorem 16.
For any fuzzy filterη of L, Xη=Xμ←μη.Proof.
For any fuzzy filterη of L, we have η⊆μ←μη and Xη⊆Xμ←μη. Now we proceed to show the other inclusion; let θ∈Xμ←μη. Then, μ←μη⊈θ. Suppose θ∉Xη; then η⊆θ. This implies μ←μη⊆μ←μθ=θ, which is impossible. Thus, θ∈Xη and hence Xη=Xμ←μη.
In the following result, we show that there is a one-to-one correspondence between the class ofμ-fuzzy filters and the lattice of all open sets in Xμ.Theorem 17.
The latticeFFμL is isomorphic with the lattice of all open sets in Xμ.Proof.
The lattice of all open sets inXμ is T,∩,∪. Define the mapping(43)f:FFμL⟶Tbyfλ=Xλ,for allλ∈Xμ.
SinceXλ=Xμ⟵μλ and μ←μλ is a μ-fuzzy filter, every open subset of Xμ is of the form Xθ for some θ∈FFμL. This shows that the map is onto.
Letfλ=fθ. Now, we need to show λ=θ. Suppose not. Then, λ≠θ, which implies that there is x∈L such that either λx<θx or θx<λx. Without loss of generality, we can assume that λx<θx. Put λx=β. Then by Corollary 3, we can find a prime μ-fuzzy filter η such that λ⊆η and ηx≤β. Thus, η∉Xλ and θ⊈η. So, η∉Xλ and η∈Xθ. This is a contradiction. Hence, λ=θ.
Now, we show thatf is homomorphism. Let λ,θ∈FFμL. Then,(44)fλ∨¯θ=Xμ←μλ∨θ=Xλ∨θ=fλ∪fθ.
Similarly,fλ∩θ=fλ∩fθ. This shows that f is a homomorphism. Hence, f is an isomorphism.
For any fuzzy subsetη of L, Xθ=η∈Xμ:θ⊈η is an open set of Xμ and Vθ=η∈Xμ:θ⊆η=Xμ−Vθ is a closed set of Xμ. In the following result, we prove the closure of a fuzzy set.Theorem 18.
For any familyℱ⊆Xμ, closure of ℱ is given by ℱ¯=V∩λ∈ℱλ.Proof.
We know that closure ofℱ is the smallest closed set containing ℱ. To prove our claim, it is enough to show that V∩λ∈ℱλ is the smallest closed set containing ℱ. Since the set of all μ-fuzzy filter is a complete distributive lattice, ∩λ∈ℱλ is a μ-fuzzy filter and V∩λ∈ℱλ is a closed set in Xμ. If η∈ℱ; then ∩μ∈ℱλ⊆η. Thus, η∈V∩λ∈ℱλ. This implies that ℱ⊆V∩λ∈ℱλ. Let Vθ be any closed set in Xμ containing ℱ. Then, θ⊆λ, for each λ∈ℱ. Thus, θ⊆∩λ∈ℱλ and V∩λ∈ℱλ⊆Vθ. So, V∩λ∈ℱλ is the smallest closed set containing ℱ. Hence, ℱ¯=V∩λ∈ℱλ.


## 5. Conclusion

In this work, we studied the concept ofμ-fuzzy filters of a distributive lattice. We proved that the set of all μ-fuzzy filters of a distributive lattice forms a complete distributive lattice isomorphic to the set of all fuzzy ideals of A+L. We observed that every μ-fuzzy filter is the intersection of all μ-fuzzy filters containing it. We also studied the space of all prime μ-fuzzy filters in a distributive lattice. Our future work will focus on α-fuzzy ideals of a C-algebra.



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*Source: 8841670-2020-12-29.xml*