# L-Fuzzy Prime Spectrums of ADLs

Authors: Natnael Teshale Amare; Srikanya Gonnabhaktula; Ch. Santhi Sundar Raj
Journal: Advances in Fuzzy Systems
 (2021)

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## Abstract

The notion of an Almost Distributive Lattice (ADL) is a common abstraction of several lattice theoretic and ring theoretic generalizations of Boolean algebra and Boolean rings. In this paper, the set of allL-fuzzy prime ideals of an ADL with truth values in a complete lattice L satisfying the infinite meet distributive law is topologized and the resulting space is discussed.


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## Body

## 1. Introduction

The concept of prime ideal is vital in the study of structure theory of distributive lattices in general and of Boolean algebras in particular [2]. In this context, we recall the work of Stone [1] on the representation of distributive lattices by algebra of sets. In fact, he proved that a lattice L is distributive if and only if any ideal of L is the intersection of all prime ideals containing it. Also, he introduced a topology on the set of all prime ideals of a given Boolean algebra B in such a way that B is isomorphic with the Boolean algebra of cl-open subsets of resulting space.Swamy and Rao [2] have introduced the notion of an Almost Distributive Lattice (ADL) which is algebra A,∧,∨,0 of type 2,2,0 satisfying all the axioms of a distributive lattice with zero except ∧ commutative, ∨ commutative, and right distributivity of ∨ over ∧. In fact, in any ADL, three conditions are equivalent.Next, Rosenfold [3] introduced the notion of fuzzy groups; many researchers are turned into fuzzifying various algebra. Santhi Sundar Raj et al. [4–6] have introduced the concepts of fuzzy prime ideals of an ADL and studied them deeply.In this paper, we introduce a topology on the set of allL-fuzzy prime ideals of an ADL A and the resulting space is called the L-fuzzy prime spectrums of A, denoted by FLspecA or X. For an L-fuzzy ideal λ of A, open subset of X is of the form Xλ=μ∈X:λ≰μ and Vλ=μ∈X:λ≤μ is a closed set. In particular, we prove that Xa:a∈S is a base for a topology on X. Furthermore, it is proved that the space FLspecA is compact and it contains a subspace homeomorphic with the spectrum of A which is dense in it. Also, it is proved that the space X is a Hausdorff space if and only if the space is a T1-space and, further, it is noted that the space X is a T1-space if and only if every L-fuzzy prime ideal is an L-fuzzy maximal ideal and L-fuzzy minimal prime ideal of A. Finally, it is proven that if A and B are isomorphic ADLs, then the space FLspecA is homeomorphic with the space FLspecB.Throughout this paper,A stands for an ADL A,∧,∨,0 with a maximal element and L stands for a complete lattice L,∧,∨,0,1 satisfying the infinite meet distributive law; that is, x∧∨y∈Sy=∨y∈Sx∧yfor anyS⊆L and x∈L.


## 2. Preliminaries

In this section, we recall some definitions and basic results mostly taken from [2, 4].Definition 1.
An algebraA=A,∧,∨,0 of type 2,2,0 is called an Almost Distributive Lattice (abbreviated as ADL) if it satisfies the following conditions for all a,b and c∈A:(1)
0∧a=0(2)
a∨0=a(3)
a∧b∨c=a∧b∨a∧c(4)
a∨b∧c=a∨b∧a∨c(5)
a∨b∧c=a∧c∨b∧c(6)
a∨b∧b=b
Any bounded below distributive lattice is an ADL. Any nonempty setX can be made into an ADL which is not a lattice by fixing an arbitrarily chosen element 0 in X and by defining the binary operations ∧ and ∨ on X by (1)a∧b=0,ifa=0,b,ifa≠0,a∨b=b,ifa=0,a,ifa≠0.
This ADLX,∧,∨,0 is called a discrete ADL.Definition 2.
LetA=A,∧,∨,0 be an ADL. For any a and b∈A, define a≤b if a=a∧b⟺a∨b=b. Then ≤ is a partial order on A with respect to which 0 is the smallest element in A.Theorem 1.
The following hold for anya,b and c in an ADL A:(1)
a∧0=0=0∧a and a∨0=a=0∨a(2)
a∧a=a=a∨a(3)
a∧b≤b≤b∨a(4)
a∧b=a⟺a∨b=b(5)
a∧b=b⟺a∨b=a(6)
a∧b∧c=a∧b∧c (i.e., ∧ is associative)(7)
a∨b∨a=a∨b(8)
a≤b⟹a∧b=a=b∧a⟺a∨b=b=b∨a(9)
a∧b∧c=b∧a∧c(10)
a∨b∧c=b∨a∧c(11)
a∧b=b∧a⟺a∨b=b∨a(12)
a∧b=infa,b⟺a∧b=b∧a⟺a∨b=supa,bDefinition 3.
LetI be a nonempty subset of an ADL A. Then I is called an ideal of A if a,b∈I⇒a∨b∈I and a∧x∈I for all x∈A.
As a consequence, for any idealI of A,x∧a∈I for all a∈I and x∈A. An element m∈A is said to be maximal if, for any x∈A, m≤x implies m=x. It can be easily observed that m is maximal if and only if m∧x=x for all x∈A.Definition 4.
LetL=L,∧,∨ and M=M,∧,∨ be lattices and let f:L⟶M be a mapping. Then f is called (1) an order homomorphism (or isotone) if a≤b in L⇒fa≤fb in M and (2) a lattice homomorphism if, for any a,b∈L, fa∧b=fa∧fb and fa∨b=fa∨fb.Theorem 2.
LetL and M be lattices and let f:L⟶M be a bijection. Then f is a lattice isomorphism if and only if both f and f−1 are order homomorphisms.Definition 5.
LetA,∧,∨,0 and A′,∧′,∨′,0′ be ADLs. A mapping f:A⟶A′ is called a homomorphism if the following are satisfied for any x and y∈A: (1) fx∨y=fx∨′fy. (2) fx∧y=fx∧′fy. (3) f0=0′.Definition 6.
LetX and Y be topological spaces and let f:X⟶Y be a mapping; then f is said to be continuous if and only if inverse image of every open set in Y is open in X.Definition 7.
LetX and Y be topological spaces and let f:X⟶Y be a mapping; then f is said to be open if and only if image of every open set in X is open in Y.Definition 8.
LetX and Y be topological spaces; then a bijection ϕ:X⟶Y is said to be a homeomorphism if it is a continuous open mapping.Definition 9.
AnL-fuzzy subset λ of A is said to be an L-fuzzy ideal of A, if λ0=1 and λx∨y=λx∧λy, for all x,y∈A.Theorem 3.
Letλ be an L-fuzzy subset of A. Then λ is an L-fuzzy ideal if and only if 1λ0=1, 2λx∨y≥λx∧λy, and 3λx∧y≥λx∨λy, for all x,y∈A.Definition 10.
LetχS denote the characteristic function of any subset S of an ADL A; that is,(2)χSx=1,ifx∈S,0,ifx∉S.Definition 11.
A properL-fuzzy ideal λ of A is called an L-fuzzy prime ideal of A if, for any x,y∈A, λx∧y=λx or λy.Theorem 4.
Letλ be a proper L-fuzzy ideal of A. Then the following are equivalent to each other: (1) For each α∈L, λα=A or λα is a prime ideal of A. 2λ is an L-fuzzy prime ideal of A. (3) For any x,y∈A, λx∧y≤λx∨λy and either λx≤λy or λy≤λx.Theorem 5.
Letλ be an L-fuzzy prime ideal of A and let 0 be a prime element in A. Then λ is an L-fuzzy minimal prime ideal of A if and only if λα is a minimal prime ideal of A, for all α∈L.Definition 12.
A properL-fuzzy ideal λ of A is called an L-fuzzy maximal ideal of A if, for each α∈L, either λα=A or λα is a maximal ideal of A.


## 3. Topological Space onL-Fuzzy Prime Ideals

In this section, we introduce the Zariski topology on the set ofL-fuzzy prime ideals of an Almost Distributive Lattice A. Our definition of L-fuzzy prime ideal offers us an appropriate setting to introduce a topology on the set of L-fuzzy prime ideals of A. A topology is introduced on the set of all L-fuzzy prime ideals of A to obtain the space called the hull-kernel topology on the set of all L-fuzzy prime ideals and denoted by FLspecA or X. First, we have the following.Theorem 6.
LetA and B be ADLs. Let p:A⟶B be a lattice homomorphism and p0=0. If λ:A⟶L and μ:B⟶L are L-fuzzy ideals of A and B, respectively, then 1p−1μ is an L-fuzzy ideal of A, 2pλ is an L-fuzzy ideal of B if p is an epimorphism, and 3pp−1μ=μ.Proof.
Definep−1μ:A⟶L and pλ:B⟶L as p−1μx=μpx for each x∈A and pλy=Supλx:px=y,x∈A for each y∈B. Then,(1)
p−1μ0=μp0=μ0=1, as μ is an L-fuzzy ideal of B. Let x,y∈A. Then,(3)p−1μx∨y=μpx∨y=μpx∨pysincepis lattice homomorphism=μpx∧μpysinceμis anL−fuzzy ideal=p−1μx∧p−1μy.Also, letx≤y in A. As p is homomorphism, we get px≤py. But then μpx≥μpy (since μ is an L-fuzzy prime ideal). That is, p−1μx≥p−1μy. Therefore, p−1μ is antitone. Thus, p−1μ is an L-fuzzy ideal of A.(2)
Clearly,pλ0=1. Let p:A⟶B be a lattice epimorphism. Let a,b∈B. Then there exists x,y∈A such that px=a and py=b. Thus, px∨y=px∨py=a∨b. Now,(4)pλa∨b=Supλz:pz=a∨b,z∈A=∨z∈p−1a∨bλz≥∨x∈p−1a,y∈p−1bλx∨y≥∨x∈p−1a,y∈p−1bλx∧λysinceλis anL−fuzzy ideal=∨x∈p−1aλx∧∨y∈p−1bλy=pλa∧pλb,Thus,pλa∨b≥pλa∧pλb. Similarly, pλa∧b≥pλa∨pλb. Let a≤b in B. Then, a∨b=b. Therefore, whenever px=a and pb=y, we have px∨y=px∨py=a∨b. Now,(5)pλa∧pλb=∨c∈p−1aλc∧∨d∈p−1bλd=∨c∈p−1a,d∈p−1bλc∧λd=∨c∨d∈p−1bλc∨dsinceλis anL−fuzzy ideal=pλb,This shows thatpλb≤pλa. Thus, pλ is an antitone map. Therefore, pλ is an L-fuzzy ideal of A.(3)
Letb∈B. Then,(6)pp−1μb=∨y∈p−1bp−1μy=∨y∈p−1bμpy=μb.Thus,pp−1μ=μ.Theorem 7.
If anL-fuzzy subset λ of A is an L-fuzzy prime ideal of A, then λ is a homomorphism from A,∧,∨,0 to L,∨,∧.Proof.
Asλ is an L-fuzzy ideal, then(7)λ0=1,λx∨y=λx∧λy,for all,x,y∈A.
Sinceλ is an L-fuzzy prime ideal of A, we have λx∧y=λx or λx∧y=λy. In either case, we get(8)λx∧y≤λx∨λy.
Also,x∧y≤y and y∧x≤x and λ is antitone (being an L-fuzzy ideal) implying that λy≤λx∧y and λx≤λy∧x=λx∧y. Thus,(9)λx∨λy≤λx∧y.
From (8) and (9), we have(10)λx∨λy=λx∧y.
Therefore, from (7) and (10), λ is a homomorphism from A,∧,∨,0 to L,∨,∧.
In Theorem6, we have proved that inverse image of an L-fuzzy ideal of an ADL A is an L-fuzzy ideal again. In the case of L-fuzzy prime ideals, we have the following.Theorem 8.
LetA and B be ADLs. Let p:A⟶B be a lattice homomorphism. If λ is an L-fuzzy prime ideal of B, then p−1λ is an L-fuzzy prime ideal of A.Proof.
By Theorem6, p−1λ is an L-fuzzy ideal of A. Let x,y∈A. Then,(11)p−1λx∧y=λpx∧y=λpx∧py=λpx,orλpy=p−1λx,orp−1λy.
Therefore,p−1λ is an L-fuzzy prime ideal of A.Theorem 9.
LetA and B be ADLs. Let p:A⟶B be a lattice isomorphism. If λ is an L-fuzzy prime ideal of A, then pλ is an L-fuzzy prime ideal of B and p−1pλ=λ.Proof.
By Theorem6 (2), pλ is an L-fuzzy ideal of B. Let a,b∈B. Then, a=px and b=py, for some x,y∈A. Therefore, px∧y=a∧b. Now, if t∈p−1a, then pt=a=px implies that t=x (since p is injective).
Therefore,(12)pλa=Supλt:pt=a,t∈A=∨t∈p−1aλt=λx,
Similarly,pλb=λy and if z∈p−1a∧b, then z=x∧y. Thus,(13)pλa∧b=Supλz:pz=a∧b,z∈A=∨z∈p−1a∧bλz=λx∧y=λx,orλy.
Thus,pλ is an L-fuzzy prime ideal of B. Also, let x∈A. Then,(14)p−1pλx=pλpx=Supλy:py=px,y∈A=∨x∈p−1pyλy,sincepis injective=λx.
Therefore,p−1pλ=λ.Example 1.
Consider the latticeA=0,a,b,c,1 whose Hasse diagram is given below:
and letB=0,a,b,c and let ∨ and ∧ be binary operations on B defined by
Then,B,∧,∨,0 is an ADL, which is not a lattice a∧b≠b∧a. Let L=0,1 be the closed unit interval of real numbers. Then L is a frame with respect to the usual ordering. Define L-fuzzy subsets μ and λ of A and B, respectively, by μ0=1 and μx=0.5, for all x≠0; λ0=1, λa=λb=0, and λc=0.5; and define a function p:A⟶B by p0=0,pb=b,pc=c,pa=p1=a. Then, we observe that μα=A if 0≤α≤0.5 and μα=0 if 0.5<α≤1. Thus, by Theorem 4, μ is an L-fuzzy prime ideal of A. Also, λ0.5=0,c and λ1=0 are prime ideals of B. Therefore, λ is an L-fuzzy prime ideal of B (by4). For any x and y∈A,(15)p0=0,px∨y=px∨py,px∧y=px∧py.
Thus,p is a lattice homomorphism. Also, x≤y⇒px≤py. Thus, p is isotone and, for each y∈B, there exists x∈A such that px=y. Hence, p is a bijection map. Therefore, p is a lattice isomorphism. By the above Theorems 6 and 8, pλ and p−1μ are L-fuzzy prime ideals of A and B, respectively, since λ and μ are L-fuzzy prime ideals.
In the following, we obtain a topological space by introducing Zariski topology on the set ofL-fuzzy prime ideals of ADLs.Definition 13.
LetA=A,∧,∨,0 be a nontrivial ADL and let X be the set of all L-fuzzy prime ideals of A. For any L-fuzzy subset Θ of A, we define(16)VΘ=λ∈X:Θ≤λ,XΘ=λ∈X:Θ≰λ=X−VΘ.
The complement ofVΘ in X = XΘ.
Now, we prove some properties ofV and X.Theorem 11.
LetΘ and σ be L-fuzzy subsets of A. Then, we have the following: (1) if Θ≤σ, then Vσ⊆VΘ and XΘ⊆Xσ; 2Vσ∪VΘ⊆Vσ∧Θ and Xσ∪XΘ⊆Xσ∧Θ; 3VΘ=VΘ and XΘ=XΘ, where Θ is the smallest L-fuzzy ideal containing Θ4Vχ0=X and Vχ1=∅.Proof.
(1)
(17)μ∈Vσ⇒σ≤μ⇒Θ≤μsinceΘ≤σ⇒μ∈VΘ.Therefore,Vσ⊆VΘ. Also,(18)ν∉Xσ⇒σ≤ν⇒Θ≤νsinceΘ≤σ⇒ν∉XΘ,Therefore,XΘ⊆Xσ.(2)
(19)μ∈Vσ∪VΘ⇒μ∈Vσ,orμ∈VΘ,⇒σ≤μ,orΘ≤μ⇒σ∧Θ≤μ,sinceσ≤μ⇒σ∧Θ≤μ∧Θ≤μ,Similarly,(20)Θ≤μ⇒σ∧Θ≤σ∧μ≤μ,⇒μ∈Vσ∧Θ,Therefore,Vσ∪VΘ⊆Vσ∧Θ.(3)
ClearlyVΘ⊆VΘ, since Θ≤Θ and by (1). On the other hand, let μ∈VΘ. Then Θ≤μ; it follows that Θ≤μ. Hence, μ∈VΘ. Therefore, VΘ=VΘ. Also, clearly XΘ⊆XΘ, since Θ≤Θ and by (1). On the other hand,(21)μ∈XΘ⇒Θ≰μ⇒Θ≰μ⇒μ∈XΘ,It follows thatXΘ⊆XΘ. Therefore, XΘ=XΘ.(4)
Letλ∈X. Then λ0=1. Therefore, Vχ0=X and Vχ1=∅ (as λ is an L-fuzzy prime ideal).Remark 1.
In general, equality does not hold in Theorem7. Equality holds if Θ and σ are crisp ideals of A. The following example shows that, in a case of L-fuzzy subsets of A, equality does not hold even if Θ and σ are L-fuzzy ideals.Example 2.
LetA=0,a,b,c and L=0,s,t,1 with 0<s<t<1 and let ∨ and ∧ be binary operations on A defined by
Then,A,∧,∨,0 is an ADL. Now define λ:A⟶L by λ0=1,λa=λb=0 and λc=s. Then λ0=A,λs=0,c, and λ1=0 are prime ideals of A. Therefore, λ is an L-fuzzy prime ideal of A. Define Θ:A⟶L and σ:A⟶L as Θ0=1,Θa=0,Θb=Θc=s, and σ0=1,σa=σb=s,σc=t. Clearly, Θ and σ are L-fuzzy ideals of A and Θ≤σ. Then, Vσ=0,1 and VΘ=0,1,a,0,c,s. From this, Vσ⊆VΘ. Also, σ∧Θ=0,1,a,0,b,s,c,s and hence Vσ∧Θ=VΘ. Now, Vσ∪VΘ=0,1,a,0,c,s⊆Vσ∧Θ but c,s∉Vσ∪VΘ. Hence, Vσ∪VΘ⊂Vσ∧Θ.
Recall that ifI is an ideal of A, then the characteristic function χI of I is an L-fuzzy ideal of A. For such L-fuzzy ideals, we have the following.Theorem 12.
LetI and J be ideals of A. Then, 1VχI∪VχJ=VχI∩J and 2XχI∪XχJ=XχI∩J.Proof.
(1)
SinceI∩J⊆I,J, χI∩J≤χI,χJ. Then, by Theorem 7 (2), we have VχI∪VχJ⊆VχI∩J. On the other hand, let μ∈VχI∩J. Then, χI∩J≤μ, and it follows that μx=1, for each x∈I∩J. If χI≰μ and χJ≰μ, then there exists x∈I,y∈J such that μx≠1 and μy≠1. But, as x∧y∈I∩J, then μx∧y=1. As μ is an L-fuzzy prime ideal of A, then 1=μx∧y≤μx∨μy. It follows that μx∨μy=1. This implies that either μx=1 or μy=1, which gives a contradiction with the choice of x and y. So, χI≤μ or χJ≤μ. Therefore, μ∈VχI or μ∈VχJ and hence μ∈VχI∪VχJ.(2)
Clearly,XχI∩J⊆XχI∪XχJ. On the other hand,(22)μ∈XχI∪XχJ⇒μ∈XχIorμ∈XχJ⇒χI≰μ,orχJ≰μ⇒χI∩χJ≰μ⇒χI∩J≰μ⇒μ∈XχI∩J.
Therefore,XχI∪XχJ⊆XχI∩J.Corollary 1.
For any subsetsS of an ADL A and letting a,b∈A, (1)
VχS=∩Vχa:a∈S(2)
Vχa∪Vχb=Vχa∧bTheorem 13.
For any subsetsS of an ADL A, we have the following:(1)
1χS=∪a∈Sχa(2)
2χS=χS(3)
3χa=χa for every a∈A(4)
4XχS=XχS(5)
5VχS=VχSProof.
ClearlyχS≤χS. Let λ be an L-fuzzy ideal of A such that χS≤λ. Then, λa=1, for all a∈S. Now, we shall prove that χS≤λ. For any t∈A, we have(23)t∈S⇒t=∨i=1nai∧x,for somex∈Aanda1,a2,…,an∈SandχSt=1.
Now,(24)λt=λ∨i=1nai∧x=λ∨i=1nai∧x=∧i=1nλai∧x≥∧i=1nλai∨λx,by…=λa1∨λx∧λa2∨λx∧⋯∧λan∨λx=1∨λx∧1∨λx∧⋯∧1∨λx=1∧1∧⋯∧1=1.
Therefore,λt≥1 and hence λt=1. Therefore, χS≤λ. This shows that χS is the smallest L-fuzzy ideal of A containing χS. Thus, χS=χS.Theorem 14.
Ifλii∈Δ is a family of L-fuzzy subsets of A, then(25)V∪i∈Δλi=∩i∈ΔVλi.Proof.
(26)μ∈V∪i∈Δλi⟺∪i∈Δλi≤μ,⟺λi≤μ,for eachi∈Δ⟺μ∈Vλi,for eachi∈Δ⟺μ∈∩i∈ΔVλi.
This shows thatV∪i∈Δλi=∩i∈ΔVλi.Theorem 15.
Letτ=XΘ:Θis anL−fuzzy subset ofA. Then the pair X,τ is a topological space.Proof.
ConsiderL-fuzzy subsets of A defined by σx=0 and Θx=1, for all x∈A. Then, Vσ=λ∈X:σ≤λ=X. Therefore, Xσ=X−Vσ=∅. Also, VΘ=λ∈X:Θ≤λ=∅. Therefore, XΘ=X−VΘ=X. Let μ and ν be L-fuzzy subsets of A. Then, Vμ∪Vν=Vμ∪Vν=Vμ∧μ. Now,(27)Xμ∩Xν=X−Vμ∩X−Vν=X−Vμ∪Vν=X−Vμ∪Vν=X−Vμ∧ν=Xμ∧ν∈τ.
Also, letλi:i∈I be nonempty collection of L-fuzzy ideals of A. Then, we have V∪λi:i∈I=∩Vλi:i∈I (by the above theorem). Now,(28)∪i∈IXλi=∪i∈IX−Vλi=X−∩i∈IVλi=X−V∪i∈Iλi=X∪i∈Iλi.
Therefore,X,τ is a topological space.Definition 14.
The topological spaceX,τ, as in Theorem 15, is called L-fuzzy prime spectrum of an ADL A and is denoted by FLspecA or X.
In the following section, we introduce base ofFLspecA.Definition 15.
LetA=A,∧,∨,0 be a nontrivial ADL and let X be the set of all L-fuzzy prime ideals of A. For any a∈A, we define Xa=Xχa. That is,(29)Xa=λ∈X:χa≰λ,and hence(30)Va=λ∈X:χa≤λ.Theorem 16.
LetX be the set of L-fuzzy prime ideals of an ADL A. Then the following hold for any a and b∈A:(1)
Xa⊆Xb⟺a⊆b⟺b∧a=a(2)
Xa=Xb⟺a∼b⟺a=b(3)
Xa=∅⟺a=0(4)
Xa=X⟺a is maximal element in A(5)
Xa∩Xb=Xa∧b=Xb∧a(6)
Xa∪Xb=Xa∨b=Xb∨aProof.
(31)Xa=∅⇒X−Xa=X⇒χa≤λ,for everyλ∈X⇒1≤λa⇒λa=1⇒a∈λ1,for everyλ∈X.
LetP be a prime ideal of A. Then, χP is an L-fuzzy prime ideal of A. Put μ=χP. Then, μ1=P. Therefore, a∈P, for every prime ideal P of A. Hence, a∈∪P=0. Therefore, a=0. Conversely, if a=0, then there is no ideal not containing 0. It follows that there is L-fuzzy prime ideal λ not containing χ0 and hence Xχ0=∅. Therefore, X0=∅.
(4) Suppose thata is maximal in A. Then, χa=χA. Then, no proper L-fuzzy ideal contains χA and hence Xχa=XχA=X. Conversely,(32)Xa=X⇒χa≰λ,for everyλ∈X⇒1≰λa⇒a∉λ1.
LetP be a prime ideal of A. Then, χP is an L-fuzzy prime ideal of A and let λ=χP. Then λ1=P and hence a∉∪P:Pis a prime ideal ofA. Therefore, a is maximal.
(5)(33)λ∈Xa∩Xb⟺χa≰λ,andχb≰λ⟺1≰λa,and1≰λb⟺a,b∉λ1⟺a∧b∉λ1,sinceλ1is prime ideal ofA⟺1≰λa∧b⟺χa∧b≰λ⟺λ∈Xa∧b.
Thus,Xa∩Xb=Xa∧b=Xb∧a (since a∧b∼b∧a).
(6)(34)λ∈Xa∪Xb⟺χa≰λ,orχb≰λ,⟺1≰λa,or1≰λb⟺a∉λ1,orb∉λ1⟺a∨b∉λ1⟺χa∨b≰λ⟺λ∈Xa∨b.
Thus,Xa∪Xb=Xa∨b=Xb∨a (since a∨b∼b∨a).Theorem 17.
For any subsetS of an ADL A and X=FLspecA, we have the following:(1)
1XχS=∪Xa:a∈S(2)
2VχS=∩Va:a∈SProof.
(1)
λ∈XχS. Then, λ∈X and χS≰λ and, hence, 1≰λt, for some t∈S. Since t∈S,t=∨i=1n∧x,for somex∈Aanda1,a2,…,an∈S. Let a=∨i=1nai. Then, a∈S. Now, we shall prove that χa≰λ. Clearly, t∈a and hence χat=1≰λt. Therefore, χa≰λ and hence λ∈Xa. On the other hand, let λ∈X such that χa≰λ, for some a∈S. Then, 1≰λt for some t∈a. This implies that χS≰λ and hence λ∈XχS. Thus, XχS=∪a∈SXa.(2)
(35)VχS=X−XχS=X−∪Xa:a∈S,since by1=∩X−Xa:a∈S=∩Va:a∈S.Theorem 18.
Xa:a∈S is a base for a topology on X.Proof.
By Theorem16 (4), Xm=X, for any maximal element m of A. Therefore, X=∪Xa:a∈A. Let a,b∈A. Then, for any λ∈X, we have(36)λ∈Xa∩Xb⟺χa≰λ,andχb≰λ,⟺1≰λa,and1≰λb⟺a,b∉λ1⟺a∧b∉λ1,sinceλ1is prime ideal ofA⟺1≰λa∧b⟺χa∧b≰λ⟺λ∈Xa∧b.
Thus,Xa∩Xb=Xa∧b=Xb∧a (since a∧b∼b∧a). Therefore, Xa:a∈A form a base for a topology on X.Theorem 19.
LetA be an ADL and X=FLspecA. Then, any closed subset of X is of the form hχI for some ideal I of A and any open subset of X is of the form XχI for some ideal I of A.Proof.
For eacha∈A, Xa is open in X and XχI=λ∈X:χI≰λ=∪a∈IXa and hence XχI is open in X. Since hχI=X−XχI, it follows that hχI is closed in X. On the other hand, let Y be a closed subset of X. Then, X−Y is open in X and hence X−Y=∪a∈SXa for some subset S of A, since Xa:a∈A is a base for a topology on X. If I=S, the ideal generated by S in A, then χS≤λ⟺χI≤λfor everyλ∈X. Now, we shall prove that X−Y=XχI. Let λ∈X. Then,(37)χI≰λ⇒χS≰λ⇒1≰λa,for somea∈A⇒χa≰λ⇒λ∈Xχa=Xa.
On the other hand,(38)λ∈X−Y⇒λ∈Xa,for somea∈S,⇒χa≰λ,⇒χS≰λ,sinceχS=∪a∈Sχa⇒χS≰λ⇒χI≰λ⇒λ∈XχI.
Therefore,X−Y=XχI and hence Y=hχI.Theorem 20.
LetA be an ADL and X=FLspecA. Then, for any Y⊆X, Y is a compact open subset of X if and only if Y=Xa for some a∈A.Proof.
Suppose thatY is compact open. Since Y is open, Y=∪a∈SXa for some S⊆A. Also, Y is compact and Xa:a∈S is a cover of Y. Then, there exists a1,a2,…,an∈S such that Y=Xa1∪Xa2∪⋯∪Xan=Xa, where a=∨i=1nai.
Conversely, suppose thatY=Xa for some a∈A. Therefore, Y is open. Now, we prove that Xa is compact. Suppose that Xa⊆∪b∈SXb,for someS⊆A. Then a∈S; otherwise, if a∉S, then there exists a prime ideal P of A such that S⊆P and a∉P. Let λ=χ__P. Then, λ is an L-fuzzy prime ideal of A, since P is prime and also λ1=P. Since a∉λ1,1≰λa, it follows that χ__a≰λ and hence λ∈Xa but λ∉Xb, for every b∈S, which is a contradiction to our assumption. Therefore, a∈S and hence a=∨i=1nbi∧x,for somex∈Aandb1,b2,…,bn∈S. This implies that Xa⊆∪i=1nXbi. Thus, Xa is compact.Corollary 2.
Every basic open setXa in FLspecA is compact.Corollary 3.
LetA be an ADL. Then, FLspecA is compact if and only if A has a maximal element.Proof.
LetX=FLspecA. Suppose that X is compact. Then, X=Xa for some a∈A (by Corollary 2) and hence a is maximal element in A. On the other hand, if A has a maximal element, say m, then X=Xm, which is compact (by 2).Theorem 21.
For any subsetS of A, S is closed in FLspecA if and only if there exists T⊆A such that S=VχT.Proof.
Suppose thatS is closed in X. Then, X−S is open in X and(39)X−S=∪Xχt|t∈T,for someT⊆A,=∪X−Vχt|t∈T=X−∩Vχt|t∈T=X−VχT.
Therefore,S=VχT, for some T⊆A. Conversely, suppose that S=VχT, for some subset T of A. Then, VχT=V∪t∈Tχt=∪t∈TVχt and hence VχT is closed in FLspecA, as Vχt is closed in S. Therefore, the subset S of A is closed in FLspecA.Theorem 22.
LetP denote the set of all prime ideals of A. Then the set D=χp:p∈P is dense in FLspecA.Proof.
LetP be a prime ideal of A.Then χp is an L-fuzzy prime ideal of A. Then D⊆X. Let λ∈X−D. Let Xχa be a basic open subset of X containing λ. Now,(40)λ∈Xχa⇒λa≠1⇒a∉λ1⇒χλ1a=0⇒χλ1∈Xχa.
Thus,χλ1∈D, since λ1 is a prime ideal of A. Therefore, Xχa contains a point of D. Thus, every member of X−D is a limit point of D. Thus, D¯=X.Theorem 23.
For any subsetS of X, S¯=VχT, where T=∩λ1:λ1∈S.Proof.
Letμ∈S. Then,(41)χTa=1⇒a∈T⇒a∈μ1⇒μa=1⇒χT≤μ.
Therefore,μ∈VχT. Thus, S⊆VχT. But then VχT is closed set containing S. Hence, S¯⊆VχT. On the other hand, let λ∈VχT. Let λ∉S. Let Xχa be a basic open set containing λ. Then,(42)λ∈Xχa⇒λ∉Vχas⇒λa≠1⇒χTa≠1,sinceλ∈VχT⇒a∉T=∩σ1:σ1∈S⇒νa≠1,for someν∈S⇒χa≰ν⇒ν∉Vχa⇒ν∈Xχa.
Thus,ν∈Xχa∩S−λ. This shows that λ is a limit point of S. Thus, VχT⊆S¯. Therefore, S¯=VχT.Corollary 4.
For any subsetS∈X, λ¯=Vχλ1 for any μ,ν∈X,μ¯=ν¯ if and only if μ1=ν1.First, let us recall that a topological spaceX is said to be T0−space if, for any x≠y∈X, there exists an open set containing x and not containing y or vice versa, that X is called a T1−space if, for any x≠y∈X, there exists open sets G and H in X such that x∈G−H and y∈H−G, and that X is called a Hausdorff space (T2−space) if, for any x≠y∈X, there exist disjoint open sets G and H in X containing x and y, respectively.Theorem 24.
For any ADLA, FLspecA is a T0−space.Proof.
Letμ≠λ∈FLspecA. That is, μ and λ are distinct L-fuzzy prime ideals of A. Then, either μ≰λ or λ≰μ. Suppose that μ≰λ. Then, there exists χa∈μ such that χa≰λ. Then λ∈Xa and μ∉Xa. Also, each Xa is open set containing λ but not μ. On the other hand, if λ≰μ, then there exists an open set containing μ but not λ. Therefore, FLspecA is a T0−space.Theorem 25.
LetA be an ADL with maximal elements. Then the following are equivalent to each other:(1)
FLspecA is a T2− space (or a Hausdorff space)(2)
FLspecA is a T1− space(3)
EveryL-fuzzy prime ideal of A is an L-fuzzy maximal ideal of A(4)
EveryL-fuzzy prime ideal of A is an L-fuzzy minimal prime ideal of AProof.
(1)
1⇒2: clear.(2)
2⇒3: suppose that FLspecA is a T1−space and μ is an L-fuzzy prime ideal of A. Since μ is closed in FLspecA, μ¯=μ. By Theorem 19, we have μ¯=hχμ1=μ, which implies that there is no L-fuzzy prime ideal of A containing μ other than μ itself. That is, μ is an L-fuzzy maximal ideal of A.(3)
3⇒4: suppose that every L-fuzzy prime ideal of A is an L-fuzzy maximal ideal of A. Suppose that μ is an L-fuzzy prime ideal of A. Let λ be an L-fuzzy prime ideal of A such that λ≤μ. By assumption, λ=μ. Therefore, μ is an L-fuzzy minimal prime ideal of A.(4)
4⇒1: suppose that every L-fuzzy prime ideal of A is an L-fuzzy minimal prime ideal of A. Let μ≠λ∈FLspecA. That is, μ and λ are distinct L-fuzzy prime ideals of A. Then, either μ≰λ or λ≰μ. Suppose that μ≰λ. Then, there exists χa∈μ such that χa≰λ. Then, λ∈Xa and μ∉Xa. Also, each Xa is open set containing λ but not μ. On the other hand, if λ≰μ, then there exists an open set containing μ but not λ. Hence, a∈μ1 and a∉λ1. By the minimality of μ1, there exists b∉μ1 such that a∧b=0. Therefore, Xa∩Xb=Xa∧b=∅. Hence, FLspecA is a Hausdorff space.Theorem 26.
LetA and B be ADLs and let p:A⟶B be a lattice homomorphism. Define g:FLspecB⟶FLspecA by gλ=p−1λ, for each λ∈FLspecB. Then, (1) g is continuous mapping. (2) If p is onto, then g is one-to-one.Proof.
Clearlyg is a well-defined map.(1)
Lethχt be any basic closed set in FLspecA and t∈A. Then,(43)g−1hχt=λ∈FLspecB|gλ∈hχt=λ∈FLspecB|χt≤gλ=λ∈FLspecB|gλt=1=λ∈FLspecB|p−1λt=1=λ∈FLspecB|λpt=1=λ∈FLspecB|χpt≤λ=hχpt.Therefore, inverse image underg of any basic open set in FLspecA is a closed set in FLspecB and hence g is continuous.(2)
Letp be onto and λ,μ∈FLspecB such that gλ=gμ. Then, for each a∈A,(44)gλa=gμa⇒p−1λa=p−1μa⇒λpa=μpa⇒λ=μ.Therefore,g is a one-to-one map.Theorem 27.
Letp:A⟶B be an isomorphism. Then, the space FLspecA is homeomorphic with the space FLspecB.Proof.
Letp:A⟶B be an isomorphism. Define the function g:FLspecB⟶FLspecA by gμ=p−1μ, for each μ∈FLspecB, and f:FLspecA⟶FLspecB by fλ=pλ, for each λ∈FLspecA. Then, g and f are well defined and inverse of each other (by Theorems 8 and 9). Thus, by the above theorem, f and g are continuous. Therefore, FLspecA and FLspecB are homeomorphisms (by Definition 12).



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