# Fuzzy Spectral Spaces and Fuzzy Congruences of a Heyting ADL

Authors: Yeshiwas Mebrat Gubena; Teferi Getachew Alemayehu
Journal: Journal of Mathematics
 (2023)

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## Abstract

In this paper, the spaceFp of fuzzy prime ideals of Heyting almost distributive lattice H is studied, and it is shown that the collection of all sets Mη is a topology on Fp, where η is a fuzzy ideal on H and Mη=θ∈Fp|η⊈θ. The only compact subset of the space Fp is given. A fuzzy congruence relation Θ on H is defined, and the homomorphism between the set of all fuzzy ideals of H and the set of all fuzzy ideals of H/Θ is established. Furthermore, we established an isomorphism between fuzzy spectrum of H and fuzzy spectrum of H/Θ.


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## Body

## 1. Introduction

The class of distributive lattices has many interesting properties, which lattices, in general, do not have. For this reason, Swamy and Rao [1] introduced the concept of an almost distributive lattice (ADL) as a common abstraction of lattice and ring theoretic generalizations of a Boolean algebra. In [1], it is proved that the commutativity of ∨, the commutativity of ∧, the right distributivity of ∨ over ∧, and the absorption law s∧t∨s=t are all equivalent to each other and whenever any one of these properties holds, an ADL Abecomes a distributive lattice. The concept of an ideal in A was introduced analogous to that in a distributive lattice, and it was observed that the set PIR of all principal ideals of A forms a distributive lattice.These results enable us to extend many existing concepts from the class of distributive lattices to the class of ADLs. Rao et al. [2] introduced the concept of Heyting almost distributive lattices (HADLs) as a generalization of Heyting algebra in the class of ADLs, and HADL is characterized in terms of the set of all its principal ideals. Rao and Ravi Kumar [3] proved that given any multiplicatively closed subset S of A disjoint from an ideal I of A, there exists a prime ideal belonging to I which is disjoint from S.Sambasiva Rao and Rao in [4] studied topological properties of the space of prime ideals of a HADL. These authors proved that there is an isomorphism between a Heyting algebra and the open base of its prime spectrum. This mapping is epimorphism, and it is isomorphism if and only if the HADL is a Heyting algebra.In [5], the concept of fuzzy set theory as a generalization of classical set theory was introduced by Zadeh. Rosenfield [6] started the pioneering work in the domain of fuzzification of algebraic objects on fuzzy groups. In particular, Bo and Wangming [7] and Santhi Sundar Raj et al. [8] have laid down the foundation for fuzzy ideals of a lattice and an ADL, respectively. In [9, 10], Alemayehu et al. introduced the concepts of belligerent fuzzy GE-filter of GE-algebras and beta-fuzzy filters of stone almost distributive lattices, respectively. Santhi Sundra Raj et al. [8] introduced the concept of fuzzy prime ideals of almost distributive lattices. In 1998, Swamy and Raju [11] introduced the concept of fuzzy ideals and fuzzy congruences of distributive lattices and showed that there is a one-to-one correspondence between the lattice of fuzzy ideals and the lattice of fuzzy congruences of A. In [12], Alaba and Addis studied the fuzzy congruence relations of an ADL A, and they gave the smallest fuzzy congruence on A such that its quotient is a distributive lattice.This paper comprises of four sections. The first two sections deal with the introductory and preliminary concepts. In Section3 of this paper, we studied some properties of fuzzy prime ideals of a HADL H, and it is shown that the collection ∑=Mη:ηis a fuzzy ideal ofH is a topology on Fp. The fact that the set Mxα=θ∈Fp|xα∉θ is the only compact subset of the space is proved. Furthermore, it is shown that an ADL A is a distributive lattice if and only if the mapping between A and Mxαxα∈Y where Y=xα|x∈A,α∈0,1 is injective. In Section 4, a fuzzy congruence relation Θ on a HADL H is defined, and the homomorphism between the set FIH of all fuzzy ideals of H and the set FIH/Θ of all fuzzy ideals of H/Θ is established, where H/Θ=Θx|x∈H. Furthermore, we proved that the mapping between fuzzy spectrum of HFspecH and fuzzy spectrum of H/ΘFspecH/Θ is bijective.


## 2. Preliminaries

In this section, we recall basic definitions and results which will be used in this article.Definition 1.
An algebraA,∨,∧,0 of type 2,2,0 is called an almost distributive lattice if it satisfies the following conditions: for all r,s and t∈A,(1)
0∧r=0(2)
r∨0=r(3)
r∧s∨t=r∧s∨r∧t(4)
r∨s∧t=r∨s∧r∨t(5)
r∨s∧t=r∧t∨s∧t(6)
r∨s∧s=sLetr,s∈A and “ ≤” be a partial ordering on A. We read r is less than or equal to s and write r≤s if r∧s=r. Equivalently, r∨s=s. If an element m is a maximal with respect to the partial ordering ≤ on A, then m is said to be maximal.Theorem 1.
LetAbe an ADL andm∈A. Then, the following are equivalent:(1)
mis maximal with respect to≤(2)
m∨s=m, for alls∈A(3)
m∧s=s, for alls∈ADefinition 2.
LetA,∨,∧,0,m be an ADL with 0 and a maximal element m. Suppose ⟶ is a binary operation on A satisfying the following conditions:(1)
r⟶r=m(2)
r⟶s∧s=s(3)
r∧r⟶s=r∧s∧m(4)
r⟶s∧t=r⟶s∧r⟶t(5)
r∨s⟶t=r⟶t∧s⟶t for all r,s,t∈A
Then,A,∨,∧,⟶,0,m is called a Heyting almost distributive lattice or simply a HADL.Lemma 1.
LetA,∨,∧,⟶,0,mbe a HADLA. Then, forr,s,t,c∈H, the following hold:(1)
r⟶t∧c=r∧c⟶t⟶c∧c(2)
r∧c=s∧c⇒r⟶t∧c=s⟶t∧c(3)
r∧c=s∧c⇒t⟶r∧c=t⟶s∧cDefinition 3.
LetA,∨,∧,0,m and A′,∨,∧,0′,m′ be two HADLs. Then, the mapping α:A⟶A′ is called a homomorphism of A into A′ if for any s,t∈A, the following conditions hold:(1)
αs∧t=αs∧αt(2)
αs∨t=αs∨αt(3)
αs⟶t=αs⟶αt(4)
α0=0′For any setA, a function ν:A⟶0,1 is called a fuzzy subset of A, where 0,1 is a unit interval of real numbers.Definition 4.
Letη be a fuzzy subset of an ADL A. For any α∈0,1, we denote the level subset ηα, i.e.,(1)ηα=x∈A:α≤ηx.Definition 5.
A fuzzy subsetη of an ADL A is said to be a fuzzy ideal of A if and only if(1)
η0=1(2)
ηs∨t=ηs∧ηt, for all s,t∈AFrom [13], remember that for each x∈L and α∈0,1, the fuzzy subset xα of L given by(2)xαb=α,ifx=b,0,if otherwise,for all b∈L is called a fuzzy point of H. In this case, an element x of L is called the support of xα and α is its value. For a fuzzy point xα of H and a fuzzy subset μ of H, we write xα∈μorxα⊆μ to say that μx≥α.


## 3. Fuzzy Prime Spectrum of HADLs

This section deals with some properties of fuzzy prime ideals of HADL. The fact that the setMxα=θ∈Fp|xα∉θ (where Fp is the class of all fuzzy prime ideals of a HADL H) is the only compact subset of the space Fp is proved. Furthermore, it is shown that an ADL A is a distributive lattice if and only if the mapping between A and Mxαα∈Y where Y=xα|x∈A,α∈0,1 is injective. Unless otherwise mentioned H stands for Heyting almost distributive lattices.Definition 6.
A fuzzy subsetη of H is called a fuzzy ideal if for all s,t∈H, it satisfies the following conditions:(1)
η0=1(2)
ηs∨t≥ηs∧ηt(3)
ηs∧t≥ηs∨ηtFrom this definition, we have got thatη is a fuzzy ideal of H if and only if one of the following conditions holds:(1)
η0=1 and ηs∨t=ηs∧ηt, for all s,t∈H(2)
η0=1 and ηs∨t≥ηs∧ηt and ηs∧t≥ηs∨ηt, for all s,t∈HDefinition 7.
A fuzzy subsetη of H is called a fuzzy filter if for all s,t∈H and a maximal element m, it satisfies the following conditions:(1)
ηm=1(2)
ηs∨t≥ηs∨ηt(3)
ηs∧t≥ηs∧ηtFrom this definition, we have got thatλ is a fuzzy filter of H if and only if one of the following conditions holds:(1)
λm=1 and λs∧t=λs∧ηt, for all s,t∈H(2)
λm=1 and λs∨t≥λs∨λt and λs∧t≥ηs∧ηt, for all s,t∈HExample 1.
LetH=0,r,s,t. Define the binary operations∨,∧, and⟶onA as follows:
Then,H,∨,∧,⟶,0 is a Heyting ADL with maximal elements r and s.
For fuzzy subsetsμ and λ of A, define μ0=1, μt=0.8 and μr=μs=0.5, and λr=λs=1, λt=0.5, and λ0=0.3. Then, it can be easily checked that μ is a fuzzy ideal of A and λ is a fuzzy filter of A.Theorem 2.
Letηbe a fuzzy ideal inHands,t∈H. Then, ηs∧t=ηt∧s.Proof.
Suppose thatη is a fuzzy ideal of H and s,t∈H. Then, ηs∧t≤ηs∧t∧s=ηt∧s∧s=ηt∧s.
With similar procedure,ηt∧s≤ηs∧t. Hence, ηt∧s=ηs∧t.Theorem 3.
Letηbe a fuzzy filter inHands,t∈H. Then, ηs∨t=ηt∨s.Remark 1.
In Example 1, we haver∧s=s, s∧r=r, r∧s≠s∧r, andμr∧s=μs∧r.We denote the class of all fuzzy ideals ofH by FIH.Theorem 4.
The setFIHforms a Heyting algebra.Proof.
Clearly,FIH is a bounded distributive lattice with greatest element χH and least element χ0. In addition to this, the intersection of a class of fuzzy ideals is a fuzzy ideal. This implies that it forms a complete lattice. For any two fuzzy ideals η,θ∈FIH, define an operation ⟶ on FIH as follows:(3)η⟶θ=supξ∈FIH|η∩ξ⊆θ.
Letμ,η and θ be fuzzy ideals of H such that η∩μ⊆θ. Now we prove that η⟶θ is the largest fuzzy ideal of H. Since μ∩η⊆θ and θ0=1, we have got that η⟶θ0=Supμ0:μ∈FIH,η∩μ⊆θ≥θ0=1. Again for any x,y∈H, η⟶θx∧η⟶θy(4)=Supμx:μ∈FIH,η∩μ⊆θ∧Supσx:σ∈FIH,η∩σ⊆θ=Supμx∧σy:μ,σ∈FIH,η∩μ⊆θ,η∩σ⊆θ≤Supμ∨σx∧μ∨σy:μ,σ∈FIH,η∩μ⊆θ,η∩σ⊆θ.
For eachμ,σ∈FIH such that η∩μ⊆θandη∩σ⊆θ,μ∨σ∈FIH and μ∨σ∩η⊆θ. Then, η⟶θx∧η⟶θy≤Supλx∧λy:λ∈FIH,η∩λ⊆θ=Supλx∨y:λ∈FIH,η∩λ⊆θ=η⟶θx∨y. Thus, η⟶θx∨y≥η⟶θx∧η⟶θy. Now we show that η⟶θx≤η⟶θx∧y and η⟶θy≤η⟶θx∧y. Thus, η⟶θx=Supμx:μ∈FIH,η∩μ⊆θ≤Supμx∧y:μ∈FIH,η∩μ⊆θ=η⟶θx∧y. Similarly, η⟶θy≤η⟶θx∧y. So, η⟶θx∧y≥η⟶θx∨η⟶θy. Hence, η⟶θ is a fuzzy ideal of H. From the definition of η⟶θ and the given condition η∩μ⊆θ, it is clear that η∩η⟶θ⊆θ. Suppose that σ is a fuzzy ideal of H containing η⟶θ such that η∩σ⊆θ. It contradicts the definition of η⟶θ. Hence, η⟶θ is the largest fuzzy ideal of H such that η∩σ⊆θ for any σ∈FIH.
Therefore,FIH,∪,∩,⟶,χ0,χH is a Heyting algebra.Lemma 2.
Letηbe a fuzzy ideal ofH,s∈H, andβ∈0,1. Ifηs≤β, then there exists a prime fuzzy idealθofHsuch thatη⊆θandθs≤β.LetFp denote the set of all fuzzy prime ideals of H. For any fuzzy ideal η of H, define Mμ=θ∈Fp|μ⊈θ, and for any fuzzy point xα, define Mxα=θ∈Fp|xα∉θ. Then, we have the following results.Theorem 5.
Letηbe a fuzzy ideal ofH. Then, xα∈ηif and only ifMxα⊆Mη.Proof.
Letxα∈η and θ∈Mxα. Then, xα∉θ. Hence, we get η⊈θ. Thus, θ∈Mη. So, Mxα⊆Mη. Conversely, assume that Mxα⊆Mη. Suppose that xα∉η. This implies that ηx<α. Then, from Lemma 2, there exists θ∈Fp such that η⊆θ and θx<α. This shows that xα∉θ so that θ∈Mxα. It follows that θ∈Mη where η⊈θ which is a contradiction. Therefore, xα∈η.Lemma 3.
For any fuzzy idealsηandθofH, the following conditions hold:(1)
η⊆θif and only ifMη⊆Mθ(2)
Mη∩Mθ=Mη∩θ(3)
Mη∪Mθ=Mη∨θ(4)
Mη⟶θ=Mθ∪Fp−MηProof.
(1)
SupposeMη⟶θ=Mθ∪Fp−Mη and θ are fuzzy ideals of H such that η⊆θ. Let λ∈Mη. This implies that η⊈λ. This implies that θ⊈λ, and so λ∈Mθ. Hence, Mη⊆Mθ. Conversely, suppose that Mη⊆Mθ. We prove that η⊆θ. Assume η⊈θ. This implies that ηx>θx for some x∈H. Put ηx=α. This implies that xα∈η and xα∉θ. By Theorem 5, we have Mxα⊆Mη and Mxα⊈Mθ. Hence, Mη⊈Mθ, which is contradiction. Hence, η⊆θ.(2)
Clearly,Mη∩θ⊆Mη∩Mθ. Conversely, let λ∈Fp. Suppose that λ∈Mη∩Mθ. Then, λ∈Mη and λ∈Mθ. This means η⊈λ and θ⊈λ where η∩θ⊈λ. Consequently, λ∈Mη∩θ. Therefore, Mη∩Mθ⊆Mη∩θ.(3)
Clearly,Mη∪Mθ⊆Mη∨θ. On the other hand, if λ∈Mη∨θ, then η∨θ⊈λ. Suppose that η⊆λ and θ⊆λ. This shows that η∨θ⊆λ which is a contradiction. Thus, η⊈λ or θ⊈λ. Following this, λ∈Mη or λ∈Mθ where λ∈Mη∪Mθ. It follows that Mη∨θ⊆Mη∪Mθ. Hence, Mη∨θ=Mη∪Mθ.(4)
Letλ∈Mη⟶θ. Then, η⟶θ⊈λ. This implies that there exists σ∈FIH such that σ⊈λ and η∩σ⊆θ. This implies that σx>λx for some x∈H.(5)Ifη⊆λ,then we getλ∉Mη⇒λ∈Mθ∪Fp−Mη⇒Mη⟶θ⊆Mθ∪Fp−MηIfη⊈λ,then there exists,y∈Hsuch thatηy>λy⇒σ∩ηx∧y≥σx∧y∧ηx∧y>σx∧ηy>λx∧λy=λx∧y⇒σ∩η⊈λSinceσ∩η⊆θ,θ⊈λand soλ∈Mθ⇒λ∈Mθ∪Fp−Mη⇒Mη⟶θ⊆Mθ∪Fp−Mη.
Conversely,Mσ⊆Mθ∪Fp−Mη for some fuzzy ideals of H. Let λ∈Fp be such that λ∈Mη∩σ.(6)⇒λ∈Mηandλ∈Mσ⇒λ∉Fp−Mηandλ∈Mσ⇒λ∈Mθ.
Thus,Mη∩σ⊆Mθ.(7)⇒Mη∩σ⊆Mθ⇒η∩σ⊆θ⇒σ⊆η⟶θ⇒Mσ⊆Mη⟶θ⇒Fp−Mη∪Mθ⊆Mη⟶θ.
Hence,Mθ∪Fp−Mη=Mη⟶θ.Next, we discus the space of fuzzy prime ideals of HADL analogous with the hull-kernel topology.Theorem 6.
LetΣ=Mμ:μis a fuzzy ideal ofH. Then, Σis a topology onFp.Proof.
Consider the fuzzy subsetsλ1,λ2 of H defined as λ1x=0 and λ2x=1 for all x∈H. Clearly, λ1 and λ2 are fuzzy ideals of H. λ1⊆η for all η∈Fp, which is impossible. Thus, Mλ1=∅. Since each η∈Fp is nonconstant, λ2⊈η for all η∈Fp. Thus, Fpλ2=Fp. This implies ∅,Fp∈Σ.
Also, for any fuzzy idealsλ1 and λ2 of H, by (2) in Lemma 3, we have Mλ1∩Mλ2=Mλ1∩λ2. This shows that Σ is closed under finite intersections. Next, let λi,i∈Ω be any family of fuzzy ideals of H. Now we prove that ∪i∈ΩMλi=M∪i∈Ωλi. Let η∈M∪i∈Ωλi; then, ∪i∈Ωλi⊈η, which implies that λi⊈η for some i∈Ω. Otherwise, if λi⊆η for each i∈Ω, it will be true that ∪i∈Ωλi⊆η. Thus, η∈∪i∈ΩMλi where M∪i∈Ωλi⊆∪i∈ΩMλi. Clearly, ∪i∈ΩMλi⊆M∪i∈Ωλi. Hence, ∪i∈ΩMλi=M∪i∈Ωλi. Therefore, Σ is closed under arbitrary unions, and hence it is topology on Fp.The topological spaceFp;Σ is called the fuzzy prime spectrum of H, and it is denoted by FspecH.Theorem 7.
Letηandμbe fuzzy subsets ofH. Then, the following conditions hold:(1)
Fpη=Fpμ⇒η=μ(2)
Fpη=FpηCorollary 1.
For anyx,y∈Handα∈0,1, the following conditions hold:(1)
Ifx≤y, thenMxα⊆Myα(2)
Mxα∩Myα=Mx∧yα(3)
Mxα∪Myα=Mx∨yα(4)
∪x∈H,α∈0,1Mxα=Fp(5)
Mxα⊆Mx∨mα(6)
Mxα=Mxα(7)
M0α=∅(8)
Mx⟶yα=Myα∪Fp−MxαFrom Corollary1, we have the following.Corollary 2.
Letℬ=Mxβ:x∈H,β∈0,1. Then,ℬforms a base for a topology onΣ.Corollary 3.
Fpis a compact space.For any fuzzy subsetσ of H, Fpσ=η∈Fp:η⊈σ is open set of Fp and Cσ=Fp−Mσ is a closed set of Fp. Also, every closed set in is of the form Cσ for all fuzzy subsets of H. Then, we have the following.Theorem 8.
LetA⊆Fp. Then, the closure ofA(i.e., A¯) is given byA¯=C∩η∈Aη.Theorem 9.
Letx∈Handβ∈0,1. Then, the setMxβforms a Heyting algebra. Moreover,Y=xβis epimorphic to this Heyting algebra.Proof.
From (2) and (3) of Corollary 1 it is clear that(8)Mo=Mxβxβ∈Y,∪,∩,∅,Fp,is a bounded distributive lattice. Now define the operation ⟶ on Mo by(9)Mxβ⟶Myβ=Mx⟶yβ.
Then, it can be routinely verified thatMo is a Heyting algebra. Define a mapping g:H⟶Mxβxβ∈Y by gx=Mxβ for any x∈H. Since θ0=1≥β,0β∉θ. Hence, g0=M0β=∅. Using Corollary 3.9, it can be easily obtained that g is a homomorphism in the sense of Heyting algebras. From the definition of g, clearly it is surjective. Therefore, Y is epimorphic to Mo.Theorem 10.
An ADLAis a distributive lattice if and only if the above mapgis an injective.Proof.
IfA is a distributive lattice, then it is a known fact that the map g is an isomorphism. Conversely, assume that g is injective. For any r,s∈A, we have always gr∧s=Mr∧sα=Mrα∩Msα=Msα∩Mrα=Ms∧rα=gs∧r. Since g is injective, we get r∧s=s∧r. Therefore, A is a distributive lattice.Let us denote the class of all HADLs byℌ and the category of all homomorphisms from an ADL A1 into an ADL A2 by HomA1,A2. Then, we have the following result.Lemma 4.
LetA1,A2∈ℌandg∈HomA1,A2. LetFp1andFp2be the sets of prime fuzzy ideals ofA1andA2, respectively. Then, for anyη∈Fp2, g−1η∈Fp1.Proof.
Letg∈HomA1,A2 and η∈Fp2,01∈A1. Then,(10)g−1η01=ηg01=η02=1.
For anyr,s∈A1,(11)g−1ηr∨s=ηgr∨s=ηgr∨gs≥ηgr∧ηgs≥g−1ηr∧g−1ηs.
Also, for anyr,s∈A1,(12)g−1ηr∧s=ηgr∧s=ηgr∧gs≥ηgr∨ηgs≥g−1ηr∨g−1ηs.
This shows thatg−1η is a fuzzy ideal of A1. Next, we show that g−1η is prime fuzzy ideal of A1. Let λ,ν∈FIA1, such that λ∩ν⊆g−1η. This implies gλ∩gν=gλ∩ν⊆η. Since η is a prime fuzzy ideal of A2, gλ⊆η and gν⊆η. This implies λ⊆g−1η and ν⊆g−1η. Hence, it concludes that g−1η∈Fp1.Theorem 11.
LetH1andH2be HADLs andg∈HomH1,H2. Then, the mappingTg:Fp2⟶Fp1defined byTgη=g−1ηfor allη∈Fp2is continuous.Proof.
From Lemma 4,Tg is well defined. For arbitrary xα, we know that Mxα is an open set in Fp1. Then, we have the following:(13)η∈Mgxα⇔gxα∉η⇔xα∉g−1η⇔g−1η∈Mxα⇔Tgη∈Mxα⇔η∈Tg−1Mxα.
Therefore, it follows thatTg−1Mxα=Mgxα, which is an open set in Fp2. Hence, Tg is a continuous mapping from Fp2 onto Fp1.


## 4. Isomorphism between the Fuzzy Spectral Spaces of a HADL and Its Fuzzy Congruence Lattice

In this paper, a fuzzy congruence relationΘ on H is defined and the homomorphism between the set FIH of all fuzzy ideals of H and the set FIH/Θ of all fuzzy ideals of H is established, where H/Θ=Θx|x∈H. Furthermore, we proved that the mapping between fuzzy spectrum of HFspecH and fuzzy spectrum of H/ΘFspecH/Θ is bijective.Definition 8.
A fuzzy equivalence relationΘ on a HADL H is a called fuzzy congruence relation on H, if the following are satisfied:(1)
Θs∧p,r∧q≥Θs,r∧Θp,q, for all p,q,r,s∈H(2)
Θs∨p,r∨q≥Θs,r∧Θp,q, for all p,q,r,s∈H(3)
Θs⟶p,r⟶q≥Θs,r∧Θp,q, for all p,q,r,s∈HThe following theorem is the characterization of Definition8.Theorem 18.
A fuzzy equivalence relationΘon HADLHis a fuzzy congruence onHif and only ifΘs,r≤Θs∧p,r∧p∧Θp∧s,p∧r∧Θp∨s,p∨r∧Θs∨p,r∨p∧Θs⟶p,r⟶p∧Θp⟶s,p⟶r,for allp,r,s∈H.Proof.
Suppose thatΘ is a congruence relation on H. Then, for any s,t,p∈H, Θs,t=Θs,t∧Θp,p≤Θs∧p,t∧p and also Θs,t=Θp,p∧Θs,t≤Θp∧s,p∧t. Hence, Θs,t≤Θs∧p,t∧p∧Θp∧s,p∧t. With similar procedure, Θs,r≤Θp∨s,p∨r∧Θs∨p,r∨p and Θs,r≤Θs⟶p,r⟶p∧Θp⟶s,p⟶r. Finally, we get Θs,r≤Θs∧p,r∧p∧Θp∧s,p∧r∧Θp∨s,p∨r∧Θs∨p,r∨p∧Θs⟶p,r⟶p∧Θp⟶s,p⟶rfor allx,y,z∈H.
Conversely, suppose thatΘs,r≤Θs∧p,r∧p∧Θp∧s,p∧r∧Θp∨s,p∨r∧Θs∨p,r∨p∧Θs⟶p,r⟶p∧Θp⟶s,p⟶r is true for allr,s,p∈H.
This impliesΘs,r≤Θs∧p,r∧p, Θs,r≤Θp∧s,p∧r, Θs,r≤Θp∨s,p∨r, Θs,r≤Θs∨p,r∨p, Θs,r≤Θs⟶p,r⟶p, Θs,r≤Θp⟶s,p⟶rfor alls,r,p∈H. Now Θs,r∧Θp,q=Θs,r∧Θp,p∧Θr,r∧Θp,q≤Θs∧p,r∧p∧Θr∧p,r∧q≤Θs∧p,r∧q by transitivity. Similarly, Θs∨p,r∨q≥Θs,r∧Θp,q and Θs⟶p,r⟶q≥Θs,r∧Θp,q. Hence, Θ is a congruence relation.Example 2.
LetH=0,r,s,t. Define the binary operations∨,∧, and⟶onA as follows:
Then,H,∨,∧,⟶,0 is a Heyting ADL with maximal elements r and s.
Define a fuzzy relationΘ on H as Θ0,0=Θr,r=Θs,s=Θt,t=Θr,s=Θs,r=1, Θr,t=Θt,r=Θs,t=Θt,s=0.7, and Θ0,r=Θr,0=Θs,0=Θ0,s=Θt,0=Θ0,t=0.5. Then, Θ is a fuzzy congruence relation on H.Lemma 5.
LetΘbe an equivalence relation onH. Then, Θis a congruence relation onHif and only if its characteristic functionχΘis a fuzzy congruence onH.Lemma 6.
A fuzzy relationΘonHis a fuzzy congruence onHif and only if every level subsetΘλofΘatλ∈0,1is a congruence relation onH.Theorem 19.
LetH,∨,∧,⟶,0,mbe a HADLHandμbe a fuzzy filter ofH. DefineΘμs,t=supμx:s∧x=t∧xfor allx∈H. Then, Θμis a fuzzy congruence relation onH.Proof.
Clearly,Θμ is reflexive and symmetric. Next, we show that Θμ is transitive.
For any HADLH, suppose that s∧x=t∧x and t∧y=r∧y for any s,t,x,y∈H. Then, s∧x∧y=t∧x∧y=x∧t∧y=x∧r∧y=r∧x∧y.(14)Θμs,t∧Θμt,r=supμx:s∧x=t∧xfor allx∈H∧supμy:t∧y=r∧yfor ally∈H=supμx∧μy:s∧x=t∧x,t∧y=r∧yfor allx,y∈H≤supμx∧y:s∧x∧y=r∧x∧y,∀x,y∈H=Θμs,r.
Thus,Θμ is transitive. Hence, Θμ is a fuzzy equivalence relation on H. Next, we prove the computability of ∧,∨,⟶.
Suppose thats∧x=r∧x for some s,r,x∈H; then, we get s∧q∧x=q∧s∧x=q∧r∧x=r∧q∧x and s∨q∧x=s∧x∨q∧x=r∧x∨q∧x=r∨q∧x for any q∈H. By Lemma 1, we have s⟶q∧x=s∧x⟶q∧x∧x=r∧x⟶q∧x∧x=r⟶q∧x for any q∈H. Now,(15)Θμs,t=supμx:s∧x=r∧x,x∈H≤supμx:s∨q∧x=r∨q∧x,x∈H=Θμs∨q,t∨q,Θμs,t=supμx:s∧x=r∧x,x∈H≤supμx:s∧q∧x=r∧q∧x,x∈H=Θμs∧q,t∧q,Θμs,t=supμx:s∧x=r∧x,x∈H≤supμx:s⟶q∧x=r⟶q∧x,t∨q=Θμs⟶q,t⟶q.
Similarly, we haveΘμs,t≤Θμq∧s,q∧t, Θμs,t≤Θμq∨s,q∧t, and Θμs,t≤Θμq⟶s,q⟶t. Thus, by Theorem 18, Θμ is fuzzy congruence relation on H.Theorem 20.
LetΘbe a fuzzy congruence relation onH. A fuzzy subsetμΘdefined byμΘr=Θr,mfor allr∈His a fuzzy filter ofH.Proof.
μΘm=Θm,m=1. For all r,s∈H,(16)μΘr∨s=Θr∨s,m=Θr∨s,m∨s≥Θr,m∧Θs,s=μΘr.
Similarly,μΘr∨s≥μΘr, and also we have μΘr∨s≥μΘr∨μΘs.(17)μΘr∧s=Θr∧s,m=Θr∧s,m∧m≥Θr,m∧Θs,m=μΘr∧μΘs.
Hence,μΘ is a fuzzy filter of H.The setH/Θ=Θt|t∈H of all congruence classes is an ADL H induced by a fuzzy congruence Θ with respect to the following operations:(18)Θr∧Θs=Θr∧s,Θr∨Θs=Θr∨s,Θr⟶Θs=Θr⟶s,where Θr=s∈H:Θr,s=1 is a congruence class of r. Clearly, we get that Θr∧s=Θs∧r and Θr∨s=Θs∨r. Here for any r,s∈H, it can be observed that Θr=Θs⟺Θs,t=1, Θs∩Θr=∅ or Θs=Θr, and H/Θ,∨,∧,⟶,Θ0,Θm is a HADL.Definition 9.
For any fuzzy idealν of a HADL H and a fuzzy ideal η of H/Θ, we define the following:(1)
ξνΘs=supνt|Θs,t=1for somet∈H(2)
ωηs=supηΘt|Θs,t=1for somet∈HLemma 7.
For any fuzzy idealνofHand a fuzzy idealηofH/Θ, we have the following:(1)
ξνΘs≥νs(2)
ωηis a fuzzy ideal inH(3)
ξνis a fuzzy ideal inH/Θ(4)
ωandξare isotonesProof.
Letν,λ∈FIH and η be a fuzzy ideal of H/Θ; then,(1)
SinceΘs,s=1, ξνΘs=supνt|Θs,t=1,for somet∈H≥νs.(19)2ωη0=supηΘt|Θ0,t=1for somet∈H≥ηΘ0=1.Hence,ωη0=1. Now take s,t∈H. Then, we get(20)ωηs∧ωηt=supηΘz1|Θs,z1=1for somez1∈H∧supηΘz2|Θt,z2=1for somez2∈H=supηΘz1∧ηΘz1|Θs,z1=1,Θt,z2=1,for somez1,z2∈H≤supηΘz1∨Θz2|Θs∨t,z1∨z2=1,for somez1,z2∈H=supηΘz1∨z2|Θs∨t,z1∨z2=1,for somez1,z2∈H=ωηs∨t.Hence,ωηs∨t≥ωηs∧ωηt. Next,(21)ωηs∨ωηt=supηΘz1|Θs,z1=1for somez1∈H∨supηΘz2|Θt,z2=1,for somez2∈H=supηΘz1∨ηΘz1|Θs,z1=1,Θt,z2=1,for somez1,z2∈H≤supηΘz1∧Θz2|Θs∧t,z1∧z2=1,for somez1,z2∈H=supηΘz1∧z2|Θs∧t,z1∧z2=1,for somez1,z2∈H=ωηs∧t.Hence,ωηs∧t≥ωηs∨ωηt. Therefore, ωη is a fuzzy ideal in H.(3)
ξνΘ0=supνt|Θ0,t=1for somet∈H≥ν0=1. This shows that ξνΘ0=1. For some s,t∈H, we get(22)ωηs∧ωηt=supηΘz1|Θs,z1=1for somez1∈H∧supηΘz2|Θt,z2=1for somez2∈H=supηΘz1∧ηΘz1|Θs,z1=1,Θt,z2=1,for somez1,z2∈H≤supηΘz1∨Θz2|Θs∨t,z1∨z2=1,for somez1,z2∈H=supηΘz1∨z2|Θs∨t,z1∨z2=1,for somez1,z2∈H=ωηs∨t.Hence,ξνΘs∨Θt≥ξνΘs∧ξνΘt. Similarly,(23)ξνΘs∨ξνΘt=supνz1|Θs,z1=1,for somez1∈H∨supνz2|Θt,z2=1for somez2∈H=supνz1∨νz2|Θs,z1=1,Θt,z2=1,for somez1,z2∈H≤supνz1∧z2|Θs∧t,z1∧z2=1,for somez1,z2∈H=ξνΘs∧t=ξνΘs∧Θt.Hence,ξνΘs∧Θt≥ξνΘs∨ξνΘt.Therefore,ξν is a fuzzy ideal in H/Θ.(4)
Letν⊆λ. Then,(24)ξνΘs=supνt|Θs,t=1for somet∈H≤supλt|Θs,t=1for somet∈H=ξλΘs.
This shows thatξν⊆ξλ. Hence, ξ is isotone. Similarly, ω is isotone.Example 3.
From Example2, congruence class of H induced by a fuzzy relationΘisΘ0=0,Θr=r,s=Θs,andΘt=t. Thus, H/Θ=Θ0,Θr,Θt. Also, for any fuzzy subsetηonH, in Example2defined asηr=ηs=0.2, ηt=0.4andη0=1is a fuzzy ideal ofH. We can easily show thatξηis a fuzzy ideal ofH/Θandωξηis a fuzzy ideal ofH.Theorem 21.
The mappingξ:FIH⟶FIH/Θis a homomorphism.Proof.
Letr,s,t∈H. Consider ν,λ∈FIH and ξ:FIH⟶FIH/Θ. Clearly, ν∩λ⊆ν,λ. Since ξ is isotone, we get ξν∩λ⊆ξν∩ξλ. Furthermore, ξν∩ξλΘr(25)=supνs|Θr,s=1∧supλt|Θr,t=1=supνs|Θr,r∧s=1∧supλt|Θr∧s,s∧t=1≤supνs∧t|Θr,r∧s=1∧supλs∧t|Θr∧s,s∧t=1=supνs∧t∧λs∧t|Θr,r∧s=1,Θr∧s,s∧t=1,=supν∩λs∧t|Θr,s∧t=1=ξν∩λΘr.
Hence,ξν∩ξλ⊆ξν∩λ. Therefore, ξν∩λ=ξν∩ξλ. Since ν,λ⊆ν∨λ, we get ξν∨ξλ⊆ξν∨λ. Now, ξν∨ξλΘr(26)=supνs|Θr,s=1∨supλt|Θr,t=1=supνs|Θr,r∧s=1∨supλt|Θr∧s,s∧t=1≤supνs∧t|Θr,r∧s=1∨supλs∧t|Θr∧s,s∧t=1=supνs∧t∨λs∧t|Θr,r∧s=1,Θr∧s,s∧t=1,=supν∨λs∧t|Θr,s∧t=1=ξν∨λΘr.
This shows thatξν⊆ξλ. Hence, ξν∨λ=ξν∨ξλ. Therefore, ξ is a homomorphism.Theorem 22.
Letνandλbe fuzzy ideals ofH. The mapν⟶ωoξνis closure operator onFIH. That is,(1)
ν⊆ωoξν(2)
ν⊆λimpliesωoξν⊆ωoξλ(3)
ωoξωoξν=ωoξνProof.
(1)
For anyr,s,t∈H, we get(27)ωoξνr=ξνΘr=supνs|Θr,s=1≥νr.Hence,ν⊆ωoξν.(2)
Letν⊆λ. Since ω and ξ are isotone, we have ωoξν⊆ωoξλ.(28)3ωoξωoξνr=ξωoξνΘr=supωoξνs|Θr,s=1for somes∈H=supsupξνΘs|Θr,s=1for somes∈H=supξνΘs||Θr,s=1for somes∈H=ξνΘs=ωoξνr.Theorem 23.
For any fuzzy idealνofH/Θ,ξoων=ν.Proof.
Clearly,ν⊆ξoων. Conversely, for any r,t∈H, Θr,t=1 if and only if Θr=Θt. Then, we get(29)ξoωνΘr=supωνs|Θr,s=1,for somes∈H=supsupνΘt|Θs,t=1|Θr,s=1=supνΘt|Θr,s=1,Θs,t=1≤supνΘt|Θr,t=1=νΘr.
This impliesξoων⊆ν. Hence, ξoων=ν.Lemma 8.
For any prime fuzzy idealνofH,ωoξν=ν.Proof.
For anyr,s,t∈H, we get the following:(30)ωoξνr=supξνΘs|Θr,s=1,=supsupνt|Θs,t=1|Θr,s=1=supνt|Θr,s=1,Θs,t=1≤supνt|Θr,t=1=νr.
Hence,ωoξν⊆ν. On the other hand, from (1) in Theorem 22, we get ν⊆ωoξν.Lemma 9.
Letν,ηbe any fuzzy ideal ofH/Θ. Then, ων∩η=ων∩ωη.Theorem 24.
The following conditions hold on H:(1)
For any prime fuzzy idealμofH,ξμis a prime fuzzy ideal inH/Θ(2)
For any prime fuzzy idealμofH/Θ, ωμis a prime fuzzy ideal inHProof.
(1)
Suppose thatμ is a prime fuzzy ideal of H and for any fuzzy ideals ν,λ of H/Θ such that ν∩λ⊆ξμ. By Lemma 1 and Theorem 4.15, ων∩ωλ⊆ωoξμ=μ. Since μ is prime fuzzy ideal, ων⊆μ or ωλ⊆μ. Again by Theorem 4.9 and Theorem 4.14, ν=ξoων⊆ξμ or λ=ξoωλ⊆ξμ. Hence, ξμ is a prime fuzzy ideal of H.(2)
Suppose thatμ is a prime fuzzy ideal of H/Θ and for any fuzzy ideals η,π of H such that η∩π⊆ωμ. Since ξ is an isotone and μ is a prime fuzzy ideal of H/Θ, ξη∩ξπ=ξη∩π⊆ξoωμ=μ. This implies ξη⊆μ or ξπ⊆μ. Thus, we have η⊆ωoξη⊆ωμ or π⊆ωoξπ⊆ωμ. Hence, ωμ is a prime fuzzy ideal of H.FSpecH/Θ denotes the set of all prime fuzzy ideals of H/Θ. Then, we have the following result.Theorem 25.
The mappingα:FSpecH⟶FSpecH/Θdefined for allμ∈FSpecHbyαμ=ξμis bijective.Proof.
Letμ,ν∈H such that μ=ν. Since ξ is isotone, αμ=ξμ=ξν=αν. Thus, α is well defined. Next, we prove that α is one-to-one. Let μ,ν∈SpecH such that αμ=αν. By Lemma 8, we have μ=ωoξμ=ωoξν=ν. Hence, α is one-to-one. Let η∈FSpecH/Θ. Then, we get η=ξoωη, where ωη is a prime fuzzy ideal in H. Therefore, α is onto and so f is bijective.


## 5. Conclusion

In this paper, we discussed the concept of the spaceFp of fuzzy prime ideals of Heyting almost distributive lattice H, and it is shown that the collection Σ=Mη:ηis a fuzzy ideal ofH is a topology on Fp, where Mη=θ∈Fp|η⊈θ. The fact that the set Mxα=θ∈Fp|xα∉θ is the only compact subset of the space Fp is proved. A fuzzy congruence relation Θ on H is defined, and the homomorphism between the set of all fuzzy ideals of H and the set of all fuzzy ideals of the class of all congruence classes of H (i.e., H/Θ) is established. Furthermore, we proved that the mapping between fuzzy spectrum of H and fuzzy spectrum of H/Θ is a bijective. We will further extend these concepts to fuzzy prime spectrum of semi-Heyting algebras and fuzzy prime spectrum of skew lattices.



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