# On Fuzzy Deductive Systems of Hilbert Algebras

Authors: Gezahagne Mulat Addis; Derso Abeje Engidaw
Journal: Journal of Mathematics
 (2020)

---

## Abstract

In this paper, we study fuzzy deductive systems of Hilbert algebras whose truth values are in a complete lattice satisfying the infinite meet distributive law. Several characterizations are obtained for fuzzy deductive systems generated by a fuzzy set. It is also proved that the class of all fuzzy deductive systems of a Hilbert algebra forms an algebraic closure fuzzy set system. Furthermore, we obtain a lattice isomorphism between the class of fuzzy deductive systems and the class of fuzzy congruence relations in the variety of Hilbert algebras.


---

## Body

## 1. Introduction

The pioneering work of Zadeh [1] on fuzzy subsets of a set has been extensively applied to many scientific fields. This concept was adapted by Rosenfeld [2] to define fuzzy subgroups of a group. Since then, many authors have been studying fuzzy subalgebras of several algebraic structures such as rings, modules, vector spaces, lattices, pseudo-complemented semilattice, MS algebras, and universal algebras (see [3–8]). More recently, fuzzy ideals of a poset were studied by Alaba et al. [9] as a generalization of those fuzzy ideals of lattices. However, Swamy and Raju [10, 11] have unified all these fuzzy algebraic notions, by introducing the general theory of algebraic fuzzy systems. Some applications of fuzzy algebras and fuzzy points are also given in the literature [12–14].It was Gougen [15] who first realized that the unit interval 0,1 is insufficient to take truth values for fuzzy statements. Swamy and Swamy [16] have suggested that complete lattices satisfying the infinite meet distributive property are the most suitable candidates to take truth values of the general fuzzy statements.On the other hand, in 1966, Diego [17] introduced the notion of Hilbert algebras and deductive systems and provided various properties. The theory of Hilbert algebras and deductive systems was further developed by Busneag in a series of papers [18, 19]. Jun and Hong introduced the notion fuzzy deductive system in a Hilbert algebra in [20], and Jun constructed its extension in [21] whose truth values are in the unit interval 0,1 of real numbers.Initiated by all the above results, in this paper, we define fuzzy deductive systems of Hilbert algebras whose truth values are in a complete lattice satisfying the infinite meet distributive law and we obtain several equivalent conditions for a fuzzy set to be a fuzzy deductive system. Fuzzy deductive systems generated by a fuzzy subset are characterized in different ways. It is also proved that the set of all fuzzy deductive systems of a Hilbert algebra forms an algebraic closure fuzzy set system with respect to the pointwise ordering. Furthermore, we study fuzzy congruence relations on Hilbert algebras and we obtain a lattice isomorphism between the set of all fuzzy deductive systems and the set of all fuzzy congruence relations of a Hilbert algebra.


## 2. Preliminaries

In this section, we recall some definitions and basic properties of deductive systems of a Hilbert algebra.Definition 1.
An algebraA,∗,1 of type 2,0 is said to be a Hilbert algebra if it satisfies the following conditions:(1)
x∗y∗x=1,(2)
x∗y∗z∗x∗y∗x∗z=1,(3)
ifx∗y=1=y∗x,thenx=y,for all x,y,z∈H and 1 is a fixed element of H.
It was proved by Diego in [17] that the class K of Hilbert algebras forms a variety. The following result is also adopted from [17].Lemma 1.
Every Hilbert algebraA satisfies the following conditions:(1)
x∗x=1,(2)
1∗x=x,(3)
x∗1=1,(4)
x∗y∗z=y∗x∗z,(5)
x∗y∗z=x∗y∗x∗z,(6)
x∗y∗y∗z∗x∗z=1,(7)
x∗y∗z∗x∗z∗y=1,for anyx,y,z∈A.IfA=1, then A is called trivial. The binary relation ≤‵′ defined in a Hilbert algebra A by x≤y if and only if x∗y=1 is a partial order on A with the greatest element 1. This order is called the natural ordering on A.The concept of a deductive system on a Hilbert algebraA was introduced by Diego [17] as follows.Definition 2.
A nonempty subsetD of a Hilbert algebra A is called a deductive system on A if it satisfies the following conditions:(1)
1∈D,(2)
x∈D,(3)
x∗y∈D,implyy∈D.
In other words, a deductive system ofA is a subset of A containing 1 and closed under a deduction. One can easily check that the lattice of deductive systems of a Hilbert algebra A is closed under arbitrary intersection so that for any subset S of A, always there exists a smallest deductive system DS of A containing S. It is called the deductive system of A generated by S. Chajda [22] has characterized DS in the following way. Define sets(1)X1=S∪1,Xn=y∈A:x∈Xn−1x∗y∈Xn−1,for each positive integer n≥2, inductively. Then,(2)DS=∪Xn:n∈ℕ,where ℕ denotes the set of positive integers.
The internal structure ofDS is also characterized by Diego in [17] as follows: DS=1 if S=∅ and for S≠∅,(3)DS=y∈A:x1∗x2∗⋯∗xn∗y⋯=1for somex1,x2,…,xn∈S.
Throughout this paper,L stands for a complete lattice satisfying the infinite meet distributive law and A stands for a Hilbert algebra unless and otherwise stated.
By anL-fuzzy subset μ of A, we mean a mapping μ:A⟶L. The set μx:x∈A is called the image of μ and is denoted by Imμ. An L-fuzzy subset μ of A is called normalized or unitary if 1∈Imμ. The class of all L-fuzzy subsets of A is denoted by LA.
For eachα∈L, the α-level subset of μ, which is denoted by μα, is a subset of A given by(4)μα=x:μx≥α.
For anyL-fuzzy subset μ of A and each x∈A, it was proved in [23] that(5)μx=∨α∈L:μx≥α.
ForL-fuzzy subsets μ and σ of A, we write μ⊆σ to mean μx≤σx for all x∈A in the ordering of L. It can be easily verified that “⊆” is a partial order on the set LA and it is called the pointwise ordering.
For eachx in A and 0≠α in L, we define xα∈LA as follows:(6)xαy=αify=x,0otherwise,for each y∈A and we call it an L-fuzzy point of A.


## 3.L-Fuzzy Deductive Systems

In this section, we defineL-fuzzy deductive system in a Hilbert algebra A and investigate some of its properties.Definition 3.
AnL-fuzzy subset μ of A is said to be an L-fuzzy deductive system on A if(1)
μ1=1,(2)
μx∧μx∗y≤μy,for allx,y∈A.
We denote byFDLA the set of all L-fuzzy deductive systems of A.
The following lemma is a straightforward verification of the definition, but useful to prove results in the latter section.Lemma 2.
Letμ be an L-fuzzy deductive system on A. Then, for any x,y,z,w∈A, the following hold:(1)
μx∗y=1impliesμx≤μy,(2)
x≤yimpliesμx≤μy,(3)
x∗y∗z=1impliesμx∧μy≤μz,(4)
μx∗z∧μz∗y≤μx∗y.Proof.
(1)
Letx,y∈A such that μx∗y=1. Then, we have the following:(7)μx=μx∧1=μx∧μx∗y≤μy.(2)
Letx,y∈A such that x≤y. Then, x∗y=1 and hence, μx∗y=μ1=1. Thus, it follows from (1) that μx≤μy.(3)
Letx,y,z∈A such that x∗y∗z=1. Then, x≤y∗z. By (2), we have μx≤μy∗z. This implies that(8)μx∧μy≤μy∗z∧μy≤μz.(4)
Letx,y,z∈A. By Lemma 1, x∗z∗z∗y∗x∗y=1. Then, it follows from (3) that μx∗z∧μz∗y≤μx∗y.Lemma 3.
AnL-fuzzy subset μ of A is an L-fuzzy deductive system of A if and only if the α-level subset μα is a deductive system of A, for all α∈L.Proof.
Letμ be an L-fuzzy deductive system of A and α∈L. Then, μ1=1≥α. Thus, 1∈μα. Again, let x,y∈A such that x∈μα and x∗y∈μα. Then,(9)μx≥α,μx∗y≥α.
This implies thatα≤μx∧μx∗y≤μy and hence y∈μα. So μα is a deductive system of A.
Conversely, suppose thatμα is a deductive system of A for all α∈L. In particular, μ1 is a deductive system. Since 1∈μ1, we have μ1≥1, but μx≤1,∀x∈A, and hence μ1≤1. Thus, we have μ1=1. Again, let x,y∈A and put α=μx∧μx∗y. This implies that x∈μα and x∗y∈μα so we have y∈μα. Therefore,(10)μy≥α=μx∧μx∗y.
Hence,μ is an L-fuzzy deductive system of A.Definition 4.
For any subsetS of A and each α∈L−1, define an L-fuzzy subset αS of A by for any x∈A,(11)αSx=1ifx∈S,αotherwise.Corollary 1.
LetS be a subset of A and 1≠α∈L. Then, S is a deductive system if and only if αS is an L-fuzzy deductive system of A.Proof.
Suppose thatS is a deductive system of A and let α∈L−1. Then, 1∈S and hence, αS1=1. Let x,y∈A. If either αSx=α or αSx∗y=α, then(12)αSx∧αSx∗y=α≤αSy.
Assume thatαSx≠α and αSx∗y≠α. Then, x,x∗y∈S. By our assumption, S is a deductive system of A so that y∈S and hence,(13)αSy=1≥αSx∧αSx∗y.
Thus,αS is an L-fuzzy deductive system of A. Conversely, suppose that αS is an L-fuzzy deductive system of A for some α∈L−1. Then, αS1=1 and hence, 1∈S. Also, let x,y∈A such that x,x∗y∈S. Then, αSx=1=αSx∗y, which gives αSx∧αSx∗y=1. αS being an L-fuzzy deductive system, we obtain αSy=1. Equivalently, y∈S and hence S is a deductive system of A. This completes the proof.Remark 1.
TheL-fuzzy deductive system αS given above is called the α-level L-fuzzy deductive system of A corresponding to the deductive system S.
Let us now define a binary operation⊙, called a product, on LA as follows: for any η,σ∈LA, η⊙σx=∨ηa∧σb:x=a∗b for all x∈A.
In the following theorem, we use this product and give a Rosenfeld-type characterization forL-fuzzy subsets of A to be an L-fuzzy deductive system.Theorem 1.
μ∈LA is an L-fuzzy deductive system of A if and only if μ1=1 and for any normalized η∈LA and any σ∈LA, η⊆μ and η⊙σ⊆μ together imply σ⊆μ.Proof.
Suppose thatμ is an L-fuzzy deductive system of A. Let η be a normalized L-fuzzy subset of A and σ is an L-fuzzy subset of A such that η⊆μ and η⊙σ⊆μ. Then, ηa=1 for some a∈A. Now, for any x∈A,(14)σx=1∧σx=ηa∧σx≤η⊙σa∗x≤μa∗x=ηa∧μa∗xsinceμy≤1=ηa,∀y∈A≤μa∧μa∗x≤μx.
Therefore,σ⊆μ.
Conversely, suppose thatμ∈LA satisfies the given condition. Now, we show that μ is an L-fuzzy deductive system of A. By hypothesis, μ1=1. Let x,y∈A. Define L-fuzzy subsets η and σ of A by for any z∈A,(15)ηz=1,ifz=1,μx,ifz=x,0,otherwise,σz=1,ifz=1,μx∗y,ifz=y,0,otherwise.
Clearlyη is a normalized and η⊆μ. Moreover, for each z∈A,(16)η⊙σz=1,ifz=1,μx∧μx∗y,ifz=x∗y,0,otherwise,which is a subset of μ. Hence, by hypothesis, we have σ⊆μ. Hence, μx∗y=σy≤μy and so μx∧μx∗y≤μy. Therefore, μ is an L-fuzzy deductive system.Notation 1.
We writeF⊂fA, to say that F is a finite subset of A. The following result is another characterization for L-fuzzy deductive systems of A.Theorem 2.
AnL-fuzzy subset μ of A is an L-fuzzy deductive system of A if and only if for any F⊂fA, μx≥∧a∈Fμa for all x∈DF.Proof.
Suppose thatμ is an L-fuzzy deductive system of A. Let F⊂fA and put α=infμa:a∈F. Then, μa≥α for all a∈F and hence F⊆μα. Clearly, by Lemma 3, μα is a deductive system. Therefore, DF⊆μα and hence, μx≥α=∧a∈Fμa for all x∈DF.
Conversely, suppose thatμ satisfies the given conditions. Now, since D∅=1, by hypothesis, we have μ1≥infμa:a∈∅=1 and hence μ1=1.
Letx,y∈A. Put F=x,x∗y. Then, it is easy to see that y∈DF. Thus, by hypothesis, we have μy≥μx∧μx∗y. Therefore, μ is an L-fuzzy deductive system of A.


## 4.L-Fuzzy Deductive Systems Generated by L-Fuzzy Subset

Lemma 4.
The classFDLA is closed under arbitrary intersection.Proof.
Letμi:i∈Δ be any family of L-fuzzy deductive systems of A. Now, we claim that ∩i∈Δμi is an L-fuzzy deductive system of A. If Δ=∅, then by logical convention, ∩i∈Δμi=1¯, where 1¯ is an L-fuzzy subset of A defined by(17)1¯x=1,for all x∈A, and clearly, it is an L-fuzzy deductive system of A. Assume that Δ≠∅. Since μi1=1, for all i∈Δ, we have ∩i∈Δμi1=infμi1:i∈Δ=1. Again, let x,y∈A. Then,(18)∩i∈Δμix∧∩i∈Δμix∗y=infμix:i∈Δ∧infμix∗y:i∈Δ=infμix∧μix∗y:i∈Δ≤infμiy:i∈Δ=∩i∈Δμiy.Therefore, ∩i∈Δμi is an L-fuzzy deductive systems of A.
This lemma confirms that the classFDLA of all L-fuzzy deductive systems of A forms a closure fuzzy set system and for any L-fuzzy subset μ of A, always there exists a smallest L-fuzzy deductive system of A containing μ which we call it the L-fuzzy deductive system of A generated by μ and is denoted by DLμ.Lemma 5.
Letμ and η be L-fuzzy subsets of A. Then,(1)
μ∈FDLA,if and only ifDLμ=μ,(2)
μ⊆η⇒DLμ⊆DLη,(3)
DLDLμ=DLμ.
It can be deduced from this lemma that the mapμ↦DLμ forms a closure operator, i.e., isotone, extensive and idempotent on the lattice LA. Moreover, L-fuzzy deductive systems of A are those closed elements of LA with respect to this closure operator.Lemma 6.
For any subsetS of A and each α∈L−1, αDS=DLαS.Proof.
We show thatαDS is the smallest L-fuzzy deductive system of A containing αS. Since DS is a deductive system of A containing S, it is clear that αDS is an L-fuzzy deductive system of A containing αS. Suppose that λ is an L-fuzzy deductive system of A such that αS⊆λ. Then, λs=1 for all s∈S. Let z be any element in A. If z∉DS, then αDSz=α≤λz. Let z∈DS. Since αS⊆λ, we have S⊆λ1. This implies that DS⊆Dλ1=λ1. So z∈λ1 and hence λz=1. Therefore, αDSz=1=λz so that αDS⊆λ. Therefore, αDS=DLαS.Corollary 2.
LetS be any subset of A and χS its characteristic function. Then, DLχS=χDS.
In the following theorem, we characterizeL-fuzzy deductive systems generated by an L-fuzzy subset in terms of their level sets.Theorem 3.
For anL-fuzzy subset λ of A, let λ^ be an L-fuzzy subset of A defined by(19)λ^x=∨α∈L:x∈Dλα,for all x∈A. Then, λ^=DLλ.Proof.
We show thatλ^ is the smallest L-fuzzy deductive system of A containing λ. Let us first show that λ^ is an L-fuzzy deductive system:(1)
λ^1=∨α∈L:1∈Dλα=1.(1)
Letx,y∈A. Then, consider(20)λ^x∧λ^x∗y=∨α∈L:x∈Dλα∧∨β∈L:x∗y∈Dλβ=∨α∧β:x∈Dλα,x∗y∈Dλβ.
If we putγ=α∧β, then we obtain λα⊆λγ and λβ⊆λγ, which gives the following:(21)Dλα⊆Dλγ,Dλβ⊆Dλγ.
Therefore,x,x∗y∈Dλγ and hence y∈Dλγ. Now it follows from the above-given equality that(22)λ^x∧λ^x∗y=∨α∧β:x∈Dλα,x∗y∈Dλβ≤∨γ∈L:y∈Dλγ=λ^y.
Therefore,λ^ is an L-fuzzy deductive system of A. It is also clear to see that λ⊆λ^. Suppose that μ is any other L-fuzzy deductive system of A such that λ⊆μ. Then, it is clear that Dλα⊆μα for all α∈L. Now, for any x∈A, consider(23)λ^x=∨α∈L:x∈Dλα≤∨α∈L:x∈μα=μx.
Therefore,λ^ is the smallest L-fuzzy deductive system containing λ and hence λ^=DLλ.
In the following theorem, we characterizeL-fuzzy deductive systems generated by an L-fuzzy subset using finitely generated crisp deductive systems.Theorem 4.
For anL-fuzzy subset λ of A, let λ¯ be an L-fuzzy subset of A defined by(24)λ¯x=∨∧a∈Fλa:x∈DF,F⊂fA,for all x∈A. Then, λ¯=DLλ.Proof.
By Theorem3, it is enough if we show that λ¯=λ^. For each x∈A, let us define two sets Hx and Gx as follows:(25)Hx=∧a∈Fλa:x∈DF,F⊂fA,Gx=α∈L:x∈Dλα.
Our claim is to show that(26)∨α:α∈Hx=∨α:α∈Gx.
In one way, we show thatHx⊆Gx. If α∈Hx, then α=∧a∈Fλa and x∈DF for some finite subset F of A. That is, a∈λα, for all a∈F and x∈DF. Hence, x∈Dλα. Then, α∈Gx and hence Hx⊆Gx. In the other way, we prove that, for each α∈Gx, there exists β∈Hx such that α≤β. Let α∈Gx. Then, x∈Dλα; that is, there exist x1,x2,…,xn∈λα such that x1∗x2∗⋯xn∗x⋯=1. This implies that ∧i=1nλai≥α.
Moreover, if we putβ=∧i=1nλxi and F=x1,x2,…,xn, then F is a finite subset of A such that x∈DF. Thus, β∈Hx such that α≤β.Corollary 3.
For eachx∈A and α∈L−0, the L-fuzzy deductive system of A generated by the fuzzy point xα is characterized as(27)DLxαz=1,ifz=1,α,ifz∈Dx−1,0,otherwise,for all z∈A.
The following theorem gives a description forα−level cuts of L-fuzzy deductive systems generated by an L-fuzzy subset.Theorem 5.
Letμ be an L-fuzzy subset of A and α∈L:(28)DLμα=∪∩γ∈MDμγ:M⊆Landα≤supM.Proof.
Let us put(29)H=∪∩γ∈MDμγ:M⊆Landα≤supM.
Ifx∈H, then x∈∩γ∈MDμγ for some M⊆L with α≤supM; i.e., x∈Dμγ for all γ∈M and α≤supM. By Theorem 3, we have the following:(30)DLμx=∨β∈L:x∈Dμβ.
Therefore,DLμx≥γ for all γ∈M. This gives DLμx≥α. Thus, x∈DLμα and hence H⊆DLμα. To prove the other inequality, let us take x∈DLμα. Then,(31)∨β∈L:x∈Dμβ≥α.
If we putM=β∈L:x∈Dμβ, then M⊆L such that α≤supM and x∈Dμγ for all γ∈M. This means that x∈∩γ∈MDμγ and α≤supM. Thus, x∈H, and hence, the proof ends.Corollary 4.
For anyL-fuzzy subset μ of A and each α∈L, Dμα⊆DLμα. Moreover, if L is a chain and μ is finite valued or equivalently if μ has sup property, then the equality holds.
In the following, we give an algebraic characterization forL-fuzzy deductive systems generated by L-fuzzy subset. Let μ be an L-fuzzy subset of A and ℕ be a set of positive integers. For each n∈ℕ, let us define L-fuzzy subsets μn of A as follows:(32)μ1x=1,ifx=1,μx,otherwise,for all x∈A, and for each n≥1 and each y∈A,(33)μn+1y=∨μnx∧μnx∗y:x∈A.
Then, it is immediate from the definition thatμn1=1 for all n≥1. Moreover, one can deduce that(34)μ1⊆μ2⊆μ3⊆⋯⊆μn⊆⋯Theorem 6.
Letμ be an L-fuzzy subset of A. Then, DLμ=∪μn:n∈N.Proof.
Putμ¯=∪μn:n∈N. Now, we claim that μ¯ is the smallest L-fuzzy deductive system containing μ. Now, let x∈A. Then,(35)μ¯x=∨μnx:n∈ℕ≥μ1x≥μx.
Thus, we haveμ⊆μ¯.
Sinceμ¯1≥μ11=1, we have μ¯1=1. Let x,y∈A and x≠1.
Then(36)μ¯x∧μ¯x∗y=∨μnx:n∈ℕ∧∨μmx∗y:m∈ℕ=∨μnx∧μmx∗y:n,m∈ℕ≤∨μkx∧μkx∗y:wherek=maxm,n,n,m∈ℕ=μk+1y≤∨μny:n∈ℕ=μ¯y.
Soμ¯ is an L-fuzzy deductive system of A. Again, let θ be any L-fuzzy deductive system of A such that μ⊆θ. We show that μn⊆θ for all n∈ℕ. We use induction on n. Let x∈A. Then, if x=1, then θ1=1=μ1x. If x≠1, then θx≥μx=μ1x. This implies that μ1⊆θ and hence it holds for n=1. Let n>1 and assume the result to be true for some n−1, i.e., μn−1⊆θ. Now, for any x∈A, we have(37)μnx=∨μn−1a∧μn−1a∗x≤∨θa∧θa∗x≤θx.
Hence,μn⊆θ. Thus, by mathematical induction, we have μn⊆θ for all n∈ℕ. Hence, μ¯=∪μn:n∈N⊆θ. Therefore, μ¯=DLμ.
The following is also another algebraic characterization ofL-fuzzy deductive system generated by an L-fuzzy subset of A.Theorem 7.
Letλ be an L-fuzzy subset of A. Then, the L-fuzzy subset λ¯ defined by: λ¯1=1 and for x≠1,(38)λ¯x=∨∧i=1nλxi:x1∗x2∗⋯xn∗x⋯=1,x1,…,xn∈A,is the L-fuzzy deductive system of A generated by λ.
The following theorem gives a method of constructingL-fuzzy deductive systems using crisp deductive systems in the sense of [10].Theorem 8.
Suppose thatHαα∈L is a family of deductive systems of A such that(39)∩α∈MHα=HsupM,for all M⊆L. Then, there is a unique L-fuzzy deductive system μ of A for which μα=Hα for all α∈L. Moreover, every L-fuzzy deductive system of A is obtained in this way only.Proof.
We first show that the mapα↦Hα is antitone, in the sense that, for each α,β∈L,α≤β⟹Hβ⊆Hα. Let α,β∈L such that α≤β. Put M=α,β. Then, supM=β. By our hypothesis Hα∩Hβ=HsupM=Hβ. Therefore, Hβ⊆Hα and hence, the map α↦Hα is antitone. Define an L-fuzzy subset μ of A by(40)μx=∨α∈L:x∈Hα,for all x∈A. Clearly, μ is well defined. Our aim is to show that μα=Hα for all α∈L. The inclusion Hα⊆μα follows easily from the definition of μ. To prove the other inclusion, let x∈μα. Then, μx≥α, i.e.,(41)∨γ∈L:x∈Hγ≥α.
If we putM=γ∈L:x∈Hγ, then M⊆L such that α≤supM and x∈Hγ for all γ∈M, i.e.,(42)x∈∩γ∈MHγ.
By our assumption, it follows thatx∈HsupM. Since α≤supM and the map α↦Hα is antitone, we obtain x∈Hα. Thus, Hα=μα. This means that μ is an L-fuzzy subset of A for which its α-level sets are Hα’s. Each Hα being a deductive system of A, it follows from Lemma 3 that μ is an L-fuzzy deductive system. The uniqueness of μ follows from the fact μα=Hα for all α∈L. The converse is straightforward.


## 5. Lattice ofL-Fuzzy Deductive Systems

Let us first recall some important results from [24]. Let C be a nonempty collection of fuzzy subsets of a nonempty set X.Definition 5.
C is said to be a closure system in LX if it is closed under arbitrary intersection of fuzzy sets; i.e., if for any subcollection D of fuzzy subsets of X in C, it holds that(43)∩μ∈Dμ∈C.
A closure system of fuzzy sets is also known as the “Moor family” of fuzzy sets.Remark 2.
IfC is a closure system in LX, then it contains the fuzzy subset 1X of X. This is because the fuzzy set 1X can be expressed as the infimum of an empty collection of fuzzy subsets of X.Theorem 9.
IfC is a closure system in LX, then C,⊆ forms a complete lattice, where ⊆ is the inclusion ordering of fuzzy sets.Definition 6.
A nonempty collectionC of fuzzy subsets of X is called inductive if every nonempty chain in C has a supremum in C.Definition 7.
A closure systemC in LX is said to be an algebraic closure fuzzy sets system if it is inductive.
As observed in the previous section, the collectionFDLA of all L-fuzzy deductive systems of A is closed under arbitrary intersection of fuzzy sets. Thus, FDLA,⊆ forms a closure fuzzy set system and hence it is a complete lattice. The following theorem summarizes this.Theorem 10.
The collectionFDLA forms a complete lattice with respect to the pointwise ordering, where the infimum and supremum of any of its subfamily μi:i∈Δ, respectively, are(44)infi∈Δμi=∩i∈Δμi,supi∈Δμi=DL∪i∈Δμi.Corollary 5.
The setFDLA forms the algebraic closure fuzzy set system on A with respect to the pointwise ordering of fuzzy sets.Proof 13.
It is enough to show thatFDLA is inductive in LA. Let μii∈Δ be a chain in FDLA. Let us put(45)η=∪i∈Δμi.
We show thatη is an L-fuzzy deductive system of A. Clearly, η1=1. Let x,y∈A. First observe that, for each i,j∈Δ, there exists k∈i,j such that μi⊆μK and μj⊆μk. Now, consider the following:(46)ηx∧ηx∗y=∨i∈Δμix∧∨j∈Δμjx∗y=∨i,j∈Δμix∧μjx∗y≤∨k∈Δμkx∧μkx∗y≤∨k∈Δμky=ηy.
Thereforeη is an L-fuzzy deductive system of A.Remark 3.
IfL is an algebraic lattice, then FCaA forms an algebraic lattice.


## 6.L-Fuzzy Congruences

By anL-fuzzy relation on A, we mean an L-fuzzy subset of A×A. For any α∈L and an L-fuzzy relation Θ on A, the set(47)Θα=x,y∈A×A:Θx,y≥α,is called the α-level relation of Θ on A.Definition 8.
AnL-fuzzy relation Θ on A is said to be(1)
Reflexive ifΘx,x=1 for all x∈A(2)
Symmetric ifΘx,y=Θy,x for all x,y∈A(3)
Transitive if for eachx,z∈A, Θx,z≥Θx,y∧Θy,z for all y∈A
A reflexive, symmetric, and transitiveL-fuzzy relation on A is called an L-fuzzy equivalence relation on A.Definition 9.
AnL-fuzzy relation Θ on A is said to be compatible, if(48)Θa∗c,b∗d≥Θa,b∧Θc,d,for all a,b,c,d∈A.
This fact is often expressed in the way thatΘ is said to have the substitution property with respect to the binary operation ∗.Definition 10.
A compatibleL-fuzzy equivalence relation on A is called an L-fuzzy congruence relation on A.
We denote byFConA the class of all L-fuzzy congruence relations on A, and it is clear that FConA is a complete lattice. For any a∈L and Θ∈FConA, define a fuzzy subset aΘ of A by(49)aΘx=Θa,x,for all x∈A. We call aΘ an L-fuzzy congruence class of Θ determined by a, and in particular, 1Θ is called the kernel of Θ. Let us put(50)AΘ=aΘ:a∈A,and define a binary operation ⊛ on A/Θ by(51)aΘ⊛bΘ=a∗bΘ.
Then, it is routine to verify thatA/Θ,⊛,1Θ is a Hilbert algebra and it is called the quotient Hilbert algebra of A modulo Θ.Lemma 7.
For eachΘ∈FConA, 1Θ is an L-fuzzy deductive system of A.Proof.
SinceΘ is reflexive, we have 1Θ1=Θ1,1=1. Let x,y∈A. Then,(52)1Θx∧1Θx∗y=Θ1,x∧Θ1,x∗y=Θ1,x∧Θy,y∧Θ1,x∗y≤Θy,x∗y∧Θ1,x∗y≤Θ1∗y,x∗y∗x∗y≤Θy,1=1Θy.
Thus,1Θ is an L-fuzzy deductive system of A.Remark 4.
It is sufficient forΘ to be reflexive and compatible only to obtain the result of Lemma 7.Lemma 8.
For eachΘ,Φ∈FConA,(53)1Θ=1Φ⟹Θ=Φ.Proof.
LetΘ,Φ∈FConA such that 1Θ=1Φ. Then, it is clear that Θα1=Φα1 for all α∈L. Let x,y∈A×A, and put Θx,y=α. Then, it is clear that Θα is a congruence relation on A containing x,y and hence x∗y,1 and y∗x,1 belong to Θα. This imply that x∗y,y∗x∈Θα1=Φα1 and so the following hold in the quotient Hilbert algebra A/Φα:(54)ΦαxΦαy=Φαx∗y=Φα1,ΦαyΦαx=Φαy∗x=Φα1.
A/ΦαSince the binary operations x∗yandy∗x form a Go¨del equivalence system in the variety of Hilbert algebras, it follows that Φαx=Φαy. Therefore, x,y∈Φα and thus, Φx,y≥α=Θx,y. Similarly, we can show that Θx,y≥Φx,y so that the equality holds.Corollary 6.
For eachΘ,Φ∈FConA,(55)1Θ⊆1Φ⟺Θ⊆Φ.Proof.
LetΘ,Φ∈FConA such that 1Θ⊆1Φ. Then, it is clear that Θα1⊆Φα1 for all α∈L. Let x,y∈A, and put Θx,y=α. Then, it is clear that Θα is a congruence relation on A containing x,y and hence x∗y,1 and y∗x,1 belong to Θα. This imply that x∗y,y∗x∈Θα1⊆Φα1 and so the following hold in the quotient Hilbert algebra A/Φα:(56)ΦαxΦαy=Φαx∗y=Φα1,ΦαyΦαx=Φαy∗x=Φα1.
A/ΦαSince the binary operations x∗yandy∗x form a Go¨del equivalence system in the variety of Hilbert algebras, it follows that Φαx=Φαy. Therefore, x,y∈Φα and hence Φx,y≥α=Θx,y. Thus Θ⊆Φ.Corollary 7.
For eachΘ,Φ∈FConA,(1)
1Θ∩Φ=1Θ∩1Φ,(2)
1Θ∨Φ=1Θ∨1Φ.
For anyL-fuzzy subset μ of A, let us define an L-fuzzy relation Θμ of A as follows:(57)Θμa,b=μa∗b∧μb∗a,for all a,b∈A.Lemma 9.
If the fuzzy relationΘμ is reflexive, then the kernel of Θμ is μ.Proof.
LetΘμ be reflexive. Then, for any x∈A, 1=Θμx,x=μx∗x∧μx∗x=μ1. Thus, μ1=1. Now, since for any x∈A,(58)1Θμx=Θμ1,x=μ1∗x∧μx∗1=μx∧μ1=μx∧1=μx.
Therefore, the kernel ofΘμ is μ.Lemma 10.
IfΘμ be an L-fuzzy equivalence relation on A, then for any a,b∈A with a≤b, it holds that(59)μb≥μa∧μb∗a.Proof.
LetΘμ be an L-fuzzy equivalence relation on A and a,b∈A such that a≤b. Then, a∗b=1. By Lemma 9, we have μ1=1=μa∗b and 1Θμ=μ. Thus,(60)μa∧μb∗a=μa∧μb∗a∧μa∗b=μa∧Θμa,b=1Θμa∧Θμa,b=Θμ1,a∧Θμa,b≤Θμ1,b=1Θμb=μb.
This proves the result.Theorem 11.
Ifμ is an L-fuzzy deductive system of A, then Θμ is an L-fuzzy congruence relation on A whose kernel is μ.Proof.
Letμ be an L-fuzzy deductive system of A. Then, for any x∈A, Θμx,x=μx∗x∧μx∗x=μ1=1. Thus, Θμ is reflexive. Again, for any x,y∈A, Θμx,y=μx∗y∧μy∗x=μy∗x∧μx∗y=Θμy,x. Thus, Θμ is symmetric. Let x,y,z∈A. Now,(61)Θμx,y∧Θμy,z=μx∗y∧μy∗x∧μy∗z∧μz∗y=μx∗y∧μy∗z∧μz∗y∧μy∗x≤μx∗z∧μz∗xby, Lemma2=Θμx,z.
Thus,Θμ is transitive. Let x,y,a,b∈A. Now using Lemma 1 and Lemma 2, we have(62)Θμx,y∧Θμa,b=μx∗y∧μy∗x∧μa∗b∧μb∗a=μy∗x∧μa∗b∧μx∗y∧μb∗a≤μx∗a∗y∗a∧μy∗a∗y∗b∧μy∗b∗x∗b∧μx∗b∗x∗a≤μx∗a∗y∗b∧μy∗b∗x∗a=Θμx∗a,y∗b.
Thus,Θμ is compatible. Therefore, Θμ is an L-fuzzy congruence relation on A and its kernel, by Lemma 9, is μ.Corollary 8.
The mapΘ↦1Θ is a lattice isomorphism between FConA and FDLA.Proof.
Lemma8 confirms that the map is one-one. Moreover, for any μ∈FDLA, it is proved in the above theorem that Θμ∈FConA such that 1Θμ=μ. Then, the map Θ⟼1Θ is onto. Furthermore, it is proved in Corollary 7 that this map is a lattice homomorphism. Therefore, it is a lattice isomorphism between FConA and FDLA.Theorem 12.
For a Hilbert Algebra, the latticeFDLA,⊆ is distributive with respect to the pointwise ordering ⊆′′.



---

*Source: 6910726-2020-09-01.xml*