# L-Fuzzy Prime Ideals in Universal Algebras

Authors: Berhanu Assaye Alaba; Gezahagne Mulat Addis
Journal: Advances in Fuzzy Systems
 (2019)

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## Abstract

In this paper, the notions ofL-fuzzy prime ideals and maximal L-fuzzy ideals of universal algebras are introduced by applying the general theory of algebraic fuzzy systems.


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## Body

## 1. Introduction

The commutator (or the product) of idealsI and J of a ring R, written as IJ, is the ideal of R generated by all products ij and ji, with i∈I and j∈J; i.e., (1)IJ=x∈R:x=Σi=1nyizi,yi∈I,zi∈J. In 1984, H. P. Gumm and A. Ursini [1] have studied the commutator (or the product) of ideals in a more general context. They have defined and characterized the commutator of ideals in universal algebras by the use of commutator terms. Later on, A. Ursini [2] applied this product to study prime ideals of universal algebras. P. Agliano [3] then studied the prime spectrum of universal algebras.The concept of fuzzy sets was first introduced by Zadeh [4] and this concept was adapted by Rosenfeld [5] to define fuzzy subgroups. Since then, many authors have been studying fuzzy subalgebras of several algebraic structures (see [6–9]). As suggested by Gougen [10], the unit interval [0,1] is not sufficient to take the truth values of general fuzzy statements. U. M. Swamy and D. V. Raju [11, 12] studied the general theory of algebraic fuzzy systems by introducing the notion of a fuzzy L- subset of a set X corresponding to a given class L of subsets of X having truth values in a complete lattice satisfying the infinite meet distributive law. Swamy and Swamy [13] defined the commutator (or the product) of L-fuzzy ideals μ and σ of a ring R as follows: (2)μ,σx=⋁⋀i=1nμyi∧σzi:x=Σi=1nyizi for all x∈R. They have used this commutator to define L-fuzzy prime ideals of rings.In [14], we have studied L-fuzzy ideals in universal algebras having a definable constant denoted by 0, where L is a complete distributive lattice satisfying the infinite meet distributive law. We gave a necessary and sufficient condition for a class of algebras to be ideal-determined. In the present paper, we define the commutator of L-fuzzy ideals in universal algebras and investigate some of its properties. Moreover, we study L-fuzzy prime ideals and maximal L-fuzzy ideals in universal algebras as a generalization of L-fuzzy prime ideals in those well-known structures: in semigroups [15], in rings [13], in semirings [16], in ternary semirings [17], in Γ-rings [18], in modules [19], in lattices [9], and in other algebraic structures.


## 2. Preliminaries

This section contains some definitions and results which will be used in this paper. For those elementary concepts on universal algebras we refer to [20, 21]. Throughout this paper A∈K, where K is a class of algebras of a fixed type Ω, and we assume that there is an equationally definable constant in all algebras of K denoted by 0. For a positive integer n, we write a→ to denote the n-tuple 〈a1,a2,…,an〉∈An.Definition 1 ([1]).
A termP(x→,y→) is said to be an ideal term in y→ if and only if P(x→,0→)=0.Definition 2 ([1]).
A nonempty subsetI of A is called an ideal of A if and only if P(a→,b→)∈I for all a→∈An,b→∈Im and any ideal term P(x→,y→) in y→.We denote the class of all ideals ofA, by I(A).Definition 3 ([1]).
A classK of algebras is called ideal-determined if every ideal I is the zero congruence class of a unique congruence relation denoted by Iδ. In this case the map I↦Iδ defines an isomorphism between the lattice of ideals and congruences on A.Definition 4 ([1, 2]).
A termt(x→,y→,z→) is said to be a commutator term in y→,z→ if and only if it is an ideal term in y→ and an ideal term in z→.Definition 5 ([1]).
In an ideal-determined variety, the commutator[I,J] of ideals I and J is the zero congruence class of the commutator congruence [Iδ,Jδ].It is characterized in [1] as follows.Theorem 6 ([1, 2]).
In an ideal-determined variety,(3)I,J=ta→,i→,j→:a→∈An,i→∈Imandj→∈Jkwheretx→,y→,z→isacommutatorterminy→,z→.For subsetsH,G of A, [H,G] denotes the product [〈H〉,〈G〉]. In particular, for a,b∈A, [〈a〉,〈b〉] is denoted by [a,b].Definition 7 ([2]).
A proper idealP of A is called prime if and only if for all I,J∈I(A)(4)I,J⊆P⇒eitherI⊆PorJ⊆P.Theorem 8 ([2]).
A proper idealP of A is prime if and only if (5)a,b⊆P⇒eithera∈Porb∈P for all a,b∈A.Throughout this paperL=(L,∧,∨,0,1) is a complete Brouwerian lattice; i.e., L is a complete lattice satisfying the infinite meet distributive law. By an L-fuzzy subset of A, we mean a mapping μ:A→L. For each α∈L, the α-level set of μ denoted by μα is a subset of A given by the following. (6)μα=x∈A:α≤μx For L-fuzzy subsets μ and ν of A, we write μ≤ν to mean μ(x)≤ν(x) in the ordering of L.Definition 9 ([22]).
For eachx∈A and 0≠α in L, the L-fuzzy subset xα of A given by (7)xαz=αifz=x0otherwise is called the L-fuzzy point of A. In this case x is called the support of xα and αis its value.For anL-fuzzy subset μ of A and an L-fuzzy point xα of A, we write xα∈μ whenever μ(x)≥α.Definition 10 ([14]).
AnL-fuzzy subset μ of A is said to be an L-fuzzy ideal of A if and only if the following conditions are satisfied: (1)
μ(0)=1, and(2)
IfP(x→,y→) is an ideal term in y→ and a→∈An,b→∈Am; then (8)μPa→,b→≥μmb→.Note that anL-fuzzy subset μ of A satisfying the two conditions in the above definition can be regarded as a normal L-fuzzy ideal in the sense of Jun et al. [23].Theorem 11 ([14]).
A classK of algebras is ideal-determined if and only if every L-fuzzy ideal μ is the zero fuzzy congruence class of a unique L-fuzzy congruence relation denoted by Θμ.


## 3. The Commutator ofL-Fuzzy Ideals

In this section, we define the commutator (or the product) ofL-fuzzy ideals in universal algebras. It is observed in [14] that an L-fuzzy subset μ of A is an L-fuzzy ideal of A if and only if every α-level set of μ is an ideal of A. Here we define the commutator of L-fuzzy ideals using their level ideals.Definition 12.
The commutator ofL-fuzzy ideals μ and σ of A denoted by [μ,σ] is an L-fuzzy subset of A defined by (9)μ,σx=⋁α∧β:α,β∈L,x∈μα,σβ for all x∈A.For eachα,β, and λ in L with λ=α∧β, the following can be verified. (10)x∈μα,σβ⇒x∈μλ,σλ So the commutator of L-fuzzy ideals can be equivalently redefined as follows. (11)μ,σx=⋁λ∈L:x∈μλ,σλ The following lemmas can be verified easily.Lemma 13.
For anyL-fuzzy ideals μ and σ of A, [μ,σ] is an L-fuzzy ideal of A such that (12)μ,σ≤μ∩σ.Lemma 14.
For any idealsI and J of A(13)χI,J=χI,χJ.In the following theorem, we give an algebraic characterization for the commutator ofL-fuzzy ideals.Theorem 15.
For eachx∈A and L-fuzzy ideals μ and σ of A(14)μ,σx=⋁μmb→∧σkc→:x=ta→,b→,c→,wherea→∈An,b→∈Am,c→∈Ak,andtx→,y→,z→isacommutatorterminy→,z→.Proof.
For eachx∈A, let us define two sets Hx and Gx as follows. (15)Hx=μmb→∧σkc→:x=ta→,b→,c→,wherea→∈An,b→∈Am,c→∈Akandtx→,y→,z→isacommutatorterminy→,z→Gx=α∈L:x∈μα,σα Clearly both Hx and Gx are subsets of L. Our claim is to see the following. (16)⋁α:α∈Hx=⋁α:α∈GxOne way of proof is to show that Hx⊆Gx. α∈Hx implies that α=μm(b→)∧σk(c→), where x=t(a→,b→,c→) for some a→∈An,b→∈Am,c→∈Ak, and some commutator term t(x→,y→,z→) in y→,z→. That is, b→∈(μα)m and c→∈(σα)k so that x∈[μα,σα]. Then α∈Gx and hence Hx⊆Gx. Another way is to prove that, for each α∈Gx, there exists β∈Hx such that α≤β. If α∈Gx, then x∈[μα,σα] so that x=t(a→,b→,c→) for some a→∈An,b→∈(μα)m, and c→∈(σα)k, where t(x→,y→,z→) is a commutator term in y→,z→. That is, μm(b→)≥α and σk(c→)≥α. If we put β=μm(b→)∧σk(c→), then β≥α and β∈Hx. This completes the proof.Notation 16.
We writeF⊂⊂A, to say that F is a finite subset of A.Theorem 17.
For eachx∈A and L-fuzzy ideals μ and σ of A(17)μ,σx=⋁⋀a∈E,b∈Fμa∧σb:x∈E,F,E,F⊂⊂A.Proof.
For eachx∈A, let us take the set Gx as in Theorem 15 and define a set Hx as follows. (18)Hx=⋀a∈E,b∈Fμa∧σb:x∈E,F,E,F⊂⊂A Our claim is to show the following. (19)⋁α:α∈Hx=⋁α:α∈GxOne way of proof is to show that Hx⊆Gx. If α∈Hx, then (20)α=⋀a∈E,b∈Fμa∧σb where E and F are finite subsets of A such that x∈[E,F]. That is, μ(a)∧σ(b)≥α for all a∈E and all b∈F. Then E⊆μα and F⊆σα so that [E,F]⊆[μα,σα]. Thus x∈[μα,σα] and hence α∈Gx. Therefore Hx⊆Gx. Another way is to prove that, for each α∈Gx, there exists β∈Hx such that α≤β. If α∈Gx, then x∈[μα,σα]. Therefore, x=t(a→,b→,c→) for some a→∈An,b→=〈b1,b2,…,bm〉∈(μα)m, and c→=〈c1,c2,…,ck〉∈(σα)k, where t(x→,y→,z→) is a commutator term in y→,z→. That is, (21)μmb→=μb1∧⋯∧μbm≥α and (22)σkc→=σc1∧⋯∧σcm≥α. If we put E={b1,b2,…,bm} and F={c1,c2,…,ck}, then E and F are both finite subsets of A such that x∈[E,F]. Moreover, if we take (23)β=⋀a∈E,b∈Fμa∧σb then β∈Hx such that α≤β. This completes the proof.Definition 18.
For eachμ∈FI(A), we define by induction (24)μ1=μ=μ1;μn+1=μn,μnandμn+1=μn,μ. An L-fuzzy ideal μ of A will be called fuzzy nilpotent (resp., fuzzy solvable) if μn=χ(0) (resp., μ(n)=χ(0)) for some n∈Z+.Lemma 19.
AnL-fuzzy subset μ of A is L-fuzzy nilpotent (resp., L-fuzzy solvable) if and only if μα is nilpotent (resp., solvable) for all α∈L.


## 4.L-Fuzzy Prime Ideals

In this section we defineL-fuzzy prime ideals and investigate some of their properties.Definition 20.
A nonconstantL-fuzzy ideal μ of A is called an L-fuzzy prime ideal if and only if (25)ν,σ≤μ⇒eitherν≤μorσ≤μ for all ν,σ∈FI(A).Notation 21.
ForL-fuzzy points xα and yβ of A, we denote [〈xα〉,〈yβ〉] by [xα,yβ].In the following theorem we characterizeL-fuzzy prime ideals using L-fuzzy points.Theorem 22.
A nonconstantL-fuzzy ideal μ of A is L-fuzzy prime if and only if for any L-fuzzy points xα and yβ of A(26)xα,yβ≤μeitherxα∈μoryβ∈μ.Proof.
Suppose thatμ satisfies the condition: (27)xα,yβ≤μ⇒eitherxα∈μoryβ∈μ for all L-fuzzy points xα and yβ of A. Let σ and θ be L-fuzzy ideals of A such that [σ,θ]≤μ. Suppose if possible that σ≰μ and θ≰μ. Then there exist x,y∈A such that σ(x)≰μ(x) and θ(y)≰μ(y). If we put α=σ(x) and β=θ(y), then xα and yβ are fuzzy points of A such that xα∈σ, but xα∉μ, and yβ∈θ, but yβ∉μ, so that [xα,yβ]≤[σ,θ]≤μ, but xα∉μ and yβ∉μ. This contradicts our hypothesis. Thus either σ≤μ or θ≤μ. Therefore μ is prime. The other way is clear.Theorem 23.
A nonconstantL-fuzzy ideal μ is an L-fuzzy prime ideal if and only if Img(μ)={1,α}, where α is a prime element in L and the set μ∗={x∈A:μ(x)=1} is a prime ideal of A.Proof.
Suppose thatμ is a prime L-fuzzy ideal. Clearly 1∈Img(μ), and since μ is nonconstant, there is some a∈A such that μ(a)<1. We show that μ(a)=μ(b) for all a,b∈A-μ∗. Let a,b∈A such that μ(a)<1 and μ(b)<1. Let us define L-fuzzy subsets σ and θ of A as follows: (28)σx=1ifx∈a0otherwise and (29)θx=1ifx=0μaotherwise for all x∈A. Then it can be verified that both σ and θ are L-fuzzy ideals of A. Moreover, for each x∈A we have (30)σ,θx=1ifx=0μaifx∈a-00otherwise so that [σ,θ]≤μ, but σ(a)=1>μ(b), so σ≰μ. Since μ is L-fuzzy prime, we get θ≤μ, which gives θ(b)≤μ(b); that is, μ(a)≤μ(b). Similarly it can be verified that μ(b)≤μ(a) so that μ(a)=μ(b) for all a,b∈A-μ∗. Thus Img(μ)={1,α} for some α≠1 in L. It remains to show that α is a prime element in L. Let β,γ∈L such that β∧γ≤α. Consider L-fuzzy subsets β¯ and γ¯ of A defined by (31)β¯x=1ifx=0βotherwise and (32)γ¯x=1ifx=0γotherwise for all x∈A. Then β¯ and γ¯ are both L-fuzzy ideals of A such that [β¯,γ¯]≤μ. Since μ is L-fuzzy prime, either β¯≤μ or γ¯≤μ so that either β≤α or γ≤α. Hence α is prime in L. Next we show that the level ideal μ∗ is prime. Put P=μ∗ and let I and J be ideas of A such that [I,J]⊆P. Then χ[I,J]≤χP≤μ. That is, [χI,χJ]≤μ. Since μ is L-fuzzy prime, either χI≤μ or χJ≤μ, implying that either I⊆P or J⊆P. Therefore P is prime. Conversely suppose that Img(μ)={1,α}, where α is a prime element in L and P=μ∗ is a prime ideal of A. Let σ and θ be L-fuzzy ideals of A such that [σ,θ]≤μ. Suppose if possible that there exist x,y∈A such that σ(x)≰μ(x) and θ(y)≰μ(y). Since μ is 2-valued, μ(x)=μ(y)<1 so that both x and y do not belong to P. Since P is prime, there exists z∈[x,y] such that z∉P; that is, μ(z)=α. Otherwise, if [x,y]⊆P, then either x∈P or y∈P. As z∈[x,y], z=t(a→,b→,c→) for some a→∈An, b→∈〈x〉m,c→∈〈y〉k, where t(u→,v→,w→) is a commutator term in v→,w→. Now consider the following. (33)α=μz≥σ,θz≥σmb→∧θkc→≥σx∧θy That is, σ(x)∧θ(y)≤α. Since α is a prime element in L, it follows that either σ(x)≤α=μ(x) or θ(y)≤α=μ(y), which is a contradiction. Therefore μ is L-fuzzy prime.LetP be a prime ideal of A and α be a prime element in L. Consider an L-fuzzy subset αP of A defined by (34)αPx=1ifx∈Pαotherwise for all x∈A. The above theorem confirms that L-fuzzy prime ideals of A are only of the form αP. This establishes a one-to-one correspondence between the class of all L-fuzzy prime ideals of A and the collection of all pairs (P,α) where P is a prime ideal in A and α is a prime element in L.Corollary 24.
LetP be an ideal of A and α a prime element in L. Then P is a prime ideal if and only if αP is an L-fuzzy prime ideal.Theorem 25.
Ifμ is an L-fuzzy prime ideal of A, then (35)μa∨μb≥⋀μx:x∈a,bfor all a,b∈A.Proof.
We use proof by contradiction. Suppose if possible the following.(36)⋀μx:x∈a,b≰μa∨μb Then μ(x)≰μ(a)∨μ(b) for all x∈[a,b]. Since μ is prime, by Theorem 23 there exists a prime element α<1 in L such that μ(a)=α=μ(b) and μ(x)=1 for all x∈[a,b], so [a,b]⊆μ∗. Since μ∗ is a prime ideal of A (see Theorem 23), we get that either a∈μ∗ or b∈μ∗. This is a contradiction. Thus the result holds.Theorem 26.
IfImg(μ)={1,α}, where α is a prime element in L and μ satisfies the condition: (37)μa∨μb≥⋀μx:x∈a,b for all a,b∈A, then μ is L-fuzzy prime.It is natural to ask ourselves, does every algebra inK have L-fuzzy prime ideals? Of course, probably no. In the following theorem we give a sufficient condition for an algebra A to have L-fuzzy prime ideals.Theorem 27.
LetA be an algebra satisfying the following. (38)x∈x,xforallx∈AIf a∈A and μ is an L-fuzzy ideal of A such that μ(a)≤α where α is an irreducible element in L, then there exists an L-fuzzy prime ideal θ of A such that (39)μ≤θandθa≤α.Proof.
PutF={σ∈FI(A):μ≤σandσ(a)≤α}. Clearly μ∈F so that F is nonempty and hence it forms a poset under the inclusion ordering of L-fuzzy sets. By applying Zorn’s lemma we can choose a maximal element, say θ, in F. Now it is enough to show that θ is prime. Suppose not. Then there exist L-fuzzy ideals σ and ν of A such that [σ,ν]≤θ but σ≰θ and ν≰θ. Put θ1=θ∨σ and θ2=θ∨ν. Then θ1 and θ2 are L-fuzzy ideals of A such that θ≨θ1 and θ≨θ2. By the maximality of θ in F both θ1 and θ2 do not belong to F. Thus (40)θ1a≰αandθ2a≰α.Since a∈[a,a] and α is ∧-irreducible element in L, we get [θ1,θ2](a)≰α, so θ(a)≰α, which is a contradiction. Therefore θ is prime.IfA is a nontrivial algebra such that a∈[a,a] for all a∈A, then it can be deduced from the above theorem that L-fuzzy prime ideals exist in A.Definition 28.
AnL-fuzzy subset λ of A is called an L-fuzzy m-system (resp., an L-fuzzy n-system) if for all a,b∈A, there exists x∈[a,b] (resp., there exists x∈[a,a]) such that (41)λx≥λa∧λbresp.λx≥λa.Lemma 29.
AnL-fuzzy subset λ of A is an L-fuzzy m-system (resp., a fuzzy n-system) if and only if the level set λα is an m-system (resp., an n-system) for all α∈L.Theorem 30.
Letμ be an L-fuzzy ideal of A and λ an L-fuzzy m-system such that μ∩λ≤α, where α is an irreducible element in L. Then there exists L-fuzzy prime ideal θ of A such that (42)μ≤θandθ∩λ≤α.Proof.
PutF={σ∈FI(A):μ≤σandσ∩λ≤α}. Clearly μ∈F so that F is nonempty, and hence it forms a poset under the inclusion ordering of L-fuzzy sets. By applying Zorn’s lemma we can choose a maximal element, say θ, in F. Now it is enough to show that θ is prime. Suppose not. Then there exist L-fuzzy ideals σ and ν of A such that [σ,ν]≤θ, but σ≰θ and ν≰θ. Put θ1=θ∨σ and θ2=θ∨ν. Then θ1 and θ2 are L-fuzzy ideals of A such that θ≨θ1 and θ≨θ2. By the maximality of θ in F both θ1 and θ2 do not belong to F so there exist a,b∈A such that (43)θ1∩λa≰αandθ2∩λb≰α. Since [σ,ν]≤θ, we have [θ1,θ2]≤θ. If x∈[a,b], then x=t(c→,u→,v→) for some c→∈An, u→∈〈a〉m, v→∈〈b〉k, and some commutator term t(x→,y→,z→) in y→,z→. Then for each x∈[a,b] the following holds. (44)θx≥θ1,θ2x≥θ1a∧θ2b Also we have θ1(a)∧θ2(b)≰α, which gives that θ(x)≰α. However, since θ∩λ≤α and α is an irreducible element in L, we get that λ(x)≤α for all x∈[a,b]. This contradicts that λ is a fuzzy m-system. Therefore θ is prime.For a nontrivial algebraA, to have an L-fuzzy m-system is a sufficient condition for A to possess L-fuzzy prime ideals.


## 5. Maximal FuzzyL-Ideals

A maximalL-fuzzy ideal of A is a maximal element in the collection of all nonconstant L-fuzzy ideals of A under the pointwise partial ordering of L-fuzzy sets.An element1≠α in L is called a dual atom if there is no β in L such that α<β<1. In other words α is maximal in L-{1}. In the following theorem we give an internal characterization of L-fuzzy maximal ideals in A.Theorem 31.
AnL-fuzzy ideal μ of A is maximal if and only if Img(μ)={1,α}, where α is a dual atom in L and the set μ∗={x∈A:μ(x)=1} is a maximal ideal of A.Proof.
Suppose thatμ is maximal. Clearly 1∈Img(μ), and since μ is nonconstant, there is some a∈A such that μ(a)<1. We first show that μ assumes exactly one value other than 1. Let x,y∈A such that μ(x)<1 and μ(y)<1. Put α=μ(x) and β=μ(y). Define L-fuzzy subsets μα∨ and μβ∨ of A as follows: (45)μα∨z=μz∨αandμβ∨z=μz∨β for all z∈A. Then it can be verified that both μα∨ and μβ∨ are L-fuzzy ideals of A such that μ≤μα∨ and μ≤μβ∨. By the maximality of μ we get that μ=μα∨ and μ=μβ∨. Thus α=β. Therefore Img(μ)={1,α} for some α∈L-{1}. Next we prove that this α is a dual atom. Let β∈L such that α<β. Define an L-fuzzy subset σ of A by (46)σz=1ifμz=1βotherwise for all z∈A. Then σ is an L-fuzzy ideal of A such that μ<σ. By the maximality of μ it yields that σ=1A; i.e., σ(z)=1 for all z∈A, so β=1. Therefore α is a dual atom. It remains to show that μ∗ is a maximal ideal of A. Clearly it is a proper ideal. Let J be a proper ideal of A such that μ∗⊆J. Define an L-fuzzy subset σ of A by (47)σz=1ifz∈Jαotherwise for all z∈A. Then σ is a nonconstant L-fuzzy ideal of A such that μ≤σ. Since μ is maximal, we get μ=σ, so μ∗=J. Therefore μ∗ is maximal among all proper ideals of A. Conversely suppose that Img(μ)={1,α}, where α is a dual atom in L and the set μ∗={x∈A:μ(x)=1} is a maximal ideal of A. Let σ be a nonconstant L-fuzzy ideal of A such that μ≤σ. Then σ(x)=1 for all x∈μ∗ and α≤σ(x) for all x∈A-μ∗. We show that σ=μ. Suppose not. Then there exists x∈A such that σ(x)>μ(x), so x∈A-μ∗. If σ(x)=1, then x∈σ∗={z∈A:σ(z)=1} and μ∗⊊σ∗⊊A. This contradicts the maximality of μ∗. Also if σ(x)<1, then α≤σ(x)<1. Again this contradicts the hypothesis that α is a dual atom. Therefore σ=μ. Hence μ is maximal.The above theorem confirms that there is a one-to-one correspondence between the class of all maximalL-fuzzy ideals and the set of all pairs (M,α) where M is a maximal ideal in A and α is a dual atom in L.Theorem 32.
IfA is an algebra in which every maximal ideal is a prime ideal, then every maximal L-fuzzy ideal is an L-fuzzy prime ideal.For instance, if[A,A]=A, then every maximal L-fuzzy ideal is an L-fuzzy prime ideal.


## 6. Conclusion

In this paper, the commutator (or the product) ofL-fuzzy ideals is defined and investigated in a more general context, in universal algebras. By the use of this commutator, L-fuzzy prime ideals in universal algebras are defined and fully characterized. Furthermore, maximal L-fuzzy ideals of universal algebra are studied.The study of fuzzy semiprime ideals in universal algebras is under investigation by the authors using the commutator ofL-fuzzy ideals. Moreover, the radical of L-fuzzy ideals in universal algebras will be studied and would be applied to characterize the central properties of L-fuzzy semiprime ideals.



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