# L-Fuzzy Semiprime Ideals of a Poset

Authors: Berhanu Assaye Alaba; Derso Abeje Engidaw
Journal: Advances in Fuzzy Systems
 (2020)

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## Abstract

In this paper, we introduce the concept ofL-fuzzy semiprime ideal in a general poset. Characterizations of L-fuzzy semiprime ideals in posets as well as characterizations of an L-fuzzy semiprime ideal to be L-fuzzy prime ideal are obtained. Also, L-fuzzy prime ideals in a poset are characterized.


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## Body

## 1. Introduction

Fuzzy set theory was introduced by Zadeh in 1965 as an extension of the classical notion of set theory [1]. In 1971, Rosenfeld wrote his seminal paper on fuzzy subgroups in [2]. This paper has provided sufficient motivations for researchers to study the fuzzy subalgebras of different algebraic structures, like rings, modules, vector-spaces, lattices, and more recently in MS-algebras, universal algebras, pseudocomplemented semilattice, and so on (see [3–26]).Zadeh defined a fuzzy subset of a nonempty setX as a function from X to unit interval 0,1 of real numbers. Goguen in [27] generalized the fuzzy subsets of X, to L-fuzzy subsets, as a function from X to a lattice L. Swamy and Swamy [5] initiated that complete lattices satisfying the infinite meet distributivity are the most appropriate candidates to have the truth values of general fuzzy statements.In the literature, we have found several types of ideals and filters of a poset which are generalizations of ideals and filters of a lattice (see [28–33]). Halaś and Rachůnek in [34] introduced the notions of prime ideals in a poset, and Khart and Mokbel [35] introduced the concept of a semiprime ideal in general poset.In [36, 37], the authors of this paper introduced several types of L-fuzzy ideals and filters of a partially ordered set whose truth values are in a complete lattice satisfying the infinite meet distributive law. In addition, in [38], we have introduced and presented certain comprehensive results on the notion of L-fuzzy prime ideals and maximal L-fuzzy ideals of a poset by applying the general theory of algebraic fuzzy systems introduced in [39, 40].Initiated by the above ideas and concepts, in this paper, we introduce and develop the concepts ofL-fuzzy semiprime ideal in a general partially ordered set. Characterizations of L-fuzzy semiprime ideals in posets as well as necessary and sufficient conditions of an L-fuzzy semiprime ideal to be L-fuzzy prime ideal are observed. Also, by introducing the concept of a μ-atom element in a poset, we obtain characterizations of L-fuzzy semiprime ideals and L-fuzzy prime ideals in a poset satisfying the descending chain condition (DCC).


## 2. Preliminaries

For the necessary concepts, terminologies, and notations, we refer to [41, 42].A pairQ,≤ is called a partially ordered set or simply a poset if Q is a nonempty set and “≤” is a partial order on Q. An element x∈Q is called a lower bound of S if x≤s for all s∈S. An upper bound is defined dually. The set of all lower bounds of S is denoted by Sl and the set of all upper bounds of S, by Su.By the setsSul and Slu, we mean Sul and Slu, respectively. For any a,b∈Q, the sets al and a,bl are denoted by al and a,bl, respectively. Furthermore, for subsets S,T of Q, S∪Tl is denoted by S,Tl and the set S∪al is denoted by S,al. Similar notations are used for the set of all upper bounds of subsets of a poset Q.For any subsetsS,T of a poset Q, we note that S⊆Sul and S⊆Slu, and if S⊆T in Q, then Su⊇Tu and Sl⊇Tl. In addition, aul=al and alu=au. An element x0 in Q is called the greatest lower bound of S or infimum of S, denoted by infS, if x0∈Sl and x≤x0∀x∈Sl. Dually, we have the concept of the least upper bound of S or supremum of S which is denoted by supS.Forx,y∈Q, we write x∧y (read as “x meet y”) in place of infx,y if it exists and x∨y (read as “x join y”) in place of supx,y if it exists. An element q0 in Q is called the smallest (respectively, the largest) element of a poset Q if q0≤x (respectively, x≤q0) for all x∈Q. The smallest (respectively, the largest) element if it exists in Q is denoted by 0 (respectively, by 1). A poset Q≤ is called bounded if it has 0 and 1.A posetQ is is said to satisfy the ascending chain condition (ACC), if every nonempty subset of Q has a maximal element. Dually, we have the concept of descending chain condition (DCC) [35].Definition 1.
(see [43]). A poset Q is called distributive if for all a,b,c∈Q,(1)a,bu,cl=a,cl,b,clul.Definition 2.
(see [24]). A subset I of a poset Q,≤ is called an ideal in Q if a,bul⊆I whenever a,b∈I.
Now, we consider the concept of a semiprime ideal introduced by Khart and Mokbel in a poset and by Rav in a lattice, as given in the following.Definition 3.
(see [28]). A proper ideal I of a poset Q is called a semiprime ideal of Q if for all x,y,z∈Q,x,yl⊆I and x,zl⊆I imply x,y,zul⊆I.
Dually, we have the concept semiprime filter of a posetQ.Definition 4.
(see [44]). A proper ideal I of a lattice X is called a semiprime ideal of X if for all x,y,z∈X,x∧y∈I and x∧z∈I together imply x∧y∨z∈I.
Dually, we have the concept semiprime filter of a latticeX.
For an idealI and an element a in a poset Q, define a set I:a by(2)I:a=x∈Q:a,xl⊆I.Definition 5.
(see [28]). An element i in a poset Q is called an I-atom with respect to an ideal I of Q if i∉I and for any x∈Q with x<i implies x∈I.
Throughout this paper,L stands for a complete lattice satisfying the infinite meet distributive law and Q stands for a poset with 0.
By anL-fuzzy subset μ of a poset Q, we mean a mapping from Q into L. We denote the set of L- fuzzy subsets of Q by LQ. For each α∈L and μ∈LQ, the α-level subset of μ, which is denoted by μα, is a subset of Q given by μα=x:μx≥α. For fuzzy subsets μ and σ of Q, we write μ⊆σ to mean μx≤σx for all x∈Q in the ordering of L. It can be easily verified that “⊆” is a partial order on the set LQ and is called the point-wise ordering. We write μ⊂σ if μ⊆σ and μ≠σ.
The following notions and results in this section are from the authors’ work in [29, 31].Definition 6.
μ∈LQ is said to be an L- fuzzy semi-ideal of Q if μ0=1 and for any a∈Q, μx≥μa, for all x∈al.Definition 7.
μ∈LQ is said to be an L- fuzzy ideal of Q if μ0=1 and, for any, a,b∈Q,(3)μx≥μa∧μb,for allx∈a,bul.
AnL-fuzzy ideal μ of Q is called a u-L-fuzzy ideal if, for any a,b∈Q, there exists x∈a,bu such that μx=μa∧μb.Lemma 1.
μ∈LQ is an L-fuzzy ideal of Q if and only if μα is an ideal of Q, for all α∈L.Lemma 2.
Ifμ is an L- fuzzy ideal of Q, then μ is anti-tone.
Note that, for anyβ in L the constant L-fuzzy subset of Q which maps all elements of Q onto β, is denoted by β¯.Definition 8.
AnL-fuzzy ideal μ of a poset Q is called proper, if μ≠1¯, where 1 is the largest element in L.Definition 9.
A properL-fuzzy ideal μ of a poset Q is called an L-fuzzy prime, if, for any a,b∈Q,(4)infμx:x∈a,bl=μa,orμb.Definition 10.
A properL-fuzzy ideal μ of a poset Q is said to be maximal if μ is a maximal element in the set of all proper L-fuzzy ideals of Q.


## 3.L-Fuzzy Semiprime Ideals of a Poset

In this section, we introduce and develop the notions of L-fuzzy semiprime ideal of a poset and give several characterizations of it. We shall begin with its definition.Definition 11.
AnL-fuzzy ideal μ of a poset Q is called an L-fuzzy semiprime ideal if for all a,b,c∈Q,(5)μz≥infμx∧μy:x∈a,bl,y∈a,cl,∀z∈a,b,cul.
The following result characterizes anyL-fuzzy semiprime ideal of Q in terms of its level subsets.Lemma 3.
AnL-fuzzy ideal μ of Q is an L-fuzzy semiprime ideal of Q if and only if μα is a semiprime ideal of Q for all α∈L.Proof.
Suppose thatμ is an L-fuzzy semiprime ideal and α∈L. Then, clearly, μα is an ideal of Q. Let a,b,c∈Q such that a,bl⊆μα and a,cl⊆μα and z∈a,b,cul. Then, μx≥α∀x∈a,bl and μy≥α∀y∈a,bl. This implies that(6)infμx:x∈a,bl≥α,infμy:y∈a,cl≥α.
Therefore,infμx∧μy:x∈a,bl,y∈a,cl≥α. Since μ is an L-fuzzy semiprime ideal and z∈a,b,cul, we have(7)μz≥infμx∧μy:x∈a,bl,y∈a,cl≥α.
This implies thatz∈μα for all z∈a,b,cul and hence, a,b,cul⊆μα. Therefore, μα is a semiprime ideal of a poset Q.
Conversely, suppose thatμα is a semiprime ideal of Q for all α∈L. Then, clearly, μ is an L- fuzzy ideal of Q. Let a,b,c∈Q and put(8)α=infμx∧μy:x∈a,bl,y∈a,cl.
Then,(9)infμx:x∈a,bl≥α,infμy:y∈a,cl≥α.
That is,μx≥α∀x∈a,bl and μy≥α∀y∈a,cl. This implies that a,bl⊆μα and a,cl⊆μα. Thus, since μα is a semiprime ideal of Q, we have a,b,cul⊆μα. Therefore, μz≥α=infμx∧μy:x∈a,blx∈a,cl for all z∈a,b,cul, and hence μ is an L-fuzzy semiprime ideal of Q.Corollary 1.
A subsetI of a poset Q is a semiprime ideal of Q if and only if its characteristic map χI is an L-fuzzy semiprime ideal of Q.Definition 12.
AnL-fuzzy ideal μ of a lattice Q is called an L-fuzzy semiprime ideal, if for all a,b,c∈Q,(10)μa∧b∨c=μa∧b∧μa∧c.
Dually, we have the concept ofL-fuzzy semiprime filter of a lattice Q.Lemma 4.
Letμ be an L-fuzzy ideal of Q. Then, for any a,b∈Q,(11)infμx:x∈a,bl=μa∧b,whenever a∧b exists in Q.The following theorem shows that an L-fuzzy semiprime ideal of a poset is a natural generalization of an L-fuzzy semiprime ideal of a lattice.Theorem 1.
LetQ,≤ be a lattice. Then, an L-fuzzy ideal of Q is an L-fuzzy semiprime ideal in the poset Q if and only if it is an L-fuzzy semiprime ideal in the lattice Q.Proof.
Letμ be an L-fuzzy semiprime ideal in the poset Q and a,b,c∈Q. Then, since a∧b∨c∈a,b,cul, we have(12)μa∧b∨c≥infμx∧μy:x∈a,bl,y∈a,cl=infμx:x∈a,bl∧infμy:y∈a,cl=μa∧b∧μa∧c.
Again, sincea∧b≤a∧b∨c, a∧c≤a∧b∨c and μ is antitone, we clearly have(13)μa∧b∨c≤μa∧b∧μa∧c.
Therefore,μ is an L-fuzzy semiprime ideal in the lattice Q.
Conversely, suppose thatμ is an L-fuzzy semiprime ideal in the lattice Q. Let a,b,c∈Q and z∈a,b,cul. Then, z≤a and z≤t, for all t∈a,bu. Since a∨b∈a,bu, we have z≤a∨b. This implies that z≤a∧b∨c and hence(14)μz≥μa∧b∨c≥μa∧b∧μa∧c=infμx:x∈a,bl∧infμy:y∈a,cl=infμx∧μy:x∈a,bl,y∈a,cl.
So,μ is an L-fuzzy semiprime ideal in the poset Q.
The following result establishes a connection betweenL-fuzzy prime ideals and L-fuzzy semiprime ideals of a poset Q.Lemma 5.
EveryL-fuzzy prime ideal of a poset Q is an L-fuzzy semiprime ideal.Proof.
Letμ be an L-fuzzy prime ideal of Q. Let a,b,c∈Q. Then since μ is an L-fuzzy prime ideal of Q, we clearly have(15)infμx:x∈a,bl=μa,orμb,infμy:y∈a,cl=μa,orμc.
Letz∈a,b,cul=al∩b,cul. Then, z≤a and z∈b,cul. Now if infμx:x∈a,bl=μa or infμy:y∈a,cl=μa, then we have(16)μz≥μa≥infμx:x∈a,bl∧infμy:y∈a,cl=infμx∧μy:x∈a,bl,y∈a,cl.
Again ifinfμx:x∈a,bl≠μa and infμy:y∈a,cl≠μa, then we have(17)infμx:x∈a,bl=μb,infμy:y∈a,cl=μc.
Now sincez∈b,cul and μ is an L-fuzzy ideal, we have(18)μz≥μb∧μc=infμx:x∈a,bl∧μy:y∈a,cl=infμx∧μy:x∈a,bl,y∈a,cl.
Hence, in either cases, we have(19)μz≥infμx∧μy:x∈a,bl,y∈a,cl,for allz∈a,b,cul.
Hence,μ is an L-fuzzy semiprime ideal of Q.Remark 1.
The converse of the above lemma is not true. For example, consider the posetQ≤ depicted in Figure 1. Define a fuzzy subset μ:Q⟶0,1 by(20)μ0=1,μa=μb=μ1=0.5.
Then,μ is an L-fuzzy semiprime ideal but not an L-fuzzy prime ideal of Q. This is because a,b∈Q and infμx:x∈a,bl=μ0=1≠0.5=μa=μb.
Now, given anL-fuzzy ideal of a poset Q and any element in Q, we define the following L-fuzzy subset of Q.Figure 1
AnL-fuzzy semi-prime ideal which is not an L -fuzzy prime ideal of Q.Definition 13.
Letμ be an L-fuzzy ideal of Q and x∈Q. Define an L-fuzzy subset μ:x of Q by(21)μ:xy=infμz:z∈x,yl,for allz∈Q.
From Definition9, observe that an L-fuzzy ideal μ of Q is an L-fuzzy prime ideal if, for any a,b∈Q,(22)μ:ab=μa,orμb.
Now, we have the following lemmas.Lemma 6.
Letμ be an L-fuzzy ideal of Q and x∈Q. Then, μ:x is an L-fuzzy semi-ideal containing μ.Proof.
Now(23)μ:x0=infμz:z∈x,0l=infμz:z=0=μ0=1.
Therefore,μ:x0=1. Again, let a∈Q and y∈al. Now(24)μ:xy=infμw:w∈x,yl≥infμw:w∈x,alsincex,yl⊆x,al=μ:xa.
Therefore,μ:x is an L-fuzzy semi-ideal. Again, for all y∈Q, we have(25)μ:xy=infμz:z∈x,yl≥μy.
Hence,μ⊆μ:x.
Note that, for anyx,y∈Q, observe that μ:xy=μ:yx.Remark 2.
For anL-fuzzy ideal μ of a poset Qμ:x need not be an L-fuzzy ideal of Q for all x∈Q. For example, consider the poset Q,≤ depicted in Figure 2.
Define a fuzzy subsetμ:Q⟶0.1 by(26)μ0=1,μa=0.8,μb=μc=μd=μe=0.2.
Then,μ is an L-fuzzy ideal of Q, and μ:d is a fuzzy subset of Q given by(27)μ:d0=μ:db=μ:dc=1,μ:da=0.8,μ:dd=μ:de=0.2.
Observe thate∈a,bul but μ:de=0.2≱0.8=μ:da∧μ:db. This implies that μ:d is not an L-fuzzy ideal of Q.Figure 2
An L -fuzzy idealμ of a poset Q for which μ:x need not to be L -fuzzy ideal for x∈Q.Lemma 7.
Letμ be an L-fuzzy ideal of a poset Q and x∈Q. Then,(28)μ:xα=μα:x,for anyα∈L.Proof.
Now, we have(29)y∈μ:xα⇔μ:xy≥α⇔infμw:w∈x,yl≥α⇔μw≥α∀w∈x,yl⇔w∈μα∀w∈x,yl⇔x,yl⊆μα⇔y∈μα:x.Thereforeμ:xα=μα:x.Lemma 8.
Letμ be an L-fuzzy ideal of a poset Q and x∈Q. Then, the following hold:(1)
infμ:xy:y∈a,bl=infμy:y∈x,a,bl(2)
infμ:xy:y∈a,bul=infμy:y∈x,a,bul(3)
μ:x=1¯ if and only if μx=1Proof.
(1)
PutA=μy:y∈x,a,bl and B=μ:xy:y∈a,bl. Now we claim that infA=infB. Let α=infA. Then,(30)α≤μy,∀y∈x,a,bl⟹x,a,bl⊆μα⟹a,bl⊆μα:x=μ:xα⟹μ:xy≥α,∀y∈a,bl⟹infμ:xy:y∈a,bl≥α.⟹infB≥infA.To prove the other side of the inequality, letβ=infB. Then,(31)β≤μ:xy,∀y∈a,bl⟹a,bl⊆μ:xβ=μβ:x⟹x,a,bl⊆μβ⟹μy≥β,∀y∈x,a,bl⟹infμy:y∈x,a,bl≥β⟹infA≥infB.Hence, the claim is true.(2)
The proof is similar to 1.(3)
Supposeμ:x=1¯. Then, μ:xy=1, for all y∈Q. Thus, in particular, μ:xx=1:(32)nowμ:xx=1⇒infμy:y∈x,xl=1⇒μy=1,∀y∈x,xl⇒μx=1…sincex∈x,xl.
Conversely suppose thatμx=1. Now since, for any y∈Q, μ:xy=infμz:z∈x,yl≥μx=1, we have μ:xy=1 for all y∈Q. Therefore, μ:x=1¯.
Now, we present a characterization of anL-fuzzy semiprime ideal of a poset Q in terms of μ:x where μ is an L-fuzzy ideal of Q and x∈Q.Theorem 2.
AnL-fuzzy ideal μ of a poset Q is an L-fuzzy semiprime ideal if and only if μ:x is an L-fuzzy ideal for all x∈Q, in fact, an L-fuzzy semiprime ideal for all x∈Q.Proof.
Letμ be an L-fuzzy semiprime ideal of Q and x∈Q. First, let us show that μ:x is an L-fuzzy ideal of Q. Since μ:x0=infμy:y∈x,0l=μ0=1, we have μ:x0=1. Again, let a,b∈Q and z∈a,bul. Then,(33)μ:xa∧μ:xb=infμw:w∈x,al∧infμu:u∈x,bl=infμw∧μu:w∈x,al,u∈x,bl≤μvforallv∈x,a,bul.
Again, sincez∈a,bul implies x,zl⊆x,a,bul, we have μ:xa∧μ:xb≤μt for all t∈x,zl. This implies that(34)μ:xa∧μ:xb≤infμt:t∈x,zl=μ:xz.
Therefore,μ:x is an L-fuzzy ideal of Q for all x∈Q. Now, we show that μ:x is an L-fuzzy semiprime ideal of Q. Let a,b,c∈Q and z∈a,b,cul. Now,(35)infμ:xu∧μ:xw:u∈a,bl,w∈a,cl=infμ:xu:u∈a,bl∧infμ:xw:w∈a,cl=infμu:u∈x,a,bl∧infμw:w∈x,a,cl=infμ:bu:u∈x,al∧infμ:cw:w∈x,al=infμ:bs∧μ:cs:s∈x,al≤infμ:bs∧μ:cs:s∈x,a,b,cul≤μ:bs∧μ:cs∀s∈x,a,b,cul=x,al∩b,cul=μ:sb∧μ:sc∀s∈x,a,b,cul=μ:ss∀s∈x,a,b,cul=μs∀s∈x,a,b,cul.
Now,z∈a,b,cul implies that x,zl⊆x,a,b,cul for all x∈Q. Thus, we have(36)infμ:xu∧μ:xw:u∈a,bl,w∈a,cl≤μs,for alls∈x,zl.
Thus,infμ:xu∧μ:xw:u∈a,bl,w∈a,cl≤infμs:s∈x,zl=μ:xz. Therefore, μ:x is an L-fuzzy semiprime ideal of Q.
Conversely, suppose thatμ:x is an L-fuzzy ideal of Q for all x∈Q. Now we show that μ is an L-fuzzy semiprime ideal of Q. Let a,b,c∈Q and z∈a,b,cul. Then,(37)infμx∧μy:x∈a,bl,y∈a,cl=infμx:x∈a,bl∧infμy:y∈a,cl=μ:ab∧μ:ac≤μ:az,for allz∈b,cul.
This implies that(38)infμx∧μy:x∈a,bl,y∈a,cl≤infμ:az:z∈a,bul=infμz:z∈a,b,cul.
Thus,(39)infμx∧μy:x∈a,bl,y∈a,cl≤μz,for allz∈a,b,cul.
Therefore,μ is an L-fuzzy semiprime ideal of Q.
The next result is a characterization of anL-fuzzy ideal to be an L-fuzzy prime ideal in a poset Q.Theorem 3.
Letμ be a proper L-fuzzy ideal of a poset Q. Then, μ is an L-fuzzy prime ideal of Q if and only if μ:a=μ for all a∈Q such that μa≠1.Proof.
Suppose thatμ is an L-fuzzy prime ideal of Q and let a∈Q such that μa≠1. Then, by Lemma 5 and Theorem 2, it is clear that μ:a is an L-fuzzy ideal of Q. Now we claim that μ:a=μ. Now, for any x∈Q, we have μ:ax=μxorμa. However, as μ:a is an L-fuzzy ideal of Q, μ:ax≠μa. Thus, μ:ax=μx for all x∈Q and hence μ:a=μ.
Conversely, suppose that the given condition holds. Leta,b∈Q. Now, we claim that(40)μ:ab=μa,orμb.
Suppose thatμ:ab≠μa. Then, infμx:x∈a,bl≰μa. This implies that μa≠1. Thus, by hypothesis, we have μ:a=μ and hence μ:ab=μb.
Therefore,μ is an L-fuzzy prime ideal of Q.
Now before we prove some other characterizations ofL-fuzzy primeness and L-fuzzy semiprimeness in the case of a poset satisfying DCC, we introduce the concept of a μ-atom of an L-fuzzy ideal μ of a poset.Definition 14.
Letμ be an L-fuzzy ideal of a poset Q and α∈L. An element i in Q is called a μ-atom with respect to α, if it satisfies the following conditions:(1)
α≰μi(2)
α≤μx whenever x<iExample 1.
Consider the poset depicted in Figure3. Define a fuzzy subset μ:Q⟶0,1 by(41)μ0=μa=1,μb=0.7,μc=0.6,μd=0.8,μa′=μb′=μc′=μd′=μ1=0.2.
Then, it is easy to see thatμ is an L-fuzzy ideal of Q and a' is a μ-atom with respect to α=0.6 in 0,1.Figure 3
A distributive poset in which every L-fuzzy ideal need not be an L-fuzzy semiprime ideal.Lemma 9.
There always exists aμ-atom for every proper L-fuzzy ideal μ in a poset Q satisfying DCC with respect to some α in L.Proof.
LetQ be a poset satisfying DCC and μ be a proper L-fuzzy ideal of Q. Then, there exists a∈Q such that μa≠1. This implies that there exists α∈L such that α≰μa. Put I=x∈Q:μx≥α. Then, since a∉I, I is a proper ideal of Q and Q−I is a nonempty subset of Q. Since Q is satisfying DCC, Q−I has a minimal element, say i, such that i≤a. Now we claim that i is a μ-atom with respect to α. Since i∈Q−I, we have α≰μi. Let x<i. Then, by the minimality of i, x∉Q−I and hence μx≥α. Hence, the claim is true.Remark 3.
Lemma9 gives a guarantee that if μ is an L-fuzzy ideal of a poset Q satisfying DCC and α≰μa for some a∈Q and α∈L, then there exists a μ-atom i in Q with respect to α such that i≤a.Lemma 10.
Any two distinctμ-atoms of an L-fuzzy ideal μ of a poset Q with respect to α∈L are incomparable.Proof.
Letμ be an L-fuzzy ideal of Q and i and j be any two distinct μ-atoms with respect to α∈L. Then, by definition, we have α≰μi and μx≥α whenever x<i and α≰μj and μy≥α whenever y<j. Now we show that i and j are incomparable. Suppose not. Then i<j or j<i, i.e., μi≥α or μj≥α, which is a contradiction to the fact that α≰μi and α≰μj. Hence, i and j are incomparable.Remark 4.
From Lemma10, we can deduce that if i and j are μ-atoms in a poset Q with respect to some α in L such that i≤j, then i=j.Lemma 11.
Letμ be an L-fuzzy semiprime ideal of a poset Q satisfying DCC. Then, μ:i is a u-L-fuzzy ideal for every μ-atom i in Q with respect to 1 in L.Proof.
Leti be a μ-atom in Q with respect to 1 in L. Since μ is an L-fuzzy semiprime ideal, by Theorem 2, μ:i is an L-fuzzy ideal of Q. Now we show that μ:i is a u- L-fuzzy ideal. Suppose on the contrary that μ:i is not a u-L-fuzzy ideal. Then, there exist a,b∈Q such that(42)μ:ia∧μ:ib≰μ:ix∀x∈a,bu.
This implies that there existsy∈i,xl such that(43)μ:ia∧μ:ib≰μy∀x∈a,bu.
Thus, by Remark3, there exists a μ-atom, say j, with respect to α=μ:ia∧μ:ib such that j≤y. Since j≤y and y∈i,xl, we have j≤i and hence μj≥μi. This implies that α≰μi. Again, let z<i. Then, μz=1≥α. Therefore, i is also a μ-atom with respect to α. Also since j≤y≤i, by Remark 4, we have j=y=i. This implies that i∈i,xl and hence i≤x for all x∈a,bu and so i∈a,bul. Since μ:i is an ideal, we have(44)α=μ:ia∧μ:ib≤μ:ii=μi.
which is a contradiction to the fact thatα≰μi. Therefore, μ:i is a u-L-fuzzy ideal.Theorem 4.
Letμ be an L-fuzzy ideal of a poset Q satisfying DCC. Then, μ is an L-fuzzy semiprime ideal of Q if and only if μ:i is an L-fuzzy ideal, in fact, an L-fuzzy prime ideal of Q for every μ-atom i∈Q with respect to 1 in L.Proof.
Letμ be an L-fuzzy semiprime ideal of a poset Q satisfying DCC and i is a μ-atom in Q with respect to 1 in L. Then, by Lemma 11, μ:i is a u-L-fuzzy ideal. Now, we have to show that μ:i is an L-fuzzy prime ideal. Since μi≠1, by Lemma 8, μ:i≠1¯. Hence, μ:i is proper. Let a,b∈Q and suppose that(45)infμ:ix:x∈a,bl≠μ:ia.
Putα=infμ:ix:x∈a,bl. Since μ:ia=infμy:y∈a,il, there exists y1 in i,al such that α≰μy1. Then, by Remark 3, there exists a μ-atom, say j, in Q with respect to α such that j≤y1. It is also clear that i is also a μ-atom with respect to α. Since j≤y1≤i, by Remark 4, we must have j=y1=i, and therefore i≤a, i.e., i,al=il. Thus, we have(46)infμ:ix:x∈a,bl=infμy:y∈i,a,bl=infμy:y∈i,bl=μ:ib.
This proves thatμ:i is an L-fuzzy prime ideal for every μ-atom i∈Q.
Conversely, suppose thatμ:i is an L-fuzzy ideal for any μ-atom i with respect to 1 in L. Let a,b,c∈Q. Now, we claim that(47)infμx∧μy:x∈a,bl,y∈a,cl≤μz,for allz∈a,b,cul.
Suppose not. Then, there existsz1∈a,b,cul=al∩b,cul such that(48)infμx∧μy:x∈a,bl,y∈a,cl≰μz1.
Hence, by Remark3, there exists a μ-atom j in Q with respect to α=infμx∧μy:x∈a,bl,y∈a,cl in L such that j≤z1. Then, by hypothesis, μ:j is an L-fuzzy ideal. Again, since j,bl⊆a,bl and j,cl⊆a,cl, we have(49)α=infμx∧μy:x∈a,bl,y∈a,cl=infμx:x∈a,bl∧infμy:y∈a,cl≤infμx:x∈j,bl∧infμy:y∈j,cl=μ:jb∧μ:jc≤μ:jjsincej∈b,cul=μj,which is a contradiction to the fact that j is a μ-atom with respect to α. Therefore, μ is an L-fuzzy semiprime ideal of Q.
The following result gives another characterization forL-fuzzy semiprime ideals to be L-fuzzy prime.Theorem 5.
Every maximalL-fuzzy semiprime ideal of a poset Q is an L-fuzzy prime ideal.Proof.
Letμ be a maximal L-fuzzy semiprime ideal of a poset Q, that is, maximal among all proper L-fuzzy semiprime ideals of a poset Q. Let a,b∈Q. Then, by Theorem 2, μ:b is an L-fuzzy semiprime ideal. Since μ⊆μ:b, by maximality of μ, we have either μ=μ:b or μ:b=1¯. If μ:b=1¯, then, by Lemma 8, μb=1. Thus,(50)infμx:x∈a,bl=μ:ba=1¯a=1=μb.
Again ifμ=μ:b, then we have infμx:x∈a,bl=μ:ba=μa. Thus, in either cases, we have(51)infμx:x∈a,bl=μaorμb,for alla,b∈Q.
Hence,μ is an L-fuzzy prime ideal of Q.
As a consequence, we have the following corollary.Corollary 2.
Letμ be a maximal L-fuzzy ideal of Q. Then, μ is an L-fuzzy semiprime ideal Q if and only if μ is an L-fuzzy prime ideal.The following is a characterization of anL-fuzzy ideal to be L-fuzzy prime ideal in terms of a μ-atom in a poset Q satisfying DCC.Theorem 6.
Letμ be an L-fuzzy ideal of a poset Q satisfying DCC. Then, μ is an L-fuzzy prime ideal Q if and only if Q has exactly one μ-atom with respect to some α in L.Proof.
Letμ be an L-fuzzy prime ideal of a poset Q satisfying DCC. Since μ is proper, by Lemma 9, there exists a μ-atom in Q with respect to some α in L. Now, we claim that Q has exactly one μ-atom with respect to α in L. Suppose not. Let i,j∈Q be any distinct μ-atoms in Q with respect to α in L. Then, by Lemma 10, i,j are incomparable and μx≥α for all x<i and μy≥α for all y<j. This implies that infμx:x∈i,jl≥α. Since infμx:x∈i,jl=μi or μj, we have μi≥α or μj≥α, which is a contradiction. Therefore, Q has exactly one μ-atom with respect to α in L.
Conversely suppose thatQ has exactly one μ-atom, say i, with respect to some α in L. Now, we show that μ is an L-fuzzy prime ideal. Since α≰μa, we have μi≠1 and hence μ is proper. Now, we show that for any a,b∈Q,(52)infμx:x∈a,bl=μa,orμb.
Suppose not. Thus, there exista,b∈Q such that(53)infμx:x∈a,bl≰μa,infμx:x∈a,bl≰μb.
Then, there existμ-atoms i,j∈Q with respect to α=infμx:x∈a,bl such that i≤a and j≤b. Then, by hypothesis, we have i=j and hence i∈a,bl. Therefore α=infμx:x∈a,bl≤μa, which is a contradiction to the fact that i is a μ-atom with respect to α=infμx:x∈a,bl. Therefore, μ is an L-fuzzy prime ideal.Lemma 12.
Letμ be a proper L-fuzzy ideal of a poset Q satisfying DCC and A=i∈Q:iisaμ−atom. Then, μ=∩i∈A:μ:i.Proof.
We show that∩i∈Aμ:i⊆μ as the converse inclusion always holds. Suppose that ∩i∈Aμ:i⊈μ. This implies that there exists a∈Q such that ∩i∈Aμ:ia≰μa. Thus, there exists a μ-atom j∈Q with respect to α=∩i∈Aμ:ia such that j≤a.
Then, we havej∈A, and hence,(54)∩i∈A:μa≤μ:ja=infμx:x∈j,al=infμx:x∈jl=μj,which is a contradiction to the fact that j is a μ-atom with respect to α=∩i∈Aμ:ia. Hence, α=∩i∈Aμ:iμ. Therefore, ∩i∈Aμ:i⊆μ.Lemma 13.
The intersection of any nonempty family ofL-fuzzy prime ideals of Q is an L-fuzzy semiprime ideal Q.Proof.
Letμi:i∈Δ be a nonempty family of L-fuzzy prime ideals of Q. Put μ=∩i∈Δμi. Then, clearly, μ is an L-fuzzy ideal of Q. Let a,b,c∈Q and z∈a,b,cul. Now,(55)infμx∧μy:x∈a,bl,y∈a,cl=infμx:x∈a,bl∧infμy:y∈a,cl≤infμix:x∈a,bl∧infμiy:y∈a,cl,for eachi∈Δ.=μiaorμib∧μic,for eachi∈Δ.≤μiz,for eachi∈Δ.
This implies that(56)infμx∧μy:x∈a,bl,y∈a,cl≤∩i∈Δ:μiz=μz,for allz∈a,b,cul.
Therefore,μ=∩i∈Δμi is an L-fuzzy semiprime ideal Q.
As an immediate consequence of Theorem4, Lemma 12, and Lemma 13 in the case of posets satisfying DCC, we obtain the following result.Theorem 7.
Letμ be a proper L-fuzzy ideal of a poset Q satisfying DCC. Then, μ is an L-fuzzy semiprime ideal of Q if and only if μ is expressed as an intersection of L-fuzzy prime ideals of Q.In the following, we characterize the distributive posets in terms ofL-fuzzy semiprime ideals in the following results.Theorem 8.
A posetQ is distributive if and only if χx of Q is an L-fuzzy semiprime ideal of Q, for each x∈Q.Proof.
Suppose thatQ is distributive poset and x∈Q. Now to show χx is an L-fuzzy semiprime ideal of Q, by Corollary 1, it is enough to show that x is a semiprime ideal of Q. Let a,b,c∈Q such that a,bl⊆x and a,cl⊆x. Let z∈a,b,cul. Then, z≤a and z∈b,cul. This implies that(57)zl=z,b,cul=z,bl,z,clul.
Sincez≤a, we have z,bl⊆a,bl⊆x and z,cl⊆a,cl⊆x.
This implies thatz,al∪z,bl⊆x. Thus, z∈zl=z,bl,z,clul⊆xul=x and hence, a,b,cul⊆x.
Therefore,χx is an L-fuzzy semiprime ideal of Q.
Conversely, suppose thatχx is an L-fuzzy semiprime ideal of Q for each x∈Q. Then, by Corollary 1, it is clear that x is semiprime ideal of Q for each x∈Q. Let a,b,c∈Q. It is enough to prove that a,b,cul⊆a,bl,a,clul, as the converse inclusion is always true. Now let x∈a,b,cul and y∈a,bl,a,clu. We claim that x≤y. Indeed, since a,bl,a,clul⊆yl, we have(58)a,bl⊆yl=y,a,cl⊆yl=y.
Then, by semiprimeness ofy, we conclude that x∈a,b,cul⊆y. Hence, x≤y for all y∈a,bl,a,clu. Therefore, x∈a,bl,a,clul, and hence,(59)a,b,cul⊆a,bl,b,clul.
This proves thatQ is a distributive poset.
Note that in a distributive poset everyL-fuzzy ideal need not be an L-fuzzy semiprime ideal. Consider the distributive poset Q depicted in Figure 3.
Define a fuzzy subsetμ:Q⟶0,1 by(60)μ0=μa=1,μa′=μb′=μc′=μd′=μ1=0.2,μb=0.6,μc=0.5,μd=0.7.
Then,μ is an L-fuzzy ideal but not an L-fuzzy semiprime ideal as d′,c′,b′∈Q and d′∈d′,b′,c′ul but(61)infμx∧μy:x∈d′,cl,y∈d′,b′l=0.5≰0.2=μd′.
As an immediate consequence of Theorem7 and Theorem 8, we have the following corollary.Corollary 3.
LetQ be a poset satisfying DCC. Then, Q is distributive if and only if, for every x∈Q, χx is representable as an intersection of L-fuzzy prime ideals of Q.


## 4. Conclusion

In this work, we introduce the notions ofL-fuzzy semiprime ideal in general posets. Characterizations of L-fuzzy semiprime ideals in posets as well as characterizations of an L-fuzzy semiprime ideal to be L-fuzzy prime ideal are obtained by introducing the concept μ-atom elements in a poset. Also, L-fuzzy prime ideals in a poset are characterized. Our future work will focus on the relations between the L-fuzzy semiprime (resp., L-fuzzy prime) ideals of a poset and the L-fuzzy semiprime (resp., L-fuzzy prime) of the lattice of all ideals of a poset. We will also extend and prove an analogue of Stone’s theorem for finite posets using L-fuzzy semiprime ideals.



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