L -Fuzzy Semiprime Ideals of a Poset

. In this paper, we introduce the concept of L -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


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]. is 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 ).
Zadeh defined a fuzzy subset of a nonempty set X 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][29][30][31][32][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 of L-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). For any subsets S, T of a poset Q, we note that S ⊆ S ul and S ⊆ S lu , and if S ⊆ T in Q, then S u ⊇ T u and S l ⊇ T l . In addition, a u { } l � a l and a l u � a u . An element x 0 in Q is called the greatest lower bound of S or infimum of S, denoted by infS, if x 0 ∈ S l and x ≤ x 0 ∀x ∈ S l . Dually, we have the concept of the least upper bound of S or supremum of S which is denoted by supS.
For x, y ∈ Q, we write x ∧ y (read as " x meet y ") in place of inf x, y if it exists and x ∨ y (read as "x join y") in place of sup x, y if it exists. An element q 0 in Q is called the smallest (respectively, the largest) element of a poset Q if q 0 ≤ x (respectively, x ≤ q 0 ) for all x ∈ Q. e 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 poset Q 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] Definition 2 (see [24]). A subset I of a poset (Q, ≤ ) is called an ideal in Q if (a, b) ul ⊆ 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, y) l ⊆ I and (x, z) l ⊆ I imply x, (y, z) u l ⊆ I.
Dually, we have the concept semiprime filter of a poset Q.
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 lattice X.
For an ideal I and an element a in a poset Q, define a set I : a by 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. roughout this paper, L stands for a complete lattice satisfying the infinite meet distributive law and Q stands for a poset with 0.
By an L-fuzzy subset μ of a poset Q, we mean a mapping from Q into L. We denote the set of Lfuzzy subsets of Q by L Q . For each α ∈ L and μ ∈ L Q , the α-level subset of μ, which is denoted by μ α , is a subset of Q given by μ α � x : μ(x) ≥ α . For fuzzy subsets μ and σ of Q, we write μ ⊆ σ to mean μ(x) ≤ σ(x) for all x ∈ Q in the ordering of L. It can be easily verified that "⊆" is a partial order on the set L Q and is called the point-wise ordering. We write μ ⊂ σ if μ ⊆ σ and μ ≠ σ.
e following notions and results in this section are from the authors' work in [29,31]. Definition 6. μ ∈ L Q is said to be an Lfuzzy semi-ideal of Q if μ(0) � 1 and for any a ∈ Q, μ(x) ≥ μ(a), for all x ∈ a l . Definition 7. μ ∈ L Q is said to be an Lfuzzy ideal of Q if μ(0) � 1 and, for any, a, b ∈ Q, (3)

Lemma 2. If μ is an Lfuzzy ideal of Q, then μ is anti-tone.
Note that, for any β in L the constant L-fuzzy subset of Q which maps all elements of Q onto β, is denoted by β.

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. An L-fuzzy ideal μ of a poset Q is called an L-fuzzy semiprime ideal if for all a, b, c ∈ Q, e following result characterizes any L-fuzzy semiprime ideal of Q in terms of its level subsets. 2 Advances in Fuzzy Systems Proof. Suppose that μ is an L-fuzzy semiprime ideal and α ∈ L. en, clearly, μ α is an ideal of Q. Let a, b, c ∈ Q such that (a, b) l ⊆ μ α and (a, c) l ⊆ μ α and z ∈ a, Dually, we have the concept of L-fuzzy semiprime filter of a lattice Q.

Lemma 4. Let μ be an L-fuzzy ideal of Q. en, for any
whenever a ∧ b exists in Q. e 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. Let (Q, ≤ ) be a lattice. en, 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. Let μ be an L-fuzzy semiprime ideal in the poset Q and a, b, c ∈ Q. en, since erefore, μ is an L-fuzzy semiprime ideal in the lattice Q.
Conversely, suppose that μ is an L-fuzzy semiprime ideal in the lattice Q.
So, μ is an L-fuzzy semiprime ideal in the poset Q.
e following result establishes a connection between L-fuzzy prime ideals and L-fuzzy semiprime ideals of a poset Q. en since μ is an L-fuzzy prime ideal of Q, we clearly have or μ(c).
Now since z ∈ (b, c) ul and μ is an L-fuzzy ideal, we have Hence, in either cases, we have Hence, μ is an L-fuzzy semiprime ideal of Q.
□ Remark 1. e converse of the above lemma is not true. For example, consider the poset (Q ≤ ) depicted in Figure 1.
en, μ is an L-fuzzy semiprime ideal but not an L-fuzzy Now, given an L-fuzzy ideal of a poset Q and any element in Q, we define the following L-fuzzy subset of Q.
Definition 13. Let μ be an L-fuzzy ideal of Q and x ∈ Q. Define an L-fuzzy subset μ : x of Q by From Definition 9, observe that an L-fuzzy ideal μ of Q is an L-fuzzy prime ideal if, for any a, b ∈ Q, Now, we have the following lemmas.

Lemma 6.
Let μ be an L-fuzzy ideal of Q and x ∈ Q. en, μ : x is an L-fuzzy semi-ideal containing μ.
en, μ is an L-fuzzy ideal of Q, and μ : d is a fuzzy subset of Q given by Observe that e ∈ (a, b) ul but is implies that μ : d is not an L-fuzzy ideal of Q. en,

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Proof. Now, we have □ Lemma 8. Let μ be an L-fuzzy ideal of a poset Q and x ∈ Q. en, the following hold: To prove the other side of the inequality, let β � infB. en, Hence, the claim is true. (2) e proof is similar to 1.

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erefore, 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, c) erefore, μ : x is an L-fuzzy semiprime ideal of Q.
Conversely, suppose that μ : x is an L-fuzzy ideal of Q for all x ∈ Q. Now we show that μ is an L-fuzzy semiprime ideal is implies that us, erefore, μ is an L-fuzzy semiprime ideal of Q. e next result is a characterization of an L-fuzzy ideal to be an L-fuzzy prime ideal in a poset Q. Proof. Suppose that μ is an L-fuzzy prime ideal of Q and let a ∈ Q such that μ(a) ≠ 1. en, by Lemma 5 and eorem 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 : a)(x) � μ(x) or μ(a). However, as μ : a is an L-fuzzy ideal of Q, (μ : a)(x) ≠ μ(a). us, (μ : a)(x) � μ(x) for all x ∈ Q and hence μ : a � μ.
Conversely, suppose that the given condition holds. Let a, b ∈ Q. Now, we claim that . is implies that μ(a) ≠ 1. us, by hypothesis, we have μ : a � μ and hence (μ : a) erefore, μ is an L-fuzzy prime ideal of Q. Now before we prove some other characterizations of L-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. Let μ 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: Advances in Fuzzy Systems Example 1. Consider the poset depicted in Figure 3. Define a fuzzy subset μ : Q ⟶ [0, 1] by en, it is easy to see that μ is an L-fuzzy ideal of Q and a' is a μ-atom with respect to α � 0.6 in [0, 1].

Lemma 9.
ere always exists a μ-atom for every proper L-fuzzy ideal μ in a poset Q satisfying DCC with respect to some α in L.
Proof. Let Q be a poset satisfying DCC and μ be a proper L-fuzzy ideal of Q.
en, there exists a ∈ Q such that μ(a) ≠ 1.
is implies that there exists α ∈ L such that α ≰ μ(a). Put I � x ∈ Q : μ(x) ≥ α . en, 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.
en, by the minimality of i, x ∉ Q − I and hence μ(x) ≥ α. Hence, the claim is true. □ Remark 3. Lemma 9 gives a guarantee that if μ is an L-fuzzy ideal of a poset Q satisfying DCC and α ≰ μ(a) 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 distinct μ-atoms of an L-fuzzy ideal μ of a poset Q with respect to α ∈ L are incomparable.
Proof. Let μ be an L-fuzzy ideal of Q and i and j be any two distinct μ-atoms with respect to α ∈ L. en, by definition, we have α ≰ μ(i) and μ(x) ≥ α whenever x < i and α ≰ μ(j) and μ(y) ≥ α whenever y < j. Now we show that i and j are incomparable. Suppose not. en i < j or j < i, i.e., μ(i) ≥ α or μ(j) ≥ α, which is a contradiction to the fact that α ≰ μ(i) and α ≰ μ(j). Hence, i and j are incomparable. Proof. Let i be a μ-atom in Q with respect to 1 in L. Since μ is an L-fuzzy semiprime ideal, by eorem 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.
en, there exist a, b ∈ Q such that is implies that there exists y ∈ (i, x) l such that us, by Remark 3, there exists a μ-atom, say j, with respect to α � (μ : i)(a) ∧ (μ : i)(b) such that j ≤ y. Since j ≤ y and y ∈ (i, x) l , we have j ≤ i and hence μ(j) ≥ μ(i). is implies that α ≰ μ(i). Again, let z < i. en, μ(z) � 1 ≥ α. erefore, i is also a μ-atom with respect to α. Also since j ≤ y ≤ i, by Remark 4, we have j � y � i. is implies that i ∈ (i, x) l and hence i ≤ x for all x ∈ (a, b) u and so i ∈ (a, b) ul . Since μ : i is an ideal, we have which is a contradiction to the fact that α ≰ μ(i). erefore, μ : i is a u-L-fuzzy ideal. Proof. Let μ 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. en, 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 μ(i) ≠ 1, by Lemma 8, μ : i ≠ 1. Hence, μ : i is proper. Let a, b ∈ Q and suppose that : y ∈ (a, i) l , there exists y 1 in (i, a) l such that α ≰ μ (y 1 ). en, by Remark 3, there exists a μ-atom, say j, in Q with respect to α such that j ≤ y 1 . It is also clear that i is also a μ-atom with respect to α. Since j ≤ y 1 ≤ i, by Remark 4, we must have j � y 1 � i, and therefore i ≤ a, i.e., (i, a) l � i l . us, we have is proves that μ : i is an L-fuzzy prime ideal for every μ-atom i ∈ Q.
Conversely, suppose that μ : 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 for all z ∈ a, (b, c) u l .
Hence, by Remark 3, there exists a μ-atom j in Q with respect to α � inf μ(x) ∧ μ(y) : x ∈ (a, b) l , y ∈ (a, c) l in L such that j ≤ z 1 . en, by hypothesis, μ : j is an L-fuzzy ideal. Again, since (j, b) l ⊆ (a, b) l and (j, c) l ⊆ (a, c) l , we have which is a contradiction to the fact that j is a μ-atom with respect to α. erefore, μ is an L-fuzzy semiprime ideal of Q. e following result gives another characterization for L-fuzzy semiprime ideals to be L-fuzzy prime. Proof. Let μ 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. en, by eorem 2, μ : b is an L-fuzzy semiprime ideal. Since μ ⊆ μ : b, by maximality of μ, we have either us, in either cases, we have Hence, μ is an L-fuzzy prime ideal of Q.
As a consequence, we have the following corollary. □ Corollary 2. Let μ be a maximal L-fuzzy ideal of Q. en, μ is an L-fuzzy semiprime ideal Q if and only if μ is an L-fuzzy prime ideal. e following is a characterization of an L-fuzzy ideal to be L-fuzzy prime ideal in terms of a μ-atom in a poset Q satisfying DCC. Theorem 6. Let μ be an L-fuzzy ideal of a poset Q satisfying DCC. en, 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. Let μ 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.
en, by Lemma 10, i, j are incomparable and μ(x) ≥ α for all x < i and μ(y) ≥ α for all y < j.
is implies that which is a contradiction. erefore, Q has exactly one μ-atom with respect to α in L.
Conversely suppose that Q 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 ≰ μ(a), we have μ(i) ≠ 1 and hence μ is proper. Now, we show that for any a, b ∈ Q, (53) en, there exist μ-atoms i, j ∈ Q with respect to α � inf μ(x) : x ∈ (a, b) l such that i ≤ a and j ≤ b.
en, by hypothesis, we have i � j and hence i ∈ (a, b) l . erefore α � inf μ(x) : x ∈ (a, b) l ≤ μ(a), which is a contradiction to the fact that i is a μ-atom with respect to α � inf μ(x) : x ∈ (a, b) l . erefore, μ is an L-fuzzy prime ideal. □ Lemma 12. Let μ be a proper L-fuzzy ideal of a poset Q satisfying DCC and A � i ∈ Q : i is a μ − atom . en, μ � ∩ i∈A : μ : i.
Proof. We show that ∩ i∈A μ : i ⊆ μ as the converse inclusion always holds. Suppose that ∩ i∈A μ : i ⊈ μ. is implies that there exists a ∈ Q such that (∩ i∈A μ : i)(a) ≰ μ(a). us, there exists a μ-atom j ∈ Q with respect to α � (∩ i∈A μ : i)(a) such that j ≤ a. en, we have j ∈ A, and hence,

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which is a contradiction to the fact that j is a μ-atom with respect to α � (∩ i∈A μ : i)(a). Hence, α � (∩ i∈A μ : i)μ. erefore, (∩ i∈A μ : i) ⊆ μ. □ Lemma 13. e intersection of any nonempty family of L-fuzzy prime ideals of Q is an L-fuzzy semiprime ideal Q.
Proof. Let μ i : i ∈ Δ be a nonempty family of L-fuzzy prime ideals of Q. Put μ � ∩ i∈Δ μ i . en, clearly, μ is an L-fuzzy ideal of Q . Let a, b, c ∈ Q and z ∈ a, (b, c) for each i ∈ Δ.
(56) erefore, μ � ∩ i∈Δ μ i is an L-fuzzy semiprime ideal Q. As an immediate consequence of eorem 4, Lemma 12, and Lemma 13 in the case of posets satisfying DCC, we obtain the following result. In the following, we characterize the distributive posets in terms of L-fuzzy semiprime ideals in the following results. (x] of Q is an L-fuzzy semiprime ideal of Q, for each x ∈ Q.