Spectrum of prime L -fuzzy ideals of an ordered semigroup

Abstract

In this paper, we redefine the concept of prime L-fuzzy ideals of an ordered semigroup so that the prime L-fuzzy ideals are not necessarily 2-valued. Then a topological space, called the spectrum of prime L-fuzzy ideals of an ordered semigroup, has been obtained and some topological properties like separation axioms, compactness, connectedness are researched. Further, a contravariant functor from the category of commutative ordered semigroups into the category of compact and connected topological spaces is gotten. Finally, we focus on the subspace which is defined in the set of all minimal prime ideals in an ordered semigroup S and show that if S is commutative, then this subspace is Hausdorff, totally disconnected and completely regular.

Keywords

Ordered semigroup Spectrum Contravariant functor

1. Introduction

An ordered semigroup is a typical combination of algebraic structures and ordered structures. Due to its various applications, many scholars have researched such ordered semigroups under different restrictions. For a comprehensive survey, readers can refer to Petrich’s book [28].

The theory of fuzzy set proposed by Zadeh in 1965 [39] opened up keen insights and applications in a wide range of scientific fields. Fuzzy sets in ordered semigroups were first discussed by Kehayopulu and Tsingelis in [14], and they defined fuzzy analogies for several notations, which are very usefull in ordered semigroups. Further, they proved that any ordered groupoid can be embedded into an ordered groupoid which has the greatest element in terms of fuzzy sets [6]. Moreover, they characterized the bi-ideals in ordered semigroups using fuzzy bi-ideals [15] and regular ordered semigroups in terms of fuzzy left, right ideals and fuzzy quasi-ideals [16]. Recently, a great amount of work has been done on fuzzy sets, fuzzy ideals of ordered semigroups in general and prime fuzzy ideals of ordered semigroups in particular [3 , 36]. In [36], X. Y. Xie and J. Tang studied prime fuzzy ideals in an ordered semigroup in detail. They defined prime fuzzy ideals and proved that a fuzzy ideal f of an ordered semigroup is prime if and only if Im (f) ≤2 and the ideal f¹ (≠ ∅) is prime. This shows that the image of prime fuzzy ideals in ordered semigroups turns out to be just two real numbers and one level set becomes trivially prime. In the ring theory, R. Kumar [22] refined the prime fuzzy ideals and gave the following characterizations: “A fuzzy ideal f of a ring R is prime if and only if the level ideals f_t = {x ∈ R : f (x) ≥ t}, where t ∈ Im (f), are prime ideals of R." This definition introduced by R. Kumar leads that prime fuzzy ideals are not necessarily 2-valued. Due to this, we redefined the prime L-fuzzy ideals so that prime L-fuzzy ideals are not necessarily 2-valued.

On the other hand, the prime spectra of bounded distributive lattices had been initiated by Stone [31] and he established the dual-equivalence of the category of bounded distributive lattices and certain category of topological spaces. Recently, the prime fuzzy spectrum of an algebraic structure (ring, semiring, hemiring, etc.) is introduced and discussed by many scholars [4 , 38]. It is increasingly discovered that the research of spectrum or the hull-kernel topology (also called Zariski topology) on the set of prime(fuzzy) ideals, plays a very important role in the areas of commutative algebra, order theory, lattice theory, algebraic geometry and topology theory.

Motivated by the study of prime fuzzy ideals in ordered semigroups and also by the prime fuzzy spectrum on an algebraic structure (ring, semiring, hemiring, etc.), we attempt to study the prime fuzzy spectrum on an ordered semigroup. In order to study more widely, we research the prime L-fuzzy spectrum, where L denotes a complete lattice. At first, we redefine the concept of prime L-fuzzy ideals of an ordered semigroup so that the prime L-fuzzy ideals are not necessarily 2-valued. And then, the prime L-fuzzy spectrum of an ordered semigroup or the hull-kernel topology is defined and some topological properties like separation axioms, compactness, connectedness are studied. The correspondence associating an ordered semigroup with its prime L-fuzzy spectrum is shown to obtain a contravariant functor from the category of commutative ordered semigroups into that of compact and connected topological spaces. Further, a subspace of the spectrum of prime L-fuzzy ideals of an ordered semigroup, defined in the set of minimal prime L-fuzzy ideals, is investigated. Moreover, if S is commutative, then this subspace is Hausdorff, totally disconnected, completely regular and has a base in which every element is clopen. Finally, some categorical property is obtained. As an application of the results of this paper, the corresponding results in semigroups (without the order structure) can be obtained.

The main task of this paper reads as follows: In Section 2, we shall briefly review topological spaces, ordered semigroups and related basic definitions. In Section 3, L-fuzzy ideals, prime L-fuzzy ideals and semiprime L-fuzzy ideals are studied. In Section 4, we introduce the spectrum of prime L-fuzzy ideals in an ordered semigroup and obtain some topological properties. Finally, the subspace defined in the set of minimal prime L-fuzzy ideals in an ordered semigroup with 0 is investigated.

2. Preliminaries

In this section, we shall recall some notions and notations used in this paper.

An ordered semigroup is a semigroup (S, ·) with a partial order ≤ on S which is compatible with the multiplication ·, i.e., x ≤ y implies x · z ≤ y · z and z · x ≤ z · y for all x, y, z ∈ S. If (S, ≤ , ·) is an ordered semigroup, it is customary to write x · y as xy for all x, y ∈ S. S is called commutative if xy = yx for any x, y ∈ S. Sometimes, we denote an ordered semigroup (S, ≤ , ·) by S. For subsets A, B of S, let (A] = {x ∈ S : x ≤ a for some a ∈ A}, [A) = {x ∈ S : x ≥ a for some a ∈ A} and AB = {ab : a ∈ A, b ∈ B}. The subset A of S is called a subsemigroup of S if for any x, y ∈ S, xy ∈ A. Let S be an ordered semigroup with a zero element θ. A zero element for S is an element θ such that θ ≤ x and sθ = θs = θ for all s in S. An identity for an ordered semigroup S is an element e such that xe = x for all x in S. Let S and T be ordered semigroups. f : S → T is called an ordered semigroup homomorphism if f is order preserving and a semigroup homomorphism.

Let S be an ordered semigroup and L be a complete lattice, 1, 0 denote the maximum element, minimum element of L, respectively. An arbitrary mapping μ : X → L is called an L-fuzzy subset of S. For any A ⊆ S, the L-fuzzy subset χ_A is defined by $χ_{A} (x) = {\begin{matrix} 1, & if x \in A, \\ 0, & otherewise . \end{matrix}$ χ_A is said to be the characteristic function of A. In particular, when A = {x}, χ_x is said to be the characteristic function of x. Let A, B ⊆ S. Then χ_A ∪ χ_B = χ_(A∪B), and χ_A ⊆ χ_B if and only if A ⊆ B.

Let μ, ν be two L-fuzzy subsets of S. Then the inclusion relation μ ⊆ ν is defined by $μ (x) \leq ν (x),$ for all x ∈ S. μ ∩ ν and μ ∪ ν are defined by $(μ \cap ν) (x) = μ (x) \land ν (x),$ $(μ \cup ν) (x) = μ (x) \lor ν (x),$ for all x ∈ S.

In this paper, if there is no further statement, L always denotes a complete lattice.

3. Prime and semiprime L-fuzzy ideals

In this section, we introduce the concepts of L-fuzzy ideals, prime L-fuzzy ideals and semiprime L-fuzzy ideals and investigate the properties of these concepts.

Definition 3.1. [11 –13] Let (S, ≤ , ·) be an ordered semigroup. A nonempty subset I of S is called a left(resp. right) ideal of S if the following holds:

SI ⊆ I(resp. IS ⊆ I);

for x ∈ I, y ∈ S, y ≤ x implies y ∈ I.

I is called an ideal of S if it is a left and right ideal of S.

Definition 3.2. Let S be an ordered semigroup. An L-fuzzy subset μ of S is called an L-fuzzy left (resp. right) ideal of S if for any x, y ∈ S

x ≤ y ⇒ μ (x) ≥ μ (y);

μ (xy) ≥ μ (y) (resp. μ (xy) ≥ μ (x)).

An L-fuzzy subset μ of S is called an L-fuzzy ideal of S if it is both an L-fuzzy left and an L-fuzzy right ideal of S.

Definition 3.3. [36] Let S be an ordered semigroup and μ be any L-fuzzy subset of S. The set $μ_{t} : = {x \in S : μ (x) \geq t}, where t \in L$ is called a level subset of μ.

Throughout this paper, for any L-fuzzy subset μ of S, μ¹ = {x ∈ S : μ (x) =1}.

Proposition 3.4. Let S be an ordered semigroup and μ be any L-fuzzy subset of S. Then μ is an L-fuzzy ideal of S if and only if its every non-empty level subset μ_t (t ∈ L) of S is an ideal of S.

Proof. Let μ be an L-fuzzy ideal of S, t ∈ L and μ_t≠ ∅. Let x ∈ μ_t, that is μ (x) ≥ t. Since μ is an L-fuzzy ideal of S, we have μ (sx) ≥ μ (x) ≥ t and μ (xs) ≥ μ (x) ≥ t for any s ∈ S. Thus sx ∈ μ_t, xs ∈ μ_t and so Sμ_t ⊆ μ_t, μ_tS ⊆ μ_t. Let x ∈ μ_t, s ∈ S with s ≤ x. Then μ (s) ≥ μ (x) ≥ t. So s ∈ μ_t. Therefore, μ_t (t ∈ L) of S is an ideal of S whenever μ_t is nonempty.

Let x, y ∈ S with x ≤ y and t = μ (y). Then y ∈ μ_t. Since μ_t is an ideal of S, we have x ∈ μ_t. So μ (x) ≥ t = μ (y). Let x, y ∈ S and a = μ (x). Then x ∈ μ_a. Since μ_a is an ideal of S, we have xy ∈ μ_a. So μ (xy) ≥ a = μ (x) and similarly, μ (xy) ≥ μ (y). Therefore, μ is an L-fuzzy ideal of S.

Corollary 3.5. Let S be an ordered semigroup and ∅ ≠ A ⊆ S. Then A is an ideal of S if and only if the characteristic function χ_A is an L-fuzzy ideal of S.

Definition 3.6. [13] Let S be an ordered semigroup. A proper ideal I of S is said to be prime if A, B ⊆ S, AB ⊆ I implies A ⊆ I or B ⊆ I.

Equivalently, x, y ∈ S, xy ∈ I implies x ∈ I ory ∈ I.

Definition 3.7. Let S be an ordered semigroup. An L-fuzzy ideal μ of S is called prime if μ ≠ χ_S and for any x, y ∈ S and t ∈ L, μ satisfies $μ (xy) \geq t \Rightarrow μ (x) \geq t or μ (y) \geq t .$ The set of all prime L-fuzzy ideals is denoted by $PL (S)$ .

Proposition 3.8. Let S be an ordered semigroup and μ (≠ χ_S) an L-fuzzy ideal of S. Then μ is prime if and only if its every non-empty level subset μ_t (t ∈ L) is either a prime ideal of S or Sitself.

Proof. Let μ be a prime L-fuzzy ideal of S, t ∈ L and S≠ μ_t ≠ ∅. By Proposition 3.4, we know that μ_t is an L-fuzzy ideal of S. Let x, y ∈ S with xy ∈ μ_t, that is, μ (xy) ≥ t. By the primeness of μ, we have μ (x) ≥ t or μ (y) ≥ t. Therefore x ∈ μ_t or y ∈ μ_t and so μ_t is a prime ideal of S.

Let x, y ∈ S, t ∈ L with μ (xy) ≥ t. Then xy ∈ μ_t. If μ_t = S, then x, y ∈ μ_t. If μ_t ≠ S, then μ_t is prime. Thus x ∈ μ_t or y ∈ μ_t. So μ (x) ≥ t or μ (y) ≥ t and therefore μ is prime.

Corollary 3.9. Let S be an ordered semigroup and P an ideal of S. Then P is prime if and only if χ_P is a prime L-fuzzy ideal of S.

It follows from Proposition 3.8.

Proposition 3.10. Let S be an ordered semigroup. An L-fuzzy ideal μ of S is prime if and only if for any x, y ∈ S, μ (xy) = μ (x) or μ (y). If L is totally ordered, in particular if L = [0, 1], then μ is prime if and only if μ (xy) = μ (x) ∨ μ (y).

Proof. Let μ be a prime L-fuzzy ideal of S and x, y ∈ S. Let t = μ (xy). Since μ is an L-fuzzy ideal of S, we have μ (xy) = t ≥ μ (x) and μ (xy) = t ≥ μ (y). By the primeness of μ, we have μ (x) ≥ t or μ (y) ≥ t. Therefore μ (x) = t = μ (xy) or μ (y) = t = μ (xy). The converse is obvious.

If L is totally ordered, then for any x, y ∈ S, μ (xy) = μ (x) or μ (y) is equivalent to μ (xy) = μ (x) ∨ μ (y). Therefore the result holds.

If a fuzzy ideal is prime according to Definition 4.1 of [36], then by Proposition 3.8, it is prime according to Definition 3.7. The following example shows that the converse is not true.

Example 3.11. (N, ≤ , ·) with the standard multiply operation is an ordered semigroup, where for any m, n ∈ N, m ≤ n if there exists k ∈ N such that m = k · n. Let p be a prime integer. Define a fuzzy set μ_p : N → [0, 1] by

$μ_{p} (x) = {\begin{matrix} 1, & if x = 0, \\ 12, & if x \in p \cdot N \ 0, \\ 13, & if x \notin p \cdot N . \end{matrix}$

By Proposition 10, μ_p is a prime L-fuzzy ideal. But obviously, μ_p is not a prime L-fuzzy ideal according to Lemma 4.3 of [36].

Definition 3.12. [13] Let S be an ordered semigroup. An ideal I of S is said to be semiprime if A ⊆ S, A² ⊆ I implies A ⊆ I.

Equivalently, a ∈ S, a² ∈ I implies a ∈ I.

Definition 3.13. Let S be an ordered semigroup. An L-fuzzy ideal μ of S is called semiprime if for all x ∈ S, t ∈ L, μ (x²) ≥ t implies μ (x) ≥ t.

Proposition 3.14. Let S be an ordered semigroup. An L-fuzzy ideal μ of S is semiprime if and only if μ (x²) = μ (x), for any x ∈ S.

Proof. This proof is parallel to that of Proposition 10.

Proposition 3.15. Let S be an ordered semigroup and μ an L-fuzzy ideal of S. Then μ is semiprime if and only if its every non-empty level subset μ_t (t ∈ L) is either a semiprime ideal of S or S itself.

Proof. This proof is parallel to that of Proposition 3.8.

The following corollaries are immediate.

Corollary 3.16. Let S be an ordered semigroup and I an ideal of S. Then I is semiprime if and only if χ_I is a semiprime L-fuzzy ideal of S.

Corollary 3.17. Let S be an ordered semigroup and μ_i (i ∈ Λ) semiprime L-fuzzy ideals of S. Then ⋂ {μ_i : i ∈ Λ} is also a semiprime L-fuzzy ideal of S. In particular, if μ_i (i ∈ Λ) are prime L-fuzzy ideals of S, then ⋂ {μ_i : i ∈ Λ} is also a semiprime L-fuzzy ideal of S.

Although every prime L-fuzzy ideal of S is semiprime, the following example shows that, the converse is not true.

Example 3.18. (N⁺, ≤ , ·) with the standard multiply operation is an ordered semigroup, where for any m, n ∈ N⁺, m ≤ n if there exists k ∈ N⁺ such that m = k · n. Let p be a prime integer. Define a fuzzy set μ : N⁺ → [0, 1] by $μ (x) = {\begin{matrix} 0, & if x \in N^{+} \ 2 \cdot N^{+}, \\ 12, & if x \in x \in 2 \cdot N^{+} \ 6 \cdot N^{+}, \\ 1, & if x \in 6 \cdot N^{+} . \end{matrix}$

Then by Proposition 3.14, μ is a semiprime L-fuzzy ideal of S, but not a prime L-fuzzy ideal.

4. Spectrum of prime L-fuzzy ideals

In this section, we introduce the spectrum of prime L-ideals of an ordered semigroup S and investigate some topological properties like separation axioms, compactness, connectedness.

Let A be a nonempty subset of an ordered semigroup S. We denote I (A) the ideal of S generated by A i.e. the smallest (under inclusion relation) ideal of S containing A and therefore I (A) = (A ∪ SA ∪ AS ∪ SAS]. For any x ∈ S, I (x) = (x ∪ Sx ∪ xS ∪ SxS]. If S is commutative, then I (A) = (A ∪ SA] = (A] ∪ (SA].

Let S be an ordered semigroup. If μ is any L-fuzzy subset of S, let $H (μ) = {ν \in PL (S) : μ \subseteq ν}$ . When μ = χ_x, where x ∈ S, we denote $H (μ)$ by $H (x)$ . So $H (x) = {μ \in PL (S) : μ (x) = 1}$ .

Proposition 4.1. Let S be an ordered semigroup and x, y ∈ S. Then

(1) $H (x) = H (χ_{I (x)})$ , $H (χ_{A}) = H (χ_{I (A)})$ for any x ∈ S, A ⊆ S;

(2) $H (x) = H (x^{n})$ for any n ∈ N⁺;

(3) $μ \subseteq ν \Rightarrow H (ν) \subseteq H (μ)$ , for any L-fuzzy subsets μ, ν of S;

(4) x ≤ y implies $H (y) \subseteq H (x)$ ;

(5) x ≤ sy, for some s ∈ S, implies $H (y) \subseteq H (x)$ ;

(6) $H (μ) \cup H (ν) \subseteq H (μ \cap ν)$ , for any L-fuzzy subsets μ, ν of S;

(7) $H (χ_{A}) ⋃ H (χ_{B}) = H (χ_{(A \cap B)})$ for any A, B ⊆ S;

(8) If {μ_i : i ∈ Λ} is a family of L-fuzzy subsets of S, then $H (⋃ {μ_{i} : I \in Λ}) = ⋂ {H (μ_{i}) : I \in Λ}$ ;

(9) If M ⊆ S, then $H (χ_{M}) = ⋂ {H (A) : A \in M}$ ;

(10) $H (x) ⋃ H (y) = H (xy)$ , for any x, y ∈ S.

Proof. (1) - (5) are immediate by the definitions.

(6) Let $ω \in H (μ) \cup H (ν)$ , that is $ω \in H (μ)$ or $ω \in H (ν)$ , and without loss of generality, assume that $ω \in H (μ)$ . This implies that μ ∩ ν ⊆ μ ⊆ ω. Thus $H (μ) \cup H (ν) \subseteq H (μ \cap ν)$ .

(7) Firstly we will show that if I₁, I₂ are ideals of S, then $H (χ_{I_{1}}) ⋃ H (χ_{I_{2}}) = H (χ_{(I_{1} \cap I_{2})})$ . Because χ_{I
₁} ∩ χ_{I
₂} = χ_{(I₁∩I₂)}, we can get $H (χ_{I_{1}}) ⋃ H (χ_{I_{2}}) \subseteq H (χ_{(I_{1} \cap I_{2})})$ by the item (6). To prove the converse, let $μ \in H (χ_{(I_{1} \cap I_{2})})$ , that is χ_{(I₁∩I₂)} ⊆ μ. Hence for any a ∈ I₁ ∩ I₂, μ (a) =1. Suppose that χ_{I
₁}notsubseteqμ and χ_{I
₂}notsubseteqμ. So there exist a₁ ∈ I₁, a₂ ∈ I₂ such that μ (a₁) ≠ 1 and μ (a₂) ≠ 1. Since I₁, I₂ are ideals of S, we have a₁a₂ ∈ I₁ ∩ I₂. Thus μ (a₁a₂) =1. As μ is an L-fuzzy ideal of S, we have μ (a₁) =1 or μ (a₂) =1, which is a contradiction. Hence χ_{I
₁} ⊆ μ or χ_{I
₂} ⊆ μ and therefore $H (χ_{(I_{1} \cap I_{2})}) \subseteq H (χ_{I_{1}}) ⋃$ $H (χ_{I_{2}})$ .

Then we will show that $H (χ_{A}) ⋃ H (χ_{B}) = H (χ_{(A \cap B)})$ . By what was just proved, $H (χ_{A}) ⋃ H (χ_{B}) = H (χ_{I (A)}) ⋃ H (χ_{I (B)})) = H (χ_{i} (A)$ ∩I (B))). Since I (A ∩ B) = I (A) ∩ I (B), we have $H (χ_{I (A) \cap I (B)})) = H (χ_{I (A \cap B)})) = H (χ_{(A \cap B)})$ . Therefore for any A, B ⊆ S, $H (χ_{A}) ⋃ H (χ_{B}) = H (χ_{(A \cap B)})$ .

(8) Assume $ν \in ⋂ {H (μ_{i}) : i \in Λ}$ . Then μ_i ⊆ ν for each i ∈ Λ, which gives ⋃ {μ_i : i ∈ Λ} ⊆ ν and $ν \in H (⋃ {μ_{i} : i \in Λ})$ . Hence $⋂ {H (μ_{i}) : i \in Λ} \subseteq H (⋃ {μ_{i} : i \in Λ})$ .

Conversely, let $ν \in H (⋃ {μ_{i} : i \in Λ})$ . Then ⋃ {μ_i : i ∈ Λ} ⊆ ν. Hence μ_i ⊆ ν for each i ∈ Λ. Consequently, $ν \in H (μ_{i})$ for each i ∈ Λ and thus $ν \in ⋂ {H (μ_{i}) : i \in Λ}$ .

(9) Since χ_M = ⋃ {χ_a : a ∈ M}, we can get the result by the item (8).

(10) Let $μ \in H (x) ⋃ H (y)$ , that is $μ \in H (x)$ or $μ \in H (y)$ , and without loss of generality, assume that $μ \in H (x)$ . This implies that μ (x) =1. Since μ is an L-fuzzy ideal of S, we have μ (xy) ≥ μ (x) =1 which means μ (xy) =1. Hence $μ \in H (xy)$ and $H (x) ⋃ H (y) \subseteq H (xy)$ .

Conversely, let $μ \in H (xy)$ , that is μ (xy) =1. By the primeness of μ, we have μ (x) =1 or μ (y) =1. Hence $μ \in H (x) ⋃ H (y)$ and therefore $H (xy) \subseteq H (x) ⋃ H (y)$ .

In general, the equality does not hold in the item (6) of Proposition 4.1. In crisp case, if μ, ν are ideals of S, then the equality holds. The following example shows that the equality does not hold even for L-fuzzy ideals.

Example 4.2. (N, ≤ , ·) with the standard multiply operation is an ordered semigroup, where for any m, n ∈ N, m ≤ n if there exists k ∈ N such that m = k · n. Let p be a prime integer. Define a fuzzy set μ : N → [0, 1] by

$μ (x) = {\begin{matrix} 1, & if x = 0, \\ 0.6, & if x \in 2 \cdot N \ 0, \\ 0.4, & if x \notin 2 \cdot N . \end{matrix}$

Define a fuzzy set ν₁ : N → [0, 1] by

$ν_{1} (x) = {\begin{matrix} 1, & if x = 0, \\ 0.7, & if x \in 4 \cdot N \ 0, \\ 0.3, & if x \notin 4 \cdot N . \end{matrix}$

Define another fuzzy set ν₂ : N → [0, 1] by

$ν_{2} (x) = {\begin{matrix} 1, & if x = 0, \\ 0.55, & if x \in 4 \cdot N \ 0, \\ 0.5, & if x \notin 4 \cdot N . \end{matrix}$

Therefore, ν₁, ν₂ are L-fuzzy ideals of S, μ is a prime L-fuzzy ideal of S and ν₁ ∩ ν₂ ⊆ μ. But as ν₁ (x) ≥ μ (x) for x ∈ 4 · N \ 0, we have ν₁ ≰ μ. Since ν₂ (x) ≥ μ (x) for x ∉ 2 · N, we have ν₂ ≰ μ. Hence $μ \in H (ν_{1} \cap ν_{2})$ , but $μ \notin H (ν_{1}) \cup H (ν_{2})$ .

In the following, we will show that the set ${H (χ_{A}) : A \subseteq S}$ forms a system of closed sets for a topology on the set $PL (S)$ of prime L-fuzzy ideals of an ordered semigroup S. If μ : S → L is any L-fuzzy subset, let $P (μ) = PL (S) \ H (μ)$ . So $P (μ) = {ν \in PL (S) : μ ⊈ ν}$ . For any x ∈ S, let $P (x) = PL (S) \ H (x)$ . So $P (x) = {μ \in PL (S) : μ (x) \neq 1}$ .

Theorem 4.3. Let S be an ordered semigroup. The set ${P (x) : x \in S}$ forms a basis for a topology on $PL (S)$ . The open sets of $PL (S)$ are precisely $P (χ_{A})$ , where A ⊆ S. This topology is completely determined by semiprime L-fuzzy ideals of S.

Proof. First we will show that $⋃ {P (x) : x \in S} = PL (S)$ . Because $⋂ {H (x) : x \in S} = {μ \in PL (S) : μ (x) = 1, \forall x \in S}$ and for any $μ \in PL (S)$ , μ ≠ χ_S, we have $⋂ {H (x) : x \in S} = \emptyset$ . So $⋃ {P (x) : x \in S} = PL (S) \ ⋂ {H (x) : x \in S} = PL (S)$ . By the item (10) of Proposition 4.1, we have $P (x) ⋂ P (y) = P (xy)$ for any x, y ∈ S. So the set ${P (x) : x \in S}$ forms a basis for a topology on $PL (S)$ . An open set in this topology is precisely $⋃ {P (x) : x \in A}$ for some A ⊆ S. So, we have $\begin{matrix} ⋃ {P (x) : x \in A} = & ⋃ {PL (S) \ H (x) : x \in A} \\ = & PL (S) \ ⋂ {H (x) : x \in A} \\ = & PL (S) \ H (χ_{A}) \\ = & P (χ_{A}) . \end{matrix}$

Obviously, if V is a closed set of this topology, then $V = H (χ_{A})$ for some A ⊆ S. Since $H (χ_{A}) = H (⋂ {μ : μ \in H (χ_{A})})$ and by Corollary 3 $⋂ {μ : μ \in H (χ_{A})$ is a semiprime L-fuzzy ideal of S, we conclude this topology is completely determined by semiprime L-fuzzy ideals of S.

Definition 4.4. Let S be an ordered semigroup. The topological space of prime L-fuzzy ideals of S defined in Theorem 4.3 will be called the spectrum of prime L-fuzzy ideals(or in short prime L-fuzzy spectrum) of S and will be denoted by L-spec (S).

Example 4.5. (1) (R, ≤ , ·) with the standard order and standard multiply operation is an ordered semigroup. Define a fuzzy set μ^αβ : R → [0, 1] by $μ^{α β} (x) = {\begin{matrix} α, & if x \leq 0, \\ β, & if x > 0 . \end{matrix}$

Since the only prime L-fuzzy ideal of R is (- ∞ , 0], we have the set of prime L-fuzzy ideals is $PL (R) = {μ^{α β} : 0 \leq β \leq α \leq 1, α, β are not equal to$ 1 at the same time .}. If x > 0, then $P (x) = PL (R)$ . If x ≤ 0, then $P (x) = {μ^{α β} \in PL (R) : α \neq 1}$ . The basis of L-spec (R) is ${P (0), P (1)}$ .

(2) (N, ≤ , ·) with the standard multiply operation is an ordered semigroup, where for any m, n ∈ N, m ≤ n if there exists k ∈ N such that m = k · n. Let p be a prime integer. Define a fuzzy set ν_p : N → [0, 1] by

$ν_{p}^{α β γ} (x) = {\begin{matrix} α, & if x = 0, \\ β, & if x \in p \cdot N \ 0, \\ γ, & if x \notin p \cdot N . \end{matrix}$ It is easy to check that $ν_{p}^{α β γ}$ is a prime L-fuzzy ideal if 0 ≤ γ ≤ β ≤ α ≤ 1 and α, β, γ are not equal to 1 at the same time. The set of all prime L-fuzzy ideals is $ℒ (N) = {ν_{p}^{α β γ} : 0 \leq γ \leq β \leq α \leq 1, α, β, γ$ are not equal to 1 at the same time .}. The basis of L-spec (R) is ${P (n) : n \in N}$ .

For any subset X of $PL (S)$ , we denote the closure of X in the space L-spec (S) by $\bar{X}$ .

Proposition 4.6. Let S be an ordered semigroup and X any subset of $PL (S)$ . Then $\bar{X} = H (χ_{⋂ {μ^{1} : μ \in X}})$ , where μ¹ = {x ∈ S : μ (x) =1}. If $μ, ν \in PL (S)$ , then $\bar{μ} = \bar{ν}$ if and only if μ¹ = ν¹. So, L-spec (S) is not a T₀ space.

Proof. Clearly, $H (χ_{⋂ {μ^{1} : μ \in X}})$ is a closed set of L-spec (S) and $X \subseteq H (χ_{⋂ {μ^{1} : μ \in X}})$ . We claim that any closed set Y containing X also contains $H (χ_{⋂ {μ^{1} : μ \in X}})$ . By Theorem 4.3, $Y = H (χ_{A})$ for some subset A ⊆ S. Since $X \subseteq H (χ_{A})$ , we have for any a ∈ A, μ ∈ X, μ (a) =1. So A ⊆ ⋂ {μ¹ : μ ∈ X} and hence χ_A ⊆ χ_{⋂{μ¹:μ∈X}}. By the item (3) of Proposition 4.1, $H (χ_{⋂ {μ^{1} : μ \in X}}) \subseteq H (χ_{A})$ . Therefore $\bar{X} = H (χ_{⋂ {μ^{1} : μ \in X}})$ .

In the classical case, Wu., Li. and Yu. [34] defined the spectrum of prime ideal of an ordered semigroup S with the zero element θ which willed be denoted by Spec (S) and researched many topological properties of this topological space. More precisely, if $P (S)$ denotes the set of all prime ideals of S, then a typical closed set of Spec (S) is of the form ${P \in P (S) : I \subseteq P}$ for some ideal I of S. In the following theorem, we establish a close relation between Spec (S) and L-spec (S) which shows Spec (S) is homeomorphic to a dense subspace of L-spec (S).

Theorem 4.7. Let S be an ordered semigroup with the zero element θ. The topological space Spec (S) is homeomorphic to a dense subspace of L-spec (S).

Proof. Let $f : P (S) \to PL (S)$ be a mapping which maps any $P \in P (S)$ to χ_P. Let $X = {χ_{P} : P \in P (S)}$ . Since P is a prime ideal of S, we have χ_P is a prime L-fuzzy ideal of S. Thus, $X \subseteq PL (S)$ . Hence a closed subset of the induced subspace X of $PL (S)$ is of the form {χ_P : A ⊆ P} for some subset A ⊆ S. Clearly, f is a homeomorphism of Spec (S) into the induced subspace X of L-spec (S).

Then we will show that $\bar{X} = PL (S)$ . By Proposition 4, $\bar{X} = H (χ_{⋂ {μ^{1} : μ \in X}}) = H (χ_{⋂ {P : P \in P (S)}})$ . For any $μ \in PL (S)$ , let μ¹ = {x ∈ S : μ (x) =1}. By Proposition 3, μ¹ is a prime ideal of S which means $μ^{1} \in P (S)$ . So $⋂ {P : P \in P (S) \subseteq μ^{1}$ and $χ_{⋂ {P : P \in P (S)}} \subseteq χ_{μ^{1}} \subseteq μ$ . Thus $μ \in H (χ_{⋂ {P : P \in P (S)}}) = \bar{X}$ . Therefore $PL (S) \subseteq \bar{X}$ and $\bar{X} = PL (S)$ .

Definition 4.8. [8, 35] If I is an ideal of an ordered semigroup S, then the radical of I, denoted by $\sqrt{I}$ , is defined by $\sqrt{I} = {s \in S : \exists n \in N^{+}, s^{n} \in I} .$

It is easily seen that $I \subseteq \sqrt{I}$ and $(\sqrt{I}] = \sqrt{I}$ .

Lemma 4.9. Let S be a commutative ordered semigroup. Then $H (χ_{A}) = \emptyset$ if and only if $\sqrt{I (A)} = S$ .

Proof. Assume that $H (χ_{A}) = \emptyset$ but $\sqrt{I (A)}$ is proper. So there exists a ∈ S but $a \notin \sqrt{I (A)}$ . Let B = {aⁿ : n ∈ N⁺}. Thus I (A)⋂ B = ∅. Let M = ⋃ {I ⊆ S : I is an ideal of S and I ⋂ B = ∅}. Clearly, M is an ideal of S and I (A) ⊆ M. Then we will show that M is prime. Let xy ∈ M but x ∉ M, y ∉ M. Hence (M⋃ I (x)) ⋂ B ≠ ∅ and (M⋃ I (y)) ⋂ B ≠ ∅. So there exist n₁, n₂ ∈ N⁺ such that a^{n
₁} ∈ (M ⋃ I (x)) ⋂ B and a^{n
₂} ∈ (M ⋃ I (y)) ⋂ B. Thus a^{n
₁} ∈ I (x) and a^{n
₂} ∈ I (y). As I (x) = (x] ∪ (Sx] for any x ∈ S, four cases are possible: (1) a^{n
₁} ≤ x, a^{n
₂} ≤ y; (2) a^{n
₁} ≤ s₁x, a^{n
₂} ≤ y for some s₁ ∈ S; (3) a^{n
₁} ≤ x, a^{n
₂} ≤ s₂y for some s₂ ∈ S; (4) a^{n
₁} ≤ s₃x, a^{n
₂} ≤ s₄y for some s₃, s₄ ∈ S. Of these, we only consider the fourth case and the other cases are similar. If (4) holds, then a^{n
₁}a^{n
₂} = a^n₁+n₂ ≤ s₃s₄xy. Since xy ∈ M, we have a^{n
₁}a^{n
₂} ∈ M which contradicts with M⋂ B = ∅. So x ∈ M or y ∈ M and therefore M is prime. By Corollary 3.9, χ_M is a prime L-fuzzy ideal of S which contradicts with $H (χ_{A}) = \emptyset$ . So $\sqrt{I (A)} = S$ .

Assume that $\sqrt{I (A)} = S$ but there exists a prime L-fuzzy ideal $μ \in H (χ_{A})$ . For any $x \in \sqrt{I (A)}$ , there exists n ∈ N⁺ such that xⁿ ∈ I (A). So xⁿ ∈ μ¹ and by the primeness of μ¹, we have x ∈ μ¹. Hence $\sqrt{I (A)} \subseteq μ^{1}$ . Since $\sqrt{I (A)} = S$ , we have μ¹ = S which is a contradiction. Therefore $H (χ_{A}) = \emptyset$ .

Lemma 4.10. [35] Let S be a commutative ordered semigroup. If I is an ideal of S, then $\sqrt{I}$ is the intersection of all prime ideals containing I.

Lemma 4.11. Let S be a commutative ordered semigroup. Then $⋂ {μ : μ \in H (χ_{A})} = χ_{\sqrt{I (A)}}$ where A ⊆ S and $H (χ_{A}) \neq \emptyset$ .

Proof. Let $x \in \sqrt{I (A)}$ , that is, there exists n ∈ N such that xⁿ ∈ I (A). For any $μ \in H (χ_{A})$ , we have A ⊆ μ¹ = {x ∈ S : μ (x) =1}. Since μ¹ is a prime ideal of S, we have I (A) ⊆ μ¹. So xⁿ ∈ I (A) ⊆ μ¹ which implies x ∈ μ¹. Hence $\sqrt{I (A)} \subseteq μ^{1}$ and $χ_{\sqrt{I (A)}} \subseteq μ$ . Therefore $χ_{\sqrt{I (A)}} \subseteq ⋂ {μ : μ \in H (χ_{A})}$ .

For any prime ideal P containing I (A) of S, we have $χ_{P} \in H (χ_{A})$ . By Lemma 4, $⋂ {μ : μ \in H (χ_{A})} \subseteq χ_{\sqrt{I (A)}}$ . Therefore $⋂ {μ : μ \in H (χ_{A})} = χ_{\sqrt{I (A)}}$ .

Theorem 4.12. Let S be a commutative ordered semigroup. For any x ∈ S, $P (x)$ is compact. If addition, S has an identity, then the topological space L-spec (S) is compact and connected.

Let ${P (a) : a \in A}$ , where A ⊆ S, be an open cover of $P (x)$ . Then $\begin{matrix} P (x) = & PL (S) \ H (x) \\ \subseteq & ⋃ {P (a) : a \in A} \\ = & PL (S) \ ⋂ {H (a) : a \in A} \\ = & PL (S) \ H (χ_{A}) . \end{matrix}$ So $H (χ_{A}) \subseteq H (x)$ . Then we will show that $x \in \sqrt{I (A)}$ . If $H (χ_{A}) = \emptyset$ , then by Lemma 4, $\sqrt{I (A)} = S$ and hence $x \in \sqrt{I (A)}$ . If $H (χ_{A}) \neq \emptyset$ , then by Lemma 4, $⋂ {μ : μ \in H (χ_{A})} = χ_{\sqrt{I (A)}}$ and $⋂ {μ : μ \in H (x)} = χ_{\sqrt{I (x)}}$ . Because $H (χ_{A}) \subseteq H (x)$ , we have $χ_{\sqrt{I (x)}} \subseteq χ_{\sqrt{I (A)}}$ . So $\sqrt{I (x)} \subseteq \sqrt{I (A)}$ . Since $x \in \sqrt{I (x)}$ , we have $x \in \sqrt{I (A)}$ . Hence there exists n ∈ N⁺ such that xⁿ ∈ I (A). Because I (A) = (A] ⋃ (SA], two cases are possible: (1) xⁿ ≤ a for some a ∈ A; (2) xⁿ ≤ sa for some s ∈ S and a ∈ A. Of these, we only consider the second case and the first case is similar. If (2) holds, then $H (A) \subseteq H (x^{n}) = H (x)$ by Proposition 4.6. So there exists a ∈ A such that $P (x) \subseteq P (A)$ . This shows that $P (x)$ is compact.

If S has an identity e, then by what was just proved, $PL (S) = P (e)$ is compact. Suppose that $A$ is a subset of $PL (S)$ which is both open and closed, but it is neither ∅ nor $PL (S)$ . Let $I = ⋂ {μ^{1} : μ \in A}$ and $J = ⋂ {μ^{1} : μ \in PL (S) \ A}$ . Since $A$ is clopen, we have $H (χ_{i}) = A$ and $H (χ_{J}) = PL (S) \ A$ by Proposition 4.6. If $μ \in PL (S)$ such that χ_(I∪J) ⊆ μ, then $μ \in H (χ_{(I \cup J)}) = H (χ_{I}) \cap H (χ_{J}) = A ⋂ (PL (S) \ A)$ , which is impossible. So $H (χ_{i} \cup χ_{J}) = \emptyset$ . By Lemma 4.9, $\sqrt{I \cup J} = S$ . Thus, e ∈ I or e ∈ J. Since I, J are ideals of S, we have I = S, $A = \emptyset$ or J = S, $PL (S) \ A = \emptyset$ , which is a contradiction. Therefore $PL (S)$ is connected.

If S and T are ordered semigroups with identities, then we set that an ordered semigroup homomorphism between S and T preserves identity.

Proposition 4.13. Let (S, ≤ , ·) , (S′, ≤ ′, ★) be ordered semigroups with e, e′ respectively and f : S → S′ be an ordered semigroup homomorphism. If μ′ is a prime L-fuzzy ideal of S′, then μ′ ∘ f mapping any x ∈ S to μ′ (f (x)) is a prime L-fuzzy ideal of S.

Proof. First we will show that μ′ ∘ f is an L-fuzzy ideal of S. Let x, y ∈ S with x ≤ y. Since f is an ordered semigroup homomorphism, we have f (x) ≤ ′f (y). As μ′ is a prime L-fuzzy ideal of S′, we have μ′ (f (x)) ≥ μ′ (f (y)). Let x, y ∈ S. So f (x · y) = f (x) ★ f (y) and hence μ′ (f (x · y)) = μ′ (f (x) ★ f (y)). Since μ′ (f (x) ★ f (y)) ≥ μ′ (f (x)) and μ′ (f (x) ★ f (y)) ≥ μ′ (f (y)). Thus μ′ (f (x · y)) ≥ μ′ (f (x)) and μ′ (f (x · y)) ≥ μ′ (f (y)). Therefore μ′ ∘ f is an L-fuzzy ideal of S.

Then we claim that μ′ ∘ f is prime. Assume that μ′ (e′) =1. Then μ′ (x′) = μ′ (x′ ★ e′) ≥ μ′ (e′) =1, for any x′ ∈ S′. That is μ′ (x′) =1 for any x′ ∈ S′ which contradicts to the primeness of P. So μ′ (e′) ≠ 1. Since f (e) = e′, we have μ′ ((f (e)) ≠ 1. Therefore μ′ ∘ f ≠ χ_S. For any x, y ∈ S, μ′ (f (x · y)) = μ′ (f (x) ★ f (y)) = μ′ (f (x)) or μ′ (f (x)). Hence μ′ ∘ f is prime.

Proposition 4.14. Let S, S′ be ordered semigroups with identities. If f : S → S′ is an ordered semigroup homomorphism, then $f^{*} : PL (S^{'}) \to PL (S)$ which maps any $μ^{'} \in PL (S^{'})$ to f^* (μ′) = μ′ ∘ f is a continuous map between L-spec (S′) and L-spec (S).

Proof. By Proposition 5.11, f^* is well-defined. For any x ∈ S, $\begin{matrix} f^{* - 1} (H (x)) = & {μ^{'} \in PL (S^{'}) : f^{*} (μ^{'}) \in H (x)} \\ = & {μ^{'} \in PL (S^{'}) : f^{*} (μ^{'}) (x) = 1} \\ = & {μ^{'} \in PL (S^{'}) : μ^{'} (f (x)) = 1} \\ = & H^{'} (f (x)) . \end{matrix}$

As $H (x)$ is a basic closed set in L-spec (S) and $H^{'} (f (x))$ is a basic closed set in L-spec (S′), we have f^* is continuous.

Let S denote the category consisting of commutative ordered semigroups with identities as objects and homomorphisms of ordered semigroups as morphisms. We use T to denote the category consisting of compact and connected topological spaces as objects and continuous mappings of topological spaces as morphisms.

Theorem 4.15. Let S, S′ be commutative ordered semigroups with identities. Then the correspondence which associates S with the topological space L-spec (S) in T and a morphism f : S → S′ in S with the morphism $f^{*} : PL (S^{'}) \to PL (S)$ in T, defines a contravariant functor from S to T.

Let S, S′, S″ be ordered semigroups with identities and f : S → S′, g : S′ → S″ be morphisms in S. By Proposition 5.11 and 5.12, f^*, g^* are well-defined and f^*, g^* are morphisms in T. Clearly, (g ∘ f) ^* = f^* ∘ g^* and if f : S → S is the identity map on S in S, then f^* is the identity map on L-spec (S). This completes the proof.

5. Spectrum of minimal prime L-fuzzy ideals

In this section, we focus on the minimal prime L-fuzzy spectrum of an ordered semigroup S with the zero element θ. Throughout this section, the symbol S will always denote an ordered semigroup with θ which contains at least one minimal prime L-fuzzy ideal.

Let S be an ordered semigroup with a zero element θ. A zero element for S is an element θ such that θ ≤ x and sθ = θs = θ for all s in S. We further assume that for any L-fuzzy ideal μ of S, μ (θ) =1. In fact, the hypothesis μ (θ) =1, for an L-fuzzy ideal f of S is not really strict because μ (θ) ≥ μ (x) for any x ∈ S. A prime L-fuzzy ideal μ of an ordered semigroup S with θ is said to be a minimal L-fuzzy prime ideal of S if there is no other prime L-fuzzy ideal properly contained in μ. We denote the set of all minimal L-fuzzy prime ideals of S by $ML (S)$ .

Let S be an ordered semigroup with θ. For any L-fuzzy subset μ of S, let $H_{M} (μ) = H (μ) \cap ML (S) = {ν \in ML (S) : μ \subseteq ν}$ and $M (μ) = ML (S) \ H_{M} (μ)$ . Further, for s ∈ S, let $H_{M} (s) = H_{M} (χ_{s})$ and $M (s) = M (χ_{s})$ . So $H_{M} (s) = {ν \in ML (S) : ν (x) = 1}$ , $M (s) = {ν \in ML (S) : ν (x) \neq 1}$ .

We study the subspace topology of $PL (S)$ on the set $ML (S)$ and denote this subspace by ML-spec (S). The topological space ML-spec (S) is called thespectrum of minimal prime L-fuzzy ideals(or in short minimal prime L-fuzzy spectrum) of S.

Proposition 5 and Proposition 5.2 in the following are consequences of the above notations.

Proposition 5.1. Let S be an ordered semigroup S with the zero element θ. The collection ${M (x) : x \in S}$ forms a basis for the open sets of ML-spec (S).

Proposition 5.2. Let S be an ordered semigroup S with the zero element θ and X be any non-empty subset of ML-spec (S). Then $\bar{X} = H_{M} (χ_{⋂ {μ^{1} : μ \in X}})$ , where μ¹ = {x ∈ S : μ (x) =1}.

Definition 5.3. [1] Let P be a poset. D ⊆ P is called a directed subset of P if for any x, y ∈ D, there exists z ∈ D such that x ≤ z, y ≤ z. P is called a dcpo if for any directed subset of P, there exists a least upper bound in P.

Lemma 5.4. [1] If P is a dcpo, then there exists a maximal element in P.

Lemma 5.5. Let S be an ordered semigroup and I be an L-fuzzy ideal of S. If A is a subsemigroup such that A∩ I = ∅, then A is contained in a subsemigroup which is maximal with respect to the property of not meeting I.

Proof. Let $C = {T is A subsemigroup of S : I \cap T = \emptyset}$ . Then $C$ is a poset with respect to the inclusion order. Assume that $D \subseteq C$ is a directed set. Then for any b ∈ S, $a \in ⋃_{D \in D} D$ with a ≤ b, there exists $D_{0} \in D$ such that a ∈ D₀. So $b \in D_{0} \subseteq ⋃_{D \in D} D$ . For any $x, y \in ⋃_{D \in D} D$ , there exists $D_{1}, D_{2} \in D$ such that x ∈ D₁, y ∈ D₂. Since $D$ is directed, there exists $D_{3} \in D$ such that D₁ ⊆ D₃, D₂ ⊆ D₃. So x, y ∈ D₃ and $xy \in D_{3} \subseteq ⋃_{D \in D} D$ . Thus $⋃_{D \in D} D \in C$ . Hence $C$ is a dcpo. Therefore, A is contained in a subsemigroup T which is maximal with respect to the property of not meeting I by Lemma 5.4.

Lemma 5.6. Let S be a commutative ordered semigroup with the zero element θ. If A is a subsemigroup of S which satisfies θ ∉ A, then there exists an ideal M which is maximal with respect to the property of not meeting A. Moreover, M is prime.

Proof. Let M = ⋃ {I is an ideal of S : I ∩ A = ∅}. Then M is a maximal ideal with respect to the property of not meeting A. For an element a in S, let I (M, a) denote the ideal generated by M and a; then I (M, a) = M ∪ (a] ∪ (Sa]. Assume that xy ∈ M but x, y ∉ M. Then there are elements s, t ∈ S such that s ∈ A ∩ I (M, x) and t ∈ A ∩ I (M, y). Since A∩ M = ∅, four cases are possible: (1) s ≤ s₁x, t ≤ s₂y for some s₁, s₂ ∈ S; (2) s ≤ s₁x, t ≤ y for some s₁ ∈ S; (3) s ≤ x, t ≤ y; (4) s ≤ x, t ≤ s₁y for some s₁ ∈ S. Of these, we need consider only the case (1). If (1) holds, then st ≤ s₁s₂xy. Because s, t ∈ A and A is a subsemigroup, we have st ∈ A. Since xy ∈ M and M is an ideal, we have s₁s₂xy ∈ M and so st ∈ M. Hence st ∈ M ∩ A, contrary to M⋂ A = ∅. Therefore, M is prime.

Theorem 5.7. Let S be a commutative ordered semigroup with the zero element θ. If μ is a minimal prime L-fuzzy ideal, then S \ μ¹ is a subsemigroup of S which is maximal with respect to the property of not containing θ.

Proof. let μ be a minimal prime L-fuzzy ideal. Then μ¹ is a prime ideal of S and S \ μ¹ is a subsemigroup of S which does not contain θ, and by Lemma 5.5, it is contained in a subsemigroup T which is maximal with respect to the property of not containing θ. Hence S \ μ¹ ⊆ T. By Lemma 5, there exists a prime ideal M such that M⋂ T = ∅, that is T ⊆ S \ M. Hence S \ μ¹ ⊆ S \ M. So M ⊆ μ¹. Since M is a prime ideal of S, we have χ_M is a prime L-fuzzy ideal of S. Since M ⊆ μ¹, we have χ_M ⊆ μ which contradicts with the minimal of μ. Therefore S \ μ¹ is a subsemigroup of S which is maximal with respect to the property of not containing θ.

Characteristics of minimal prime L-fuzzy ideals of an ordered semigroup are given in the following theorem.

Theorem 5.8. Let S be a commutative ordered semigroup with the zero element θ. If μ is a minimal prime L-fuzzy ideal of S, then for any x ∈ μ¹ there exist n ∈ N, y ∈ S \ μ¹ such that yxⁿ = θ.

Proof. Let μ be a minimal prime L-fuzzy ideal containing I. By Theorem 5, S \ μ¹ is a subsemigroup of S which is maximal with respect to the property of not containing θ. Let $T = {{yx}^{n} : n \in N, y \in S \ μ^{1}}$ where, for any y ∈ S \ μ¹, yx⁰ = y. Then T is a subsemigroup of S and S \ μ¹ ⊆ T. Since for any y ∈ S \ μ¹, yx ∈ T but yx ∈ μ¹, i.e., yx ∉ S \ μ¹, we have T properly contains S \ μ¹. By the maximal property of S \ μ¹, we have θ ∈ T. So there exist n ∈ N, y ∈ S \ μ¹ such that yxⁿ = θ.

Proposition 5.9. Let S be an ordered semigroup with the zero element θ. For any x, y ∈ S, $M (x) \cap M (y) = M (xy)$ .

Proof. It is obvious.

Theorem 5.10. Let S be a commutative ordered semigroup with the zero element θ. Then

(1) $M (x)$ is clopen in ML-spec (S) for any x ∈ S;

(2) ML-spec (S) is a T₂ space;

(3) ML-spec (S) is totally disconnected;

(4) ML-spec (S) is completely regular.

Proof. (1) We have already observed that $M (x)$ is a basic open set. Let $μ \in ML (S)$ but $μ \notin M (x)$ . Then x ∈ μ¹, and so by Theorem 5.8, there exist n ∈ N, y ∈ S \ μ¹ such that yxⁿ = θ. Hence $M ({yx}^{n}) = \emptyset$ . By Proposition 5 $M ({yx}^{n}) = M (y) \cap M (x)$ . So $M (y) \subseteq ML (S) \ M (x)$ . Because y ∈ S \ μ¹, we have $μ \in M (y)$ . So $μ \in M (y) \subseteq ML (S) \ M (x)$ . This shows that $M (x)$ is closed.

(2) Let $μ, ν \in ML (S)$ with μ ≠ ν. We claim that μ¹ ≠ ν¹. If μ¹ = ν¹, then χ_{μ
¹} = χ_{ν
¹} ⊆ μ and χ_{μ
¹} = χ_{ν
¹} ⊆ μ which contradict with the minimal property of μ and ν. Therefore μ¹ ≠ ν¹. Without loss of generality, assume that μ¹ ⊈ ν¹. Let x ∈ ν¹ but x ∉ μ¹. Then $μ^{1} \in M (x)$ and $ν^{1} \in ML (S) \ M (x)$ . By the item (1), $ML (S)$ is a T₂ space.

(3) It follows from the proof of the item (2).

(4) Let $μ_{0} \in ML (S)$ and A be a closed set of $ML (S)$ not containing μ₀. Then there exists s₀ ∈ S such that $μ_{0} \in M (s_{0})$ and $M (s_{0}) ⋂ A = \emptyset$ by Proposition 5.1. Define $f : ML (S) \to [0, 1]$ by $f (μ) = {\begin{matrix} 1, & if μ \in M (s_{0}) \\ 0, & otherwise . \end{matrix}$ By the item (1), f is a continuous function such that f (μ₀) = {1} and f (A) = {0}. Thus $ML (S)$ is completely regular.

Proposition 5.11. Let (S, ≤ , ·) , (S′, ≤ ′, ★) be ordered semigroups with θ, e, θ′, e′ respectively and f : S → S′ be an ordered semigroup homomorphism and preserves prime ideals. If μ′ is a minimal prime L-fuzzy ideal of S′, then μ′ ∘ f mapping any x ∈ S to μ′ (f (x)) is a minimal prime L-fuzzy ideal of S.

Proof. By Proposition 5.1, μ′ ∘ f is a prime L-fuzzy ideal of S. Suppose that ν is a prime ideal of S such that ν ⊆ μ′ ∘ f. Then ν¹ ⊆ (μ′ ∘ f) ¹. Hence for any x ∈ ν¹, μ′ (f (x)) =1, that is μ′ (f (ν¹)) =1. Since f preserves prime ideals, we have f (ν¹) is a prime ideal of S′. Since f (ν¹) ∈ μ′¹, we have χ_f(ν¹) ⊆ μ′. Since μ′ is a minimal prime ideal, we have χ_f(ν¹) = μ′. So $μ^{'} (x^{'}) = {\begin{matrix} 1, & if x^{'} \in f (ν^{1}), \\ 0, & otherwise . \end{matrix}$ And hence $μ^{'} \circ f (x) = μ^{'} (f (x)) = {\begin{matrix} 1, & if x \in ν^{1}, \\ 0, & otherwise . \end{matrix}$ Because ν ⊆ μ′ ∘ f, we have ν = μ′ ∘ f. Therefore μ′ ∘ f is a minimal prime L-fuzzy ideal of S.

Proposition 5.12. Let S, S′ be ordered semigroups with zero elements and identities. If f : S → S′ is an ordered semigroup homomorphism and preserves prime ideals, then $f^{*} : ML (S^{'}) \to ML (S)$ mapping any $μ^{'} \in ML (S^{'})$ to f^* (μ′) = μ′ ∘ f is a continuous map between ML-spec (S′) and ML-spec (S).

Proof. By Proposition 5.11, f^* is well-defined. For any x ∈ S, $\begin{matrix} f^{* - 1} (H_{M} (x)) = & {μ^{'} \in ML (S^{'}) : f^{*} (μ^{'}) \in H_{M} (x)} \\ = & {μ^{'} \in ML (S^{'}) : f^{*} (μ^{'}) (x) = 1} \\ = & {μ^{'} \in ML (S^{'}) : μ^{'} (f (x)) = 1} \\ = & H_{M}^{'} (f (x)) . \end{matrix}$

As $H_{M} (x)$ is a basic closed set in ML-spec (S) and $ℋ_{M}^{'} (f (x))$ is a basic closed set in ML-spec (S′), we have f^* is continuous.

Let $S$ denote the category consisting of commutative ordered semigroups with zero elements and identities as objects and homomorphisms of ordered semigroups and preserve prime ideals as morphisms. We use $T$ to denote the category consisting of totally disconnected and completely regular topological spaces as objects and continuous mappings of topological spaces as morphisms.

Theorem 5.13. Let S, S′ be commutative ordered semigroups with zero elements and identities. Then the correspondence which associates S with the topological space ML-spec (S) in $T$ and a morphism f : S → S′ in $S$ with the morphism $f^{*} : ML (S^{'}) \to ML (S)$ in $T$ , defines a contravariant functor from $S$ to $T$ .

Proof. Let S, S′, S″ be ordered semigroups with zero elements and identities and f : S → S′, g : S′ → S″ be morphisms in $S$ . By Proposition 5.11 and 5.12, f^*, g^* are well-defined and f^*, g^* are morphisms in $T$ . Clearly, (g ∘ f) ^* = f^* ∘ g^* and if f : S → S is the identity map on S in $S$ , then f^* is the identity map on ML-spec (S). This completes the proof.

Footnotes

Acknowledgments

We would like to thank the anonymous referees for their careful reading and valuable comments which have improved the quality of this paper.

This work is supported by National Natural Science Foundation of China (Nos. 11771134, 11701540).

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