A novel study on fuzzy ideals and fuzzy filters of ordered *-semigroups

1 Introduction

It is well known that an ordered semigroup is a semigroup together with a partial order that is compatible with the semigroup operation. Ordered semigroups have many applications in the theory of sequential machines, formal languages and error-correcting codes. Many authors, especially Kehayopulu [1], Kehayopulu and Tsingelis [2], Satyanarayana [11] and Xie [14], studied such semigroups with some restrictions. In 1978, Nordahl and Scheiblich [9] considered a unary operation * on semigroups and introduced the concept of regularity on *-semigroups. As a generalization of ordered semigroups, Wu [13] imposed the *-operation on ordered semigroups under the assumption of order preserving.

On the other hand, a fuzzy subset f of a given set S (or a fuzzy set in S) is described as an arbitrary function f : S → [0, 1], where [0, 1] is the usual closed interval of real numbers. This important concept of fuzzy sets was first introduced by Zadeh [17] in 1965, which provides a natural framework for generalizing some basic notions of algebra, e.g. set theory, group theory, semigroup theory and so on. For instance, Rosenfeld [10] investigated fuzzy sets in the context of group theory and formulated the concept of a fuzzy subgroup of a group. In [3], Kuroki introduced fuzzy sets in semigroup theory, and then introduced fuzzy semiprime ideals in semigroup. The applications of fuzzy technology in information processing is already important and it will certainly increase in importance in the future.

Fuzzy sets in an ordered semigroup were first researched by Kehayopulu and Tsingelis in [5], then they defined and studied the theory of fuzzy sets on ordered semigroups for several notations, for example, see [6–8], which have proven useful in the theory of ordered semigroups. Furthermore, Tang and Xie [12] investigated the regular ordered semigroups in terms of generalized fuzzy ideals. It is well known that fuzzy ideals and fuzzy filters of ordered semigroups with special properties always play important roles in the study of ordered semigroups. In the present paper, motivated by the study of ordered semigroups in terms of fuzzy subsets, we shall study the fuzzy ideals and fuzzy filters of ordered *-semigroups, and extend some results of ordered semigroups to ordered *-semigroups.

Let us describe the organization of the present paper. After an introduction, in Section 2 we recall some basic definitions and results of ordered *-semigroups which will be used throughout this paper and define a unary operation * on the set of all fuzzy subsets of an ordered *-semigroup S, which is a key notion to depict prime, weakly prime and semiprime fuzzy ideals of S in the sequel. In Section 3, we study the concepts of prime fuzzy, weakly prime fuzzy and semiprime fuzzy ideals in ordered *-semigroups in detail, and establish the relationships among these three types of fuzzy ideals. In Section 4, we force our attention on the intra-regular ordered *-semigroups. In particular, some characterizations of this class of ordered *-semigroups in terms of fuzzy ideals are provided. The last section is devoted to define and study the fuzzy filters of an ordered *-semigroup S. In particular, the relationships between the filters and the fuzzy filters in ordered *-semigroups are given.

2 Preliminaries and some notations

In this section, we investigate below necessary notions and present a few auxiliary results that will be used throughout the paper.

As we know, an ordered semigroup (S, ·, ≤) is a semigroup (S, ·) with an order relation “≤” such that a ≤ b implies xa ≤ xb and ax ≤ bx for any x ∈ S. In the following, we shall recall the notion of ordered *-semigroups from [13].

Definition 2.1. An ordered semigroup S with a unary operation * : S → S is called an ordered *-semigroup if it satisfies:

(∀ x ∈ S)   (x^*) ^* = x^*.

Such a unary operation * is called an involution. If for any a, b ∈ S with a ≤ b, we have a^* ≤ b^*, then * is called an order preserving involution.

Example 2.2. Let S = {a, b, c, d} be an ordered semigroup. The multiplication “·”, the order “≤” and the corresponding Hasse diagram are given below (see [16]). Define the involution * by a^* = a,  b^* = b,  c^* = d and d^* = c. We can easily check that (S, ·, *, ≤) is an ordered *-semigroup with order preserving involution *.

≤ : = { ( a , a ) , ( a , b ) , ( b , b ) , ( a , c ) , ( c , c ) , ( a , d ) , ( d , d ) } .

Throughout this paper, we denote by S an ordered *-semigroup. For a nonempty subset H of S, we define ( H ] : = { x ∈ S | ( ∃ h ∈ H ) x ≤ h } .

If H = {a}, we write (a] instead of ({a}]. By a subsemigroup of S we mean a nonempty subset A of S such that A² ⊆ A.

A function f from S to the real closed interval [0, 1] is called a fuzzy subset of S. The fuzzy subset “1” of S is defined as follows: ( ∀ x ∈ S ) 1 : S → [ 0 , 1 ] , x ↦ 1 ( x ) : = 1 .

The set of all fuzzy subsets of S is denoted by F (S). In particular, the element “0” of F (S) also is a fuzzy subset of S defined by ( ∀ x ∈ S ) 0 : S → [ 0 , 1 ] , x ↦ 0 ( x ) : = 0 .

Let f, g ∈ F (S) be two fuzzy subsets of S. Then, for all x ∈ S, the inclusion relation f ⊆ g is defined by f (x) ≤ g (x), and f ∩ g and f ∪ g are defined by ( f ∩ g ) ( x ) = min ( f ( x ) , g ( x ) ) = f ( x ) ∧ g ( x ) , ( f ∪ g ) ( x ) = max ( f ( x ) , g ( x ) ) = f ( x ) ∨ g ( x ) .

One can easily show that (F (S), ⊆, ∩, ∪) forms a complete lattice with the maximum element 1 and the minimum element 0.

Let S be an ordered *-semigroup. For any x ∈ S, we define A _x  : = {(y, z) ∈ S × S| x ≤ yz}. The product f ∘ g of f and g is defined by ( f ∘ g ) ( x ) = { ⋁ ( y , z ) ∈ A x [ f ( y ) ∧ g ( z ) ] if A x ≠ ∅ , 0 if A x = ∅ , for all x ∈ S. It is well known that this operation “∘” is associative and (F (S), ∘, ⊆) is an ordered semigroup. Furthermore, we define a unary operation * on F (S) by ( ∀ x ∈ S ) f * : S → [ 0 , 1 ] , x ↦ f * ( x ) : = f ( x * ) .

Then, f^* is a fuzzy subset of S and it is easy to calculate that, for any f, g ∈ F (S) and all x ∈ S, the unary operation * such that ( f * ) * ( x ) = f ( x ) , ( f ∘ g ) * ( x ) = ( g * ∘ f * ) ( x ) , and f ⊆ g { implies f * ⊆ g * .

That is, (F (S), ∘, *, ⊆) forms an ordered *-semi-group with the order preserving involution *.

Proposition 2.3.Let f, g ∈ F (S) be two fuzzy subsets of an ordered *-semigroup S. Then the following statements hold:

(f ∪ g) ^* = f^* ∪ g^*.

Proof. Straightforward. □

Let S be an ordered *-semigroup and A a nonempty subset of S. We denote by f _A the characteristic function of A, that is, the mapping of S into [0, 1] defined by

f A ( x ) = { 1 , if x ∈ A , 0 , if x ∉ A .

Then f _A is a fuzzy subset of S. A fuzzy subset f of S is called a fuzzy subsemigroup of S if for all x, y ∈ S, f (xy) ≥ f (x) ∧ f (y). A fuzzy subset f of S is called a fuzzy left (resp. right) ideal of S if for any x, y ∈ S

x ≤ y ⇒ f (x) ≥ f (y).

We can easily check that the condition (2) is equivalent to 1 ∘ f ⊆ f (resp. f ∘ 1 ⊆ f). A fuzzy subset f of S is called a fuzzy ideal of S if it is both a fuzzy left and a fuzzy right ideal of S.

Proposition 2.4. Let S be an ordered *-semigroup with order preserving involution *. Then f^* is a fuzzy ideal for any fuzzy ideal f of S.

Proof. Let f be a fuzzy ideal of S and x, y ∈ S such that x ≤ y. Then x^* ≤ y^* and f^* (x) = f (x^*) ≥ f (y^*) = f^* (y). Furthermore, f^* (xy) = f (y^*x^*) ≥ f (y^*) = f^* (y). Similarly, f^* (xy) ≥ f^* (x). Thus, f^* is a fuzzy ideal of S. □

In 2008, Xie and Tang [15] introduced the concept of an ordered fuzzy point of an ordered semigroup, and characterizes fuzzy ideals generated by ordered fuzzy points of an ordered semigroup.

Definition 2.5. (Definition 3.1, [15]) Let S be an ordered semigroup, a ∈ S and λ ∈ [0, 1]. An ordered fuzzy pointa _λ of S is defined by the rule that a λ ( x ) = { λ , if x ∈ ( a ] , 0 , if x ∉ ( a ] .

It is accepted that a _λ is a mapping from S into [0, 1], then an ordered fuzzy point of S is a fuzzy subset of S. For any fuzzy subset f of S, we still write a _λ  ∈ f instead of a _λ  ⊆ f. We denote by I (a _λ ) the fuzzy (two-sided) ideal of S generated by a _λ . It is not difficult to check that I (a _λ ) = a _λ  ∪ 1 ∘ a _λ  ∪ a _λ  ∘ 1 ∪1 ∘ a _λ  ∘ 1 (see [15]). As follows, we also define a unary * on ordered fuzzy points by ( ∀ x ∈ S ) a λ * ( x ) = { λ , if x ∈ ( a * ] , 0 , if x ∉ ( a * ] .

Definition 2.6. Let S be an ordered *-semigroup. A fuzzy subset f of S is said to be strongly convex if f = (f], where (f] is defined by the rule that ( f ] ( x ) = ⋁ x ≤ y f ( y ) .

Proposition 2.7. Let S be an ordered *-semigroup and f a strongly convex fuzzy subset. Then f = ⋃ y λ ∈ f y λ .

Proof. By Proposition 3.4 in [15], the proof is straightforward verification, and thus we omit the details. □

Proposition 2.8.Let S be an ordered *-semigroup. Then the following statements are true:

f ⊆ (f] for any f ∈ F (S).

Proof. The detailed proofs come from Proposition 3.5 in [15]. □

The reader is referred to [14] for notation andterminology not defined in this paper.

3 Prime fuzzy ideals of ordered *-semigroups

In this section we define and characterize the prime, weakly prime and semiprime fuzzy ideals in an ordered *-semigroup, and investigate their related properties.

Definition 3.1. Let P be a fuzzy ideal of an ordered *-semigroup S. Then P is said to be prime fuzzy ideal if for any ordered fuzzy points a _λ , b_μ ∈ S  (λ, μ ∈ (0, 1]), a _λ  ∘ b_μ ∈ P implies a λ * ∈ P or b μ * ∈ P .

Definition 3.2. Let S be an ordered *-semigroup and f a fuzzy ideal of S. f is called weakly prime if for any two fuzzy ideals g, h of S such that g ∘ h ⊆ f implies g^* ⊆ f or h^* ⊆ f.

Definition 3.3. A fuzzy ideal P of an ordered *-semigroup S is called a semiprime fuzzy ideal if for any ordered fuzzy point a _λ  ∈ S  (∀ λ ∈ (0, 1]), a λ ∘ a λ = a λ 2 ∈ P implies a λ * ∈ P .

In the following we shall study the relationships among the prime fuzzy ideals, weakly prime fuzzy ideals and semiprime fuzzy ideals in ordered *-semigroups.

Lemma 3.4. Let f be a fuzzy ideal of an ordered *-semigroup S. Then f is prime if and only if for any strongly convex fuzzy subset g and h of S, g ∘ h ⊆ f implies that g^* ⊆ f or h^* ⊆ f.

Proof. Let g and h be strongly convex fuzzy subsets of S such that g ∘ h ⊆ f. If g^*⊈f, by Proposition 2.7, there exists an ordered fuzzy point a λ * ∈ g *  (that is, a _λ  ∈ g) such that a λ * ∉ f . For any b_μ ∈ h, we have a _λ  ∘ b_μ ∈ g ∘ h ⊆ f. Since f is a prime fuzzy ideal, we have b μ * ∈ f . Therefore, h * = ⋃ b μ * ∈ h * b μ * ⊆ f . □

Lemma 3.5. A fuzzy ideal P of an an ordered *-semigroup S is semiprime if and only if for any strongly convex fuzzy subset f of S, f ∘ f ⊆ P implies f^* ⊆ P.

Proof. Since the proof is similar to the proof of Lemma 3.4, we omit it. □

Theorem 3.6. Let S be an ordered *-semigroup. A fuzzy ideal P of S is prime if and only if P is semiprime and weakly prime.

Proof. Let P be a prime fuzzy ideal of S. Then, by Lemmas 3.4 and 3.5, we claim that P is weakly prime and semiprime fuzzy ideal of S.

Conversely, assume that P is a fuzzy ideal of S which is weakly prime and semiprime. Let a _λ , b_μ be ordered fuzzy points of S such that a _λ  ∘ b_μ ∈ P.Then ( b μ ∘ 1 ∘ a λ ) ∘ ( b μ ∘ 1 ∘ a λ ) = b μ ∘ 1 ∘ a λ ∘ b μ ∘ 1 ∘ a λ ⊆ 1 ∘ ( a λ ∘ b μ ) ∘ 1 ⊆ 1 ∘ P ∘ 1 ⊆ P .

By Lemma 3.5 and P is semiprime, we deduce that (b_μ ∘ 1 ∘ a _λ ) ^* ⊆ P. Hence ( 1 ∘ a λ * ∘ 1 ) ∘ ( 1 ∘ b μ * ∘ 1 ) ⊆ 1 ∘ ( a λ * ∘ 1 ∘ b μ * ) ∘ 1 = 1 ∘ ( b μ ∘ 1 ∘ a λ ) * ∘ 1 ⊆ 1 ∘ P ∘ 1 ⊆ P .

By Proposition 3.7 in [15], 1 ∘ a λ * ∘ 1 and 1 ∘ b μ * ∘ 1 are fuzzy ideals of S. Note that P is weakly prime. Thus, ( 1 ∘ a λ * ∘ 1 ) * ⊆ P or ( 1 ∘ b μ * ∘ 1 ) * ⊆ P . That is, 1 ∘ a _λ  ∘ 1 ⊆ P or 1 ∘ b_μ ∘ 1 ⊆ P. To show that P is prime, we just need to prove that 1 ∘ a _λ  ∘ 1 ⊆ P implies that a λ * ∈ P . Since other part can be proved similarly. If 1 ∘ a _λ  ∘ 1 ⊆ P, then ( I ( a λ ) ) 3 ⊆ ( a λ ∪ 1 ∘ a λ ∪ a λ ∘ 1 ∪ 1 ∘ a λ ∘ 1 ) 2 ∘ I ( a λ ) ⊆ 1 ∘ ( a λ ∪ 1 ∘ a λ ∪ a λ ∘ 1 ∪ 1 ∘ a λ ∘ 1 ) ∘ I ( a λ ) ⊆ ( 1 ∘ a λ ∪ 1 ∘ a λ ∘ 1 ) ∘ ( a λ ∪ 1 ∘ a λ ∪ a λ ∘ 1 ∪ 1 ∘ a λ ∘ 1 ) ⊆ ( 1 ∘ a λ ∪ 1 ∘ a λ ∘ 1 ) ∘ 1 ⊆ 1 ∘ a λ ∘ 1 ⊆ P .

Thus, (I (a _λ )) ^* ⊆ P or (I (a _λ )) ^* ∘ (I (a _λ )) ^* ⊆ P since P is weakly prime and I (a _λ ), (I (a _λ )) ² are fuzzy ideals of S. If (I (a _λ )) ^* ⊆ P, then a λ * ∈ ( I ( a λ ) ) * ⊆ P . Suppose that (I (a _λ )) ^* ∘ (I (a _λ )) ^* ⊆ P. Then a λ * ∘ a λ * ∈ ( I ( a λ ) ) * ∘ ( I ( a λ ) ) * ⊆ P . Hence a _λ  ∈ P since P is semiprime. It follows that a _λ  ∘ a _λ  ∈ P which implies that a λ * ∈ P . □

Theorem 3.7. Let S be a commutative ordered *-semigroup and P a fuzzy ideal of S. Then P is prime if and only if P is weakly prime.

Proof. If P is a prime fuzzy ideal of S, then it is obvious that P is weakly prime. Conversely, suppose that P is weakly prime such that a _λ  ∘ b_μ ∈ P for any ordered fuzzy points a _λ and b_μ. Since S is commutative, we have I ( a λ ) ∘ I ( b μ ) ⊆ ( a λ ∪ 1 ∘ a λ ∪ a λ ∘ 1 ∪ 1 ∘ a λ ∘ 1 ) ∘ ( b μ ∪ 1 ∘ b μ ∪ b μ ∘ 1 ∪ 1 ∘ b μ ∘ 1 ) ⊆ a λ ∘ b μ ∪ 1 ∘ a λ ∘ b μ ⊆ P ∪ 1 ∘ P ⊆ P .

Notice that P is weakly prime, we conclude that (I (a _λ )) ^* ⊆ P or (I (b_μ)) ^* ⊆ P. Hence, a λ * ∈ P or b μ * ∈ P , that is, P is prime. □

Proposition 3.8.Let S be an ordered *-semigroup with order preserving involution *. Then the following statements are equivalent:

f^* ∘ f^* = f for any fuzzy ideal f of S.

Proof. (1) ⇒ (2). Since f^* and g^* are fuzzy ideals of S, we have f ∘ g ⊆ f ∘ 1 ⊆ f = f * ∘ f * ⊆ f * .

Similarly, we have f ∘ g ⊆ g^*. Thus f ∘ g ⊆ f^* ∩ g^*. On the other hand, by Propositions 2.4 and 2.8(6), f^* ∩ g^* is a fuzzy ideal of S. This implies that f * ∩ g * = ( f * ∩ g * ) * ∘ ( f * ∩ g * ) * = ( f ∩ g ) ∘ ( f ∩ g ) ⊆ f ∘ g .

Consequently, we have f^* ∩ g^* = f ∘ g.

(2) ⇒ (3). By Proposition 2.4, (I (a _λ )) ^* and (I (b_μ)) ^* are fuzzy ideals. Then, it is easy to prove that I (a _λ ) ∩ I (b_μ) = (I (a _λ )) ^* ∘ (I (b_μ)) ^*.

(3) ⇒ (4). We only need to prove that ( I ( a λ ) ) * = I ( a λ * ) . Obviously, a λ * ∈ ( I ( a λ ) ) * . Hence I ( a λ * ) ⊆ ( I ( a λ ) ) * . On the other hand, let x _γ  ∈ (I (a _λ )) ^*. Then x γ * ∈ I ( a λ ) = a λ ∪ 1 ∘ a λ ∪ a λ ∘ 1 ∪ 1 ∘ a λ ∘ 1 . Thus, x γ * = a λ or x γ * ∈ a λ ∘ 1 or x γ * ∈ 1 ∘ a λ or x γ * ∈ 1 ∘ a λ ∘ 1 . This means that x γ ∈ a λ * ∪ a λ * ∘ 1 ∪ 1 ∘ a λ * ∪ 1 ∘ a λ * ∘ 1 = I ( a λ * ) .

That is, ( I ( a λ ) ) * ⊆ I ( a λ * ) . Consequently, (I (a _λ )) ^* = I ( a λ * ) .

(4) ⇒ (5). If we can prove the following two facts: I ( a λ ) = ( I ( a λ * ) ) 6 ∘ I ( a ) and ( I ( a λ * ) ) 6 ∘ I ( a λ ) ⊆ S ∘ a λ * ∘ S ∘ a λ * ∘ S , then we obtain that a λ ∈ I ( a λ ) ⊆ S ∘ a λ * ∘ S ∘ a λ * ∘ S and complete the proof.

(i) By hypothesis, we have I ( a λ ) = I ( a λ * ) ∘ I ( a λ * ) = I ( a λ ) ∘ I ( a λ ) ∘ I ( a λ ) ∘ I ( a λ ) = ( I ( a λ * ) ) 6 ∘ I ( a λ ) .

(ii) By using a similar argument as the proof of Theorem 3.6, we have (I (a _λ )) ³ ⊆ 1 ∘ a _λ  ∘ 1. So ( I ( a ) ) 5 = ( I ( a λ ) ) 3 ∘ I ( a λ ) ∘ I ( a λ ) ⊆ 1 ∘ a λ ∘ 1 ∘ ( a λ ∪ a λ ∘ 1 ∪ 1 ∘ a λ ∪ 1 ∘ a λ ∘ 1 ) ∘ 1 ⊆ 1 ∘ a λ ∘ 1 ∘ a λ ∘ 1 .

Therefore, ( I ( a λ * ) ) 5 ⊆ 1 ∘ a λ * ∘ 1 ∘ a λ * ∘ 1 . We conclude that ( I ( a λ * ) ) 6 ∘ I ( a λ ) ⊆ 1 ∘ a λ * ∘ 1 ∘ a * ∘ 1 ∘ I ( a λ * ) ∘ I ( a λ ) ⊆ 1 ∘ a λ * ∘ 1 ∘ a * ∘ 1 ∘ 1 ⊆ 1 ∘ a λ * ∘ 1 ∘ a * ∘ 1 .

(5) ⇒ (6). The proof is similar as the proof of Proposition 3.7 (7) in [15], hence we omit the details.

(6) ⇒ (1). By hypothesis, there exist x, y, z ∈ S such that a ≤ xa^*ya^*z. It means that A _t ≠ ∅ for any t ∈ S. If f is a fuzzy ideal of S, then f (a) ≥ f (xa^*ya^*z). Thus, ( f * ∘ f * ) ( a ) = ⋁ a ≤ pq [ f * ( p ) ∧ f * ( q ) ] ≥ f * ( xa * y ) ∧ f * ( a * z ) = f ( y * ax * ) ∧ f ( z * a ) ≥ f ( a ) ∧ f ( a ) = f ( a ) .

That is, f ⊆ f^* ∘ f^*. Conversely, for any t ∈ S, ( f * ∘ 1 ) ( t ) = ⋁ t ≤ ab [ f * ( a ) ∧ 1 ( b ) ] = ⋁ t ≤ ab { f * ( a i ) | i ∈ I } .

On the other hand, since t ≤ a _i b _i for each i ∈ I, we get that f ( t ) ≥ f ( a i b i ) ≥ f ( a i ) ≥ f ( xa i * ya i * z ) ≥ f ( a i * ) = f * ( a i ) .

Hence, f ( t ) ≥ ⋁ t ≤ ab { f * ( a i ) | i ∈ I } = ( f * ∘ 1 ) ( t ) . That is, f^* ∘ 1 ⊆ f. Therefore, f^* ∘ f^* ⊆ f^* ∘ 1 ⊆ f. Consequently, f^* ∘ f^* = f as required. □

Theorem 3.9. Let S be an ordered *-semigroup with order preserving involution *. The fuzzy ideals of S are weakly prime if and only if f^* = f ∘ f for any fuzzy ideal f of S and any two fuzzy ideals are comparable under the inclusion relation ⊆.

Proof. Suppose that the fuzzy ideals of S are weakly prime. Let f, g be any fuzzy ideal of S. Notice that g^* is an fuzzy ideal and f ∘ g^* is weakly prime. Thus, f ∘ g^* ⊆ f ∘ g^* implies that f^* ⊆ f ∘ g^* or g ⊆ f ∘ g^*. If f^* ⊆ f ∘ g^*, then f^* ⊆ 1 ∘ g^* ⊆ g^*, i.e., f ⊆ g. On the other hand, if g ⊆ f ∘ g^*, then g ⊆ f ∘ 1 ⊆ f. The conclusion follows that f and g are comparable.

Next, we claim that f^* = f ∘ f. Since f ∘ f is weakly prime and f ∘ f ⊆ f ∘ f, we have f^* ⊆ f ∘ f. Note that, f ∘ f ⊆ f ∘ 1 ⊆ f implies f^* ⊆ f. So f ⊆ f^* and f ∘ f ⊆ f^* ∘ f^* ⊆ f^*. Therefore, f ∘ f = f^*.

Conversely, let f, g and P be fuzzy ideals of S with f ∘ g ⊆ P. Since f^* = f ∘ f, we have f^* ∩ g^* = f ∘ g by Proposition 3.8. Notice that f and g are comparable. If f ⊆ g, then f^* ⊆ g^*, whence f^* = f^* ∩ g^* = f ∘ g ⊆ P. If g ⊆ f, then by a similar argument, we deduce that g^* ⊆ P. Thus, P is weakly prime. □

Proposition 3.10. Let S be an ordered *-semigroup with order preserving involution *. If the fuzzy ideals of S are semiprime, then

I (a _λ ) =1 ∘ a _λ  ∘ 1 for any ordered fuzzy point a _λ of S.

Proof. (1). Let a _λ be an ordered fuzzy point of S. Notice that 1 ∘ a _λ  ∘ 1 is an fuzzy ideal of S, and thus it is semiprime. So a λ 2 ∘ a λ 2 = a λ 4 ∈ 1 ∘ a λ ∘ 1 implies that a λ * ∘ a λ * = ( a λ 2 ) * ∈ 1 ∘ a λ ∘ 1 . That is, a _λ  ∈ 1 ∘ a _λ  ∘ 1 and I (a _λ ) ⊆1 ∘ a _λ  ∘ 1. On the other hand, 1 ∘ a _λ  ∘ 1 ⊆ a _λ  ∪ a _λ  ∘ 1 ∪1 ∘ a _λ  ∪ 1 ∘ a _λ  ∘ 1 = I (a _λ ). Thus I (a _λ ) =1 ∘ a _λ  ∘ 1.

(2). For any ordered fuzzy points a _λ , b_μ ∈ S, a _λ  ∘ b_μ ∈ I (a _λ ) ∘1 ⊆ I (a _λ ). Thus I (a _λ  ∘ b_μ) ⊆ I (a _λ ). By a similar argument, we deduce that I (a _λ  ∘ b_μ) ⊆ I (b_μ). Hence, I (a _λ  ∘ b_μ) ⊆ I (a _λ ) ∩ I (b_μ).

Conversely, let x _γ  ∈ I (a _λ ) ∩ I (b_μ). Then x _γ  ∈ 1 ∘ a _λ  ∘ 1 ∩1 ∘ b_μ ∘ 1. Notice that (b_μ ∘ 1 ∘ a _λ ) ∘ (b_μ ∘ 1 ∘ a _λ ) ⊆1 ∘ a _λ  ∘ b_μ ∘ 1 ⊆ I (a _λ  ∘ b_μ) and I (a _λ  ∘ b_μ) is semiprime. We can obtain that (b_μ ∘ 1 ∘ a _λ ) ^* ⊆ I (a _λ  ∘ b_μ). Hence, x γ * ∘ x γ * ∈ ( 1 ∘ a λ ∘ 1 ) * ∘ ( 1 ∘ b μ ∘ 1 ) * = ( 1 ∘ a λ * ∘ 1 ) ∘ ( 1 ∘ b μ ∘ 1 ) ⊆ 1 ∘ ( a λ * ∘ 1 ∘ b μ * ) ∘ 1 = 1 ∘ ( b μ ∘ 1 ∘ a λ ) * ∘ 1 ⊆ I ( a λ ∘ b μ ) .

This means that x _γ  ∈ I (a _λ  ∘ b_μ), that is, I (a _λ ) ∩ I (b_μ) ⊆ I (a _λ  ∘ b_μ). □

4 Characterizations of intra-regular ordered *-semigroups

As a generalization of the concept of intra-regular ordered semigroups, Wu [13] introduced the concept of intra-regular ordered *-semigroups. In this section, we force our attention on this class of ordered *-semigroups. In particular, we provide some characterizations of intra-regular ordered *-semigroups in terms of fuzzy ideals.

Definition 4.1. (Definition 2.9, [13]) An ordered *-semigroup S is called intra-regular if a ∈ (Sa^*a^*S] for any a ∈ S.

Example 4.2. Consider a set S = {a, b, c, d} with the following multiplication “·”, the order “≤” and the corresponding Hasse diagram:

≤ : = { ( a , a ) , ( b , a ) , ( b , b ) , ( c , c ) , ( d , a ) , ( d , d ) } .

Now define the involution * by a^* = a,  b^* = d,  c^* = c and d^* = b, We can easily check that (S, ·, *, ≤) is an intra-regular ordered *-semigroup with the order preserving involution *.

We now give a characterization of intra-regular ordered *-semigroups by the properties of fuzzy ideals.

Lemma 4.3.Let S be an ordered *-semigroup. Then the following statements are equivalent:

S is intra-regular.

Proof. (1) ⇒ (2). Notice that R^* is a left ideal and L^* is a right ideal of S. Since S is intra-regular, for any a ∈ R^* ∩ L^*, there exist x, y ∈ S such that a ≤ xa * a * y = ( xa * ) ( a * y ) ∈ ( SL ) ( RS ) ⊆ LR .

Thus, a ∈ (LR] as required.

(2) ⇒ (1). Suppose that R^* ∩ L^* ⊆ (LR] for every left ideal L and every right ideal R of S. Then, by hypothesis, a * ∈ R * ( a ) ∩ L * ( a ) ⊆ ( L ( a ) R ( a ) ] = ( ( a ∪ Sa ] ( a ∪ aS ] ] ⊆ ( ( a ∪ Sa ) ( a ∪ aS ) ] = ( a 2 ∪ Sa 2 ∪ a 2 S ∪ Sa 2 S ] , where L (a) (resp. R (a)) is the left (resp. right) ideal of S generated by a. This implies that, if a * ≤ a 2 = ( ( a * ) * ) 2 ≤ ( ( a 2 ) * ) 2 = ( a * ) 4 ∈ Sa * a * S , then, by Proposition 2.11 in [13], we have a^* ∈ (Sa^*a^*S] = (Sa²S]. If a^* ≤ xa² for some x ∈ S. Then a * ≤ xa 2 = x ( a * ) * a ≤ x ( xa 2 ) * a = x ( a * a * x * ) a ∈ ( Sa * a * S ] = ( Sa 2 S ] .

If a^* ≤ a²y for some y ∈ S. Then, by a similar argument as above, we can deduce that a^* ∈ (Sa^*a^*S] = (Sa²S]. If a^* ≤ ua²v for some u, v ∈ S. Then a^* ∈ (Sa²S] as follows. Hence, we can always obtain that a^* ∈ (Sa²S]. Therefore, a ∈ (Sa^*a^*S] and S is intra-regular. □

Lemma 4.4. (Lemma 2.5, [16]) Let S be an ordered *-semigroup and ∅ ≠ A ⊆ S. Then A is a left (resp. right) ideal of S if and only if the characteristic function f _A of A is a fuzzy left (resp. right) ideal of S.

Now we give a characterization of intra-regular ordered *-semigroups in terms of fuzzy left ideals and fuzzy right ideals.

Theorem 4.5.Let S be an ordered *-semigroup. Then the following statements are equivalent:

S is intra-regular.

Proof. (1) ⇒ (2). Let f and g be a fuzzy right ideal and a fuzzy left ideal of a intra-regular ordered *-semigroup S, respectively. For any a ∈ S, there exist x, y ∈ S such that a ≤ xa^*a^*y. That is, (xa^*, a^*y) ∈ A _a . Thus ( f * ∩ g * ) ( a ) = f * ( a ) ∧ g * ( a ) = f ( a * ) ∧ g ( a * ) ≤ g ( xa * ) ∧ f ( a * y ) ≤ ⋁ ( y , z ) ∈ A a [ g ( y ) ∧ f ( z ) ] = ( g ∘ f ) ( a ) .

This means that f^* ∩ g^* ⊆ g ∘ f.

(2) ⇒ (1). Suppose that R and L are a right ideal and a left ideal of S, respectively. Then, by Lemma 4.4, f _{R ^*} and f _{L ^*} are a fuzzy left ideal and a fuzzy right ideal since R^* is a left ideal of S and L^* is a right ideal of S. Let a^* ∈ R^* ∩ L^*. Then, by hypothesis and notice that f R * = f R * , f L * = f L * , we have ( f R * ∘ f L * ) * ( a * ) ≥ ( f L * * ∩ f R * * ) * ( a * ) = ( f L * ∩ f L * ) ( a * ) = min { f L * ( a * ) , f R * ( a * ) } = 1 .

On the other hand, since (f _{R ^*}  ∘ f _{L ^*} ) ^* is a fuzzy subset of S, we get that (f _{R ^*}  ∘ f _{L ^*} ) ^* (a^*) ≤1 for any a^* ∈ S. Hence, we obtain ⋁ ( y , z ) ∈ A a * [ f L ( y ) ∧ f R ( z ) ] = ( f L * * ∘ f R * * ) ( a * ) = ( f R * ∘ f L * ) * ( a * ) = 1 .

This means that there exist b, c ∈ S such that a^* ≤ bc and f _L  (b) = f _R  (c) =1. Thus a^* ≤ bc ∈ LR, and so R^* ∩ L^* ⊆ (LR]. By Lemma 4.3, S is intra-regular. □

Proposition 4.6. An ordered *-semigroup S is intra-regular if and only if f (a^*) = f (a²) for any a ∈ S and every fuzzy ideal f of S.

Proof. (⇒). Suppose that f is a fuzzy ideal of S and a ∈ S. Since S is intra-regular, there exist x, y ∈ S such that a ≤ xa^*a^*y. Thus f ( a * ) ≥ f ( y * a 2 x * ) ≥ f ( y * a 2 ) ≥ f ( a 2 ) ≥ f ( xa * a * yxa * a * y ) ≥ f ( a * y ) ≥ f ( a * ) .

This implies that f (a^*) = f (a²).

(⇐). We denote by I (a²) the ideal of S generated by a², from the fact of Lemma 4.4, f_I(a²) is a fuzzy ideal of S. By hypothesis, we have f_I(a²) (a^*) = f_I(a²) (a²) =1, that is, a^* ∈ I (a²) = (a² ∪ Sa² ∪ a²S ∪ Sa²S]. Hence, by a similar argument as the proof of Lemma 4.3, we deduce that S is intra-regular. □

Proposition 4.7. Let S be an intra-regular ordered *-semigroup. Then, for any a, b ∈ S and each fuzzy ideal f of S, we have f ((ab) ^*) = f (ba).

Proof. Let f be a fuzzy ideal of S and a, b ∈ S. Then, by Proposition 4.6, we obtain that f ( ( ab ) * ) = f ( ( ab ) 2 ) = f ( a ( ba ) b ) ≥ f ( ba ) = f ( ( ba ) * ( ba ) * ) = f ( a * ( b * a * ) b * ) ≥ f ( b * a * ) = f ( ( ab ) * ) .

That is f ((ab) ^*) = f (ba). □

5 Fuzzy filters of ordered *-semigroups

In the current section, we introduce the concept of fuzzy filters of ordered *-semigroups, and investigate their related properties. In particular, we discuss the relationships between the filters and the fuzzy filters in ordered *-semigroups.

Definition 5.1. (Definition 3.1, [13]) Let S be an ordered *-semigroup. A subsemigroup F of S is called a filter if

(∀ a, b ∈ S)  ab ∈ F ⇒ a^* ∈ F and b^* ∈ F.

Example 5.2. Consider the ordered *-semigroup S given in Example 4.2. We can easily verify that F = {c} is a filter of S.

Definition 5.3. Let S be an ordered *-semigroup. A fuzzy subsemigroup f of S is called a fuzzy filter of S if

(∀ x, y ∈ S)  x ≤ y ⇒ f (x) ≤ f (y),

Example 5.4. Consider the ordered *-semigroup S given in Example 4.2. Now let f be a fuzzy subset of S such that f (a) = f (b) = f (d) =0.5 and f (c) =0.7. Then it is easy to check that f is a fuzzy filter of S.

Lemma 5.5. [4] Let S be an ordered *-semigroup. A nonempty subset A of S is a subsemigroup of S if and only if the characteristic function f _A of A is a fuzzy subsemigroup of S.

Theorem 5.6. Let S be an ordered *-semigroup. A nonempty subset F of S is a filter of S if and only if the characteristic function f _F of F is a fuzzy filter of S.

Proof. Let F be a filter of S. For any x, y ∈ S, we have f F ( xy ) ≤ f F * ( x ) ∧ f F * ( y ) . In fact, if xy ∈ F, then f _F  (xy) =1 and x^*, y^* ∈ F. In this case, we get that f F * ( x ) = f F ( x * ) = 1 and f F * ( y ) = 1 . Thus, f F ( xy ) = 1 ≤ f F * ( x ) ∧ f F * ( y ) = 1 . If xy ∉ F, then f F ( xy ) = 0 ≤ f F * ( x ) ∧ f F * ( y ) always holds. Furthermore, let x ≤ y. If x ∈ F, then, by the definition of filter, we have y ∈ F. Hence, f _F  (x) =1 ≤ f _F  (y) =1.If x ∉ F, then we can always get that f _F  (x) =0 ≤ f _F  (y). Consequently, we obtain that f _F  (x) ≤ f _F  (y). Note that, by Lemma 5.5, f _F is a fuzzy subsemigroup of S. Therefore, f _F is a fuzzy filter of S.

Conversely, suppose that f _F is a fuzzy filter of S. Let xy ∈ F for any x, y ∈ S. Then f F ( xy ) = 1 ≤ min { f F * ( x ) , f F * ( y ) } . Since f F * is a fuzzy subset of S, this means that f F * ( x ) = 1 = f F * ( y ) . That is, f _F  (x^*) = f _F  (y^*) =1 and so x^*, y^* ∈ F. In addition, for any a ∈ F, b ∈ S, let a ≤ b. Then f _F  (a) ≤ f _F  (b). Notice that f _F  (a) =1, so f _F  (b) =1. Thus, we get b ∈ F. Finally, F is a subsemigroup of S by Lemma 5.5. Hence, F is a filter of S. □

Definition 5.7. Let f be any fuzzy subset of an ordered *-semigroup S. The set f _t  = {x ∈ S| f (x) ≥ t}, where t ∈ [0, 1], is called a level subset of f.

Lemma 5.8. Let S be an ordered *-semigroup and f a fuzzy subset of S. Then f is a fuzzy subsemigroup of S if and only if the level subset f _t   (t ∈ (0, 1]) of f is a subsemigroup of S whenever f _t ≠ ∅.

Proof. Straightforward. □

Theorem 5.9. Let S be an ordered *-semigroup and f a fuzzy subset of S. Then f is a fuzzy filter of S if and only if the level subset f _t   (t ∈ (0, 1]) of f is a filter of S whenever f _t ≠ ∅.

Proof. Let f be a fuzzy filter of S and x ≤ y for any x ∈ f _t , y ∈ S. Then f (x) ≤ f (y). Since x ∈ f _t , we have t ≤ f (x) ≤ f (y), that is, y ∈ f _t . Furthermore, let xy ∈ f _t for any x, y ∈ S. Then t ≤ f (xy) ≤ min {f^* (x), f^* (y)} = min {f (x^*), f (y^*)}. Thus, we get that f (x^*) ≥ t and f (y^*) ≥ t. This implies that x^*, y^* ∈ f _t . Note that f _t is a subsemigroup of S by Lemma 5.8. Hence, f _t is a filter of S.

Conversely, let the level subset f _t   (≠ ∅) of f be a filter of S. For any x, y ∈ S, suppose that x ≤ y and t = f (x). Then x ∈ f _t . Since f _t is a filter of S, we have y ∈ f _t . Thus f (y) ≥ t = f (x). Moreover, for any x, y ∈ S, let λ = f (xy). Then xy ∈ f _λ , which implies that x^* ∈ f _λ and y^* ∈ f _λ . So f^* (x) ∧ f^* (y) = f (x^*) ∧ f (y^*) ≥ λ = f (xy). On the other hand, by Lemma 5.8, f is a fuzzy subsemigroup of S. Therefore, f is a fuzzy filter of S. □

Acknowledgments

This work was supported by the National Natural Science Foundation (No. 11371177) and the University Natural Science Project of Anhui Province (No. KJ2015A161).