A new type of fuzzy semiprime subsets in ordered semigroups

1. Introduction

The theory of fuzzy sets proposed by Zadeh [27] in 1965, has achieved a great success in various fields. By generalizing the concept of the fuzzy subsets, the study of the fuzzy algebraic structures has started with the introduction of the concept of fuzzy subgroups by Rosenfeld in his pioneering work [21]. Later on, Fuzzy sets in semigroup theory introduced by Kuroki [15], and also he initiated the study of fuzzy ideals, fuzzy bi-ideals, fuzzy generalized bi-ideals and fuzzy semiprime bi-ideals of a semigroup [16–18]. Kehayopulu and Tsingelis [6], introduced the concept of fuzzy sets in ordered semigroups. In [7], Kehayopulu and Tsingelis introduced the concept of fuzzy bi-ideals in ordered semigroups and characterized bi-ideals, left and right simple, the completely regular, and the strongly regular ordered semigroups in terms of fuzzy bi-ideals. In [22], Shabir and Khan, introduced the concept of a fuzzy generalized bi-ideal of ordered semigroups and characterized different classes of ordered semigroups by using fuzzy generalized bi-ideals. In [5], Jun et al. generalized the concept of fuzzy bi-ideals of ordered semigroups and introduced the notion of (α, β)-fuzzy generalized bi-ideal in ordered semigroups, and also characterized regular and intra-regular ordered semigroups in terms of (∈ , ∈ ∨ q)-fuzzy generalized bi-ideals. As a generalization of the concept of (α, β)-fuzzy generalized bi-ideal in ordered semigroups, the concept of (∈ , ∈ ∨ q _k )-fuzzy generalized bi-ideals in ordered semigroups introduced by Khan et al. in [13], and also the concept of (∈ , ∈ ∨ q _k )-fuzzy left (resp. right) ideals and the upper/lower parts of (∈ , ∈ ∨ q _k )-fuzzy bi-ideals (resp. right ideals, left ideals) were defined in ordered semigroups. In [25], Shabir et al. characterized the regular semigroups by (∈ , ∈ ∨ q _k )-fuzzy ideals. Recently, Khan et al. [10], introduced the notions of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left ideals, (∈ , ∈ ∨ (k^*, q _k ))-fuzzy right ideals and (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideals of ordered semigroups, respectively, and also (k^*, k)-lower parts of these generalized fuzzy ideals have been obtained. In 1982 [18], Kuroki introduced the concept of fuzzy semiprime subsets of a semigroup. The concept of semiprime fuzzy quasi-ideals in ordered semigroups was first studied by Shabir and Khan [23] and characterizations of completely regular ordered semigroups have been investigated. Kehayopulu et al. [8], characterized fuzzy left (resp. right) semiprime ideals of ordered semigroups.

Motivated by the work of Kuroki [18], Shabir and Khan [23] and Kehayopulu et al. [8], in the present analysis, we established some new types of fuzzy semiprime subsets in ordered semigroups. As a follow up, we introduced the concept of an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset in an ordered semigroup, and investigated the properties of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideals by concerning the (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subsets [2]. Moreover, we characterize completely regular ordered semigroups in terms of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideals and their (k^*, k)-lower parts. In this respect, we prove that: An ordered semigroup S is completely regular if and only if each (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal η of S is semiprime. We also prove that S is completely regular if and only if η k k * _ ⪯ η 2 ( ∘ ) k k * S ( ∘ ) k k * η 2 for each fuzzy subset η of S . The rest of paper is organized as follows: Section 2 summarizes some definitions and properties related to ordered semigroups, ideals, fuzzy subsets, fuzzy subsemigroups which are needed to develop our main results. In Section 3, the notions of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subsets of ordered semigroups are studied. Section 4 is devoted to the study of (k^*, k)-lower part of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subsets. In Section 5, we have given a conclusion containing the directions of some future work. The paper ends with a list of references.

2. Preliminaries

Throughout the present paper, for the convenience, S will stand for an ordered semigroup. To develop our main results, we need the following notions and results.

Let S be a non-empty set. The triplet (S, · , ≤) is called an ordered semigroup or po-semigroup if (S, ·) is a semigroup and (S, ≤) is a partially ordered set such that r ≤ s ⇒ rt ≤ st and tr ≤ ts for all r, s, t ∈ S.

For a subset T of S, the subset (T] of S is defined as (T] = {s ∈ S | s ≤ t forsome t ∈ T} . For any non-empty subsets C and D of an ordered semigroup S, we have: (1) C ⊆ (C]; (2) ((C]] = (C]; (3) If C ⊆ D, then (C] ⊆ (D]; (4) (C] (D] ⊆ (CD] and (5) ((C] (D]] = (CD]. The product of C and D is defined as CD = {cd | c ∈ C and d ∈ D} .

A subset T (≠ ∅) of S is said to be a:

subsemigroup of S if for all x, y ∈ T,  xy ∈ T;

Let a be an element of S. Then by L (a) , R (a) , J (a) and B (a), we denote the left ideal, the right ideal, the ideal and the generalized bi-ideal of S generated by a respectively. It is easy to verify that L (a) = (a ∪ Sa] ,  R (a) = (a ∪ aS] ,  J (a) = (a ∪ Sa ∪ aS ∪ SaS] and B (a) = (a ∪ aSa] .

An ordered semigroup S is said to be regular (resp. left regular, right regular) [9] if for each x ∈ S, there exist y ∈ S such that x ≤ xyx  (resp .  x ≤ yxx,  x ≤ xxy); and S is said to be completely regular [9] if for each x ∈ S, there exist a ∈ S such that x ≤ x²ax².

A mapping η from S to real closed interval [0, 1] is said to be a fuzzy subset of S (or fuzzy set of S). For any subset T (≠ ∅) of S, the characteristic function η _T [22] of T is defined as follows: η T ( t ) = { 1 if t ∈ T , 0 if t ∉ T .(1) For any two fuzzy subsets η and ξ of S, η ∩ ξ, η ∪ ξ and η ∘ ξ (see [11]) are defined as follows: ( η ∩ ξ ) ( s ) = min { η ( s ) , ξ ( s ) } = η ( s ) ∧ ξ ( s ) ( η ∪ ξ ) ( s ) = max { η ( s ) , ξ ( s ) } = η ( s ) ∨ ξ ( s ) for each s ∈ S and ( η o ξ ) ( s ) = { ∨ ( y , z ) ∈ A s { η ( y ) ∧ ξ ( z ) } if A s ≠ ∅ , 0 if A s = ∅ ,(2) where A _s is a relation on S defined as A _s  = {(y, z) ∈ S × S | s ≤ yz} . We define an order relation ⪯ on the set of all fuzzy subsets of S by η ⪯ ξ ⇔ η ( s ) ≤ ξ ( s ) for each s ∈ S .

If η, ξ are fuzzy subsets of S such that η ⪯ ξ, then for each fuzzy subset λ of S, η ∘ λ ⪯ ξ ∘ λ and λ ∘ η ⪯ λ ∘ ξ.

Any ordered semigroup S itself is a fuzzy subset of S (see [11]) such that S (s) =1 for each s ∈ S.

A fuzzy subset η of S is said to be a fuzzy subsemigroup [6] of S if η (rs) ≥ min {η (r) , η (s)} for all r, s ∈ S and η is said to be a fuzzy left (right) ideal [6] of S if; (1) r ≤ s ⇒ η (r) ≥ η (s) and; (2) η (rs) ≥ η (s) (η (rs) ≥ η (r)) for all r, s ∈ S. If a fuzzy subset is both a fuzzy left and fuzzy right ideal of S, then it is said to be a fuzzy ideal. A fuzzy subsemigroup η of S is said to be a fuzzy bi-ideal [10] of S if; (1) r ≤ s ⇒ η (r) ≥ η (s) and (2) η (rst) ≥ min {η (r) , η (t)} for all r, s, t ∈ S.

For more details of fuzzy subsets and their related notions, the reader is referred to [1, 26].

For any a ∈ S and u ∈ (0, 1] the ordered fuzzy point a _u of S is defined by a u ( s ) = { u , if s ∈ ( a ] , 0 , if s ∉ ( a ] .(3) Thus a _u is a fuzzy subset of S. For any fuzzy subset η of S, we shall also denote a _u  ⊆ η by a _u  ∈ η in the sequel. Then a _u  ∈ η if and only if η (a) ≥ u.

An ordered fuzzy point a _u of S, for any k^* ∈ (0, 1], is said to be (k^*, q)-quasi-coincident with a fuzzy subset η of S [10], written as a _u  (k^*, q) η, if η (a) + u > k^* . Let 0 ≤ k < k^* ≤ 1. For an ordered fuzzy point x _u , we say that

x _u  (k^*, q _k ) η if η (x) + u + k > k^*;

A fuzzy subset η of S is said to be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy subsemigroup of S [10] if x _u  ∈ η and y _v  ∈ η imply (xy) _min{u,v} ∈ ∨ (k^*, q _k ) η for all u, v ∈ (0, 1] and x, y ∈ S.

A fuzzy subset η of S is said to be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left (resp. right) ideal of S [10] if for each u ∈ (0, 1] and x, y ∈ S:

x ≤ y, y _u  ∈ η ⇒ x _u  ∈ ∨ (k^*, q _k ) η, and

η is said to be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy ideal if it is both an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left and right ideal of S.

A fuzzy subset η of S is said to be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal [10] of S if for each u, v ∈ (0, 1] and x, y, z ∈ S:

x ≤ y, y _u  ∈ η ⇒ x _u  ∈ ∨ (k^*, q _k ) η, and

Theorem 2.1.[10] A fuzzy subset η of S is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S if and only if

x ≤ y ⇒ η ( x ) ≥ min { η ( y ) , k * - k 2 } , and

Theorem 2.2.[10] Let B be a generalized bi-ideal of S and η be a fuzzy subset of S defined by η ( x ) = { k * - k 2 , if x ∈ B ; 0 , if x ∉ B .(4)Then

η is a ((k^*, q) , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S.

Theorem 2.3. [10] A fuzzy subset η of S is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S if and only if U ( η ; u ) ( ≠ ∅ ) ( u ∈ ( 0 , k * - k 2 ] ) is a generalized bi-ideal of S.

Theorem 2.4.[10]If η is a nonzero (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S, then the set η 0 = { x ∈ S | η ( x ) > 0 }(5)is a generalized bi-ideal of S.

Lemma 2.5. [10] Let η be a fuzzy subset of an ordered semigroup S. Then η is a strongly convex fuzzy subset of S if and only if x ≤ y implies η (x) ≥ η (y), for each x, y ∈ S.

Lemma 2.6. [9]An ordered semigroup S is completely regular if and only if the bi-ideals of S are semiprime.

Lemma 2.7. [10] If η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S, then η k k * _ is a fuzzy generalized bi-ideal of S.

A fuzzy subset η of S is said to be a semiprime [23] if η (a) ≥ η (a²) for each a ∈ S.

Definition 3.1. A fuzzy subset η of S is said to be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime if x u 2 ∈ η ⇒ x u ∈ ∨ ( k * , q k ) η(6) for each u ∈ (0, 1] and x ∈ S.

Remark 3.2. Clearly fuzzy semiprime subsets of the ordered semigroup are (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subsets but conversely (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subsets are not fuzzy semiprime subsets. We justify it by following:

Then, (S, · , ≤) is an ordered semigroup. Now define a fuzzy subset η of S as: η (w) =0.5,  η (x) =0.4,  η (y) =0.7,  η (z) =0.3 and take k^* = 0.9,  k = 0.3. Then η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S but not a fuzzy semiprime subset because η (z) ngeqη (z²).

Theorem 3.4. A fuzzy subset η of S is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S if and only if η ( x ) ≥ min { η ( x 2 ) , k * - k 2 }(7) for each x ∈ S.

Proof. Let η be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S. If η ( x ) < min { η ( x 2 ) , k * - k 2 } for some x ∈ S, then there exist u ∈ (0, 1] such that η ( x ) < u ≤ min { η ( x 2 ) , k * - k 2 } . So x u 2 ∈ f , but x u ∈ ∨ ( k * , q k ) ¯ η , a contradiction. Hence η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } .

Conversely assume that η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } for all x ∈ S and let x u 2 ∈ η (u ∈ (0, 1]). Then η (x²) ≥ u. So η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } ≥ min { u , k * - k 2 } . If u ≤ k * - k 2 , then η (x) ≥ u implies that x _u  ∈ η. Again, if u > k * - k 2 , then η ( x ) ≥ k * - k 2 . So η ( x ) + u > k * - k 2 + k * - k 2 = k * - k , which implies that x _u  (k^*, q _k ) η. Hence x _u  ∈ ∨ (k^*, q _k ) η.

Lemma 3.5. The characteristic function η _T of T (≠ ∅) ⊆ S which is defined as η ( x ) = { k * - k 2 , if x ∈ T ; 0 , if x ∉ T , is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S if and only if T is a semiprime subset of S.

Proof. Let x ∈ S and u ∈ (0, 1] be such that x u 2 ∈ η . Then x² ∈ T, η (x²) ≥ u. Since T is a semiprime subset of S, we have x ∈ T. Thus η ( x ) = k * - k 2 . If u ≤ k * - k 2 , then η (x) ≥ u, so we have x _u  ∈ η. If u > k * - k 2 , then η ( x ) + u > k * - k 2 + k * - k 2 = k * - k . So x _u  (k^*, q _k ) η. Therefore x _u  ∈ ∨ (k^*, q _k ) η.

Conversely assume that x ∈ S and u ∈ (0, 1] be such that x² ∈ T. Then η T ( x 2 ) = k * - k 2 . As η _T is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime, we have η T ( x ) ≥ min { η T ( x 2 ) , k * - k 2 } = k * - k 2 .

Since η T ( x ) ≤ k * - k 2 . Therefore η T ( x ) = k * - k 2 implies x ∈ T. Hence T is semiprime. □

Corollary 3.6. The characteristic function η _B of B (≠ ∅) ⊆ S which is defined as η ( x ) = { k * - k 2 , if x ∈ B ; 0 , if x ∉ B , is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideal of S if and only if B is a semiprime generalized bi-ideal of S.

Proof. Follows by Theorem 2.2 and Lemma 3.5. □

Theorem 3.7. A fuzzy subset η of S is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S if and only if U ( η ; u ) ( ≠ ∅ ) ( u ∈ ( 0 , k * - k 2 ] ) is a semiprime subset of S.

Proof. Let x ∈ S such that x² ∈ U (η ; u), where u ∈ ( 0 , k * - k 2 ] . Then η (x²) ≥ u. By Theorem 3.4, η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } ≥ min { u , k * - k 2 } = u . Therefore x ∈ U (η ; u). Hence U (η ; u) is a semiprime subset of S.

Conversely suppose that U (η ; u) (≠ ∅) is a semiprime subset of S for all u ∈ ( 0 , k * - k 2 ] . Take any x ∈ S and suppose to the contrary that η ( x ) < min { η ( x 2 ) , k * - k 2 } . Then η ( x ) < u ≤ min { η ( y ) , k * - k 2 } , for some u ∈ ( 0 , k * - k 2 ] . This implies that x² ∈ U (η ; u), but x ∉ U (η ; u), a contradiction. Thus η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } for all x ∈ S. Hence, by Theorem 3.4, η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S. □

Corollary 3.8. A fuzzy subset η of S is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideal of S if and only if U ( η ; u ) ( ≠ ∅ ) ( u ∈ ( 0 , k * - k 2 ] ) is a semiprime generalized bi-ideal of S.

Proof. Follows by Theorem 2.3 and Theorem 3.7. □

Lemma 3.9. If η is a nonzero (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S, then the set η 0 = { x ∈ S | η ( x ) > 0 } is a semiprime subset of S.

Proof. Let η be a nonzero (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S. Let x ∈ S such that x² ∈ η₀. Then η (x²) >0. Since η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S, we have η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } > 0 , thus we have η (x) >0. Therefore x ∈ η₀. Hence η₀ is a semiprime subset of S. □

Corollary 3.10. If η is a nonzero (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideal of S, then the set η 0 = { x ∈ S | η ( x ) > 0 } is a semiprime generalized bi-ideal of S.

Proof. Follows by Theorem 2.4 and Lemma 3.9. □

Definition 3.11. For any fuzzy subset η of S the set [ η ] u = { x ∈ S | x u ∈ ∨ ( k * , q k ) η } , where u ∈ (0, 1] , is said to be an (∈ ∨ (k^*, q _k ))-level subset (see [11]) of η.

Theorem 3.12. Let η be a fuzzy subset of S. Then η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S if and only if the (∈ ∨ (k^*, q _k ))-level subset [η]  _u of η is a semiprime subset of S, for each u ∈ (0, 1].

Proof. Suppose η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semi-prime subset of S. To show that [η]  _u is a semiprime subset of S, take any x² ∈ [η]  _u . Then x u 2 ∈ ∨ ( k * , q k ) η implies η (x²) ≥ u or η (x²) + u + k > k^*. Since η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S, by Theorem 3.4, η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } .

Case (i). Let η (x²) ≥ u. If u > k * - k 2 , then η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } ≥ min { u , k * - k 2 } = k * - k 2 , implying x _u  ∈ ∨ (k^*, q _k ) η. If u ≤ k * - k 2 , then η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } ≥ min { u , k * - k 2 } = u , and so x _u  ∈ η. Hence x _u  ∈ ∨ (k^*, q _k ) η.

Case (ii). Let η (x²) + u + k > k^*. If u > k * - k 2 , then η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } > min { ( k * - u - k ) , k * - k 2 } = k * - u - k , that is η (x) + u + k > k^*, and thus x _u  ∈ ∨ (k^*, q _k ) η. If u ≤ k * - k 2 , then η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } ≥ min { ( k * - u - k ) , k * - k 2 } = u , and so x _u  ∈ η. Therefore x _u  ∈ ∨ (k^*, q _k ) η. Thus, in each case, we have x _u  ∈ ∨ (k^*, q _k ) η, and so x ∈ [η]  _u . Hence [η]  _u is a semiprime subset of S.

Conversely, let [η]  _u be a semiprime subset of S. If η ( x ) < min { η ( x 2 ) , k * - k 2 } for some x ∈ S. Then there exist u ∈ ( 0 , k * - k 2 ] such that η ( x ) < u ≤ min { η ( x 2 ) , k * - k 2 } . Now x² ∈ [η]  _u , which implies that x ∈ [η]  _u . Hence η (x) ≥ u or η (x) + u + k > k^*, which is not possible. Therefore η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } for each x ∈ S. □

Similarly the following Theorem follows:

Theorem 3.13. Let η be a strongly convex fuzzy subset of S. Then η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S if and only if [η]  _u , the (∈ ∨ (k^*, q _k ))-level subset of η is a generalized bi-ideal of S, for each u ∈ (0, 1].

Corollary 3.14. Let η be any strongly convex fuzzy subset of S. Then η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideal of S if and only if the (∈ ∨ (k^*, q _k ))-level subset [η]  _u of η is a semiprime generalized bi-ideal of S, for each u ∈ (0, 1].

Proof. Follows by Theorem 3.12 and Theorem 3.13. □

Definition 3.15. Let η be a fuzzy subset of S. For any x ∈ S, we define I _x the subset of S as follows: I x = { y ∈ S | η ( y ) ≥ min { η ( x ) , k * - k 2 } } .

Lemma 3.16. Let η be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S. Then I _x is a semiprime subset of S for each x ∈ S.

Proof. Let η be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S. Take an element a of S such that a² ∈ I _x . Then η ( a 2 ) ≥ min { η ( x ) , k * - k 2 } . Since η be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S, we have η ( a ) ≥ min { η ( a 2 ) , k * - k 2 } . Therefore η ( a ) ≥ min { min { η ( x ) , k * - k 2 } , k * - k 2 } = min { η ( x ) , k * - k 2 } . Thus a ∈ I _x . Hence I _x is semiprime. □

Similarly the following Lemma follows:

Lemma 3.17. Let η be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. Then I _x is a generalized bi-ideal of S for each x ∈ S.

Corollary 3.14. Let η be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideal of S. Then I _x is a semiprime generalized bi-ideal of S for each x ∈ S.

Proof. Follows by Lemma 3.16 and Lemma 3.17. □

Theorem 3.19. An ordered semigroup S is completely regular if and only if each (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal η of S is semiprime.

Proof. Let η be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S and x ∈ S. Since S is completely regular, there exist a ∈ S. such that x ≤ x²ax². As η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S, by Theorem 2.1, we have η ( x ) ≥ min { η ( x 2 ax 2 ) , k * - k 2 } ≥ min { η ( x 2 ) , η ( x 2 ) , k * - k 2 } = min { η ( x 2 ) , k * - k 2 } .

Thus, η ( x ) ≥ min { η ( x 2 ) , k * - k 2 } . Hence η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideal of S.

Conversely take any x ∈ S. Consider the generalized bi-ideal B (x²) = (x² ∪ x²Sx²] of S generated by x² (x ∈ S). Then, by Theorem 2.2, η_B(x²) is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. By hypothesis,

η B ( x 2 ) ( x ) ≥ min { η B ( x 2 ) ( x 2 ) , k * - k 2 } .

Since x² ∈ B (x²), η B ( x 2 ) ( x 2 ) = k * - k 2 . Therefore η B ( x 2 ) ( x ) ≥ k * - k 2 . As η B ( x 2 ) ( x ) ≤ k * - k 2 , η B ( x 2 ) ( x ) = k * - k 2 . Thus x ∈ B (x²) implies x ≤ x² or x ≤ x²ax². If x ≤ x², then x ≤ x⁵ implies x ∈ (x²Sx²]. Similarly, if x ≤ x²ax², then x ∈ (x²Sx²]. Hence S is completely regular. □

Corollary 3.20. In an ordered semigroup S generalized bi-ideals are semiprime if and only if (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideals of S are semiprime.

Proof. Suppose that generalized bi-ideals of S are semiprime. Then by Lemma 2.6, S is completely regular. Thus by Theorem 3.19, (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideals of S are semiprime.

Converse follows by Theorem 3.19. □

Definition 4.1. The (k^*, k)-lower part η k k * _ of a fuzzy subset η of S is defined [10] as follows: η k k * _ ( x ) = min { η ( x ) , k * - k 2 }

for all x ∈ S and 0 ≤ k < k^* ≤ 1. Clearly, η k k * _ is a fuzzy subset of S.

For any non-empty subset A of S and fuzzy subset η of S, ( η A _ ) k k * , the (k^*, k)-lower part of the characteristic function η _A , will be denoted by ( η k k * _ ) A in the sequel.

Definition 4.2. For any fuzzy subsets η and ξ of S, the η ( ∩ ) k k * ξ , η ( ∪ ) k k * ξ and η ( ∘ ) k k * ξ are defined as follows [10]: ( η ( ∩ ) k k * ξ ) ( x ) = min { ( η ∩ ξ ) ( x ) , k * - k 2 } ( η ( ∪ ) k k * ξ ) ( x ) = min { ( η ∪ ξ ) ( x ) , k * - k 2 } ( η ( ∘ ) k k * ξ ) ( x ) = min { ( η ∘ ξ ) ( x ) , k * - k 2 } for all x ∈ S and 0 ≤ k < k^* ≤ 1. Then, clearly, η ( ∩ ) k k * ξ , η ( ∪ ) k k * ξ and η ( ∘ ) k k * ξ are all fuzzy subsets of S.

Remark 4.3. [10] Let η and ξ be any fuzzy subsets of S. Then

( η k k * _ ) k k * = η k k * _   and η k k * _ ⊆ η ;

Theorem 4.4. Let S be an ordered semigroup and η be a fuzzy subset of S. We consider the following statements:

η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S.

Proof. (1) ⇒ (2). Suppose η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S. Let ξ be a fuzzy subset of S such that ξ ( ∘ ) k k * ξ ⪯ η . Clearly ξ k k * ≤ ξ ( ∘ ) k k * ξ . Indeed:

( ξ ( ∘ ) k k * ξ ) ( x 2 )

= min { { ⋁ ( r , s ) ∈ A x 2 min { ξ ( r ) , ξ ( s ) } } , k * - k 2 }

≥ min { min { ξ ( x ) , ξ ( x ) } , k * - k 2 }

= min { ξ ( x ) , k * - k 2 }

= ξ k k * ( x ) .

Since ξ ( ∘ ) k k * ξ ⪯ η , we have ξ k k * ( x ) ≤ η ( x 2 ) . Thus min { ξ k k * ( x ) , k * - k 2 } ≤ min { η ( x 2 ) , k * - k 2 } ≤ η ( x ) . As min { ξ k k * ( x ) , k * - k 2 } = ξ k k * ( x ) , ξ k k * ( x ) ≤ η ( x ) . Hence ξ k k * ⪯ η . □

Remark 4.5. In general, converse of Theorem 4.4 is not true that is the implication (2) ⇒ (1) does not hold in general. To justify it, we present the following example.

Example 4.6. Let S be the ordered semigroup as in Example 3.3. Consider the fuzzy subsets as follows:

η ( a ) = { 0.5 if a = w , 0.4 if a = x , 0.7 if a = y , 0.1 if a = z ,

and

ξ ( a ) = { 0.2 if a ∈ { w , x } , 0.3 if a = y , 0.1 if a = z .

Take k^* = 0.8 and k = 0.2. It is easy to verify that ξ ( ∘ ) k k * ξ ⪯ η implies ξ k k * ⪯ η but η is not a (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S.

It is natural to ask under what conditions the implication (2) ⇒ (1) is satisfied. This is an open problem to find in future?

Lemma 4.7. The (k^*, k)-lower part ( η k k * _ ) A of a characteristic function η _A of A (≠ ∅) is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset of S if and only if A is a semiprime subset of S.

Proof. Proof is similar to the proof of Lemma 3.5. □

Corollary 4.8. The (k^*, k)-lower part ( η k k * _ ) B of the characteristic function η _B of a generalized bi-ideal B is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideal of S if and only if B is a semiprime generalized bi-ideal of S.

Proof. Follows by Lemma 2.7 and Lemma 4.7. □

Lemma 4.9. If η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideal of S, then η k k * _ is a fuzzy semiprime generalized bi-ideal of S.

Proof. Let x ∈ S. As η is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideal of S, then we have η k k * _ ( x ) = min { η ( x ) , k * - k 2 } ≥ min { min { η ( x 2 ) , k * - k 2 } , k * - k 2 } = min { η k k * _ ( x 2 ) , k * - k 2 } = η k k * _ ( x 2 ) . Thus η k k * _ is an fuzzy semiprime subset of S. Hence by the Lemma 2.7, η k k * _ is an fuzzy semiprime generalized bi-ideal of S. □

Theorem 4.10. In any ordered semigroup S, the following are equivalent:

S is completely regular;

Proof. (1) ⇒ (2). Let S be a completely regular ordered semigroup and a ∈ S . Then there exist x ∈ S such that a ≤ a²xa² it follows that (a²x, a²) ∈ A _a . Now we have

( η 2 ( ∘ ) k k * S ( ∘ ) k k * η 2 ) ( a )

= min { { ⋁ ( r , s ) ∈ A a min { ( η 2 ( ∘ ) k k * S ) ( r ) , η 2 ( s ) } } ,

k * - k 2 }

≥ min { min { ( η 2 ( ∘ ) k k * S ) ( a 2 x ) , η 2 ( a 2 ) } , k * - k 2 } .

As a²x ≤ a²x, (a², x) ∈ A _{a ² x} , we have

( η 2 ( ∘ ) k k * S ) ( a 2 x )

= min { ( η 2 ∘ S ) ( a 2 x ) , k * - k 2 }

= min { { ⋁ ( r , s ) ∈ A a 2 x min { η 2 ( r ) , S ( s ) } } , k * - k 2 }

≥ min { η 2 ( a 2 ) , k * - k 2 } .

As a² ≤ a², (a, a) ∈ A _{a ²} , we have

min { η 2 ( a 2 ) , k * - k 2 }

= min { min { ( η ∘ η ) ( a 2 ) , k * - k 2 } , k * - k 2 } = min { ( η ∘ η ) ( a 2 ) , k * - k 2 }

≥ min { η ( a ) , k * - k 2 }

= η k k * _ ( a ) .

Therefore

( η 2 ( ∘ ) k k * S ( ∘ ) k k * η 2 ) ( a )

≥ min { min { ( η 2 ( ∘ ) k k * S ) ( a 2 x ) , η 2 ( a 2 ) } , k * - k 2 }

≥ min { min { min { η 2 ( a 2 ) , k * - k 2 } , η 2 ( a 2 ) } , k * - k 2 }

= min { η 2 ( a 2 ) , k * - k 2 }

≥ η k k * _ ( a ) .

Thus η k k * _ ⪯ η 2 ( ∘ ) k k * S ( ∘ ) k k * η 2 .

(2) ⇒ (1). Let a ∈ S. Since ( η k k * _ ) a is a fuzzy subset of S, by hypothesis, k * - k 2 = ( η k k * _ ) a ( a ) ≤ ( ( ( η k k * _ ) a ) 2 ( ∘ ) k k * S ( ∘ ) k k * ( ( η k k * _ ) a ) 2 ) ( a ) . Then there exist (p, q) ∈ A _a such that ( ( ( η k k * _ ) a ) 2 ( ∘ ) k k * S ) ( p ) ≠ 0 and ( ( η k k * _ ) a ) 2 ) ( q ) ≠ 0 . Therefore there exist (x, y) ∈ A _p and (r, s) ∈ A _q such that ( ( η k k * _ ) a ) ) 2 ( x ) ≠ 0 and ( η k k * _ ) a ( r ) ≠ 0 , ( η k k * _ ) a ( s ) ≠ 0 . Also there exits (u, v) ∈ A _x such that ( η k k * _ ) a ( u ) ≠ 0 , ( η k k * _ ) a ( v ) ≠ 0 . Thus u = a = v and r = a = s. Since (p, q) ∈ A _a and (x, y) ∈ A _p , (u, v) ∈ A _q , we have a ≤ pq ≤ xyrs ≤ uvyrs = a²ya². Hence S is completely regular. □

Lemma 4.11. If S is completely regular, then η k k * _ = η 2 ( ∘ ) k k * S ( ∘ ) k k * η 2 for each (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal η of S.

Proof. Let η be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. By Theorem 4.10, we have η k k * _ ⪯ η 2 ( ∘ ) k k * S ( ∘ ) k k * η 2 . It is sufficient to show that η 2 ( ∘ ) k k * S ( ∘ ) k k * η 2 ⪯ η k k * _ . Now we have ( η 2 ( ∘ ) k k * S ( ∘ ) k k * η 2 ) ( a ) = min { { ⋁ ( r , s ) ∈ A a min { ( η 2 ( ∘ ) k k * S ) ( r ) , η 2 ( s ) } } , k * - k 2 } .(8)

Since

( η 2 ( ∘ ) k k * S ) ( r ) = min { { ⋁ ( u 1 , v 1 ) ∈ A r min { η 2 ( u 1 ) , S ( v 1 ) } } , k * - k 2 } = min  {⋁ _{(u1,v1)∈A r}  min {⋁ _{(u2,v2)∈A u 1}  min {η (u₂) , η ( v 2 ) } , k * - k 2 } , k * - k 2 } = ⋁ ( u 2 v 2 , v 1 ) ∈ A r min { η ( u 2 ) , η ( v 2 ) , k * - k 2 }

and

η² (s)

= min { ( η ∘ η ) ( s ) , k * - k 2 }

= min { ⋁ ( p , q ) ∈ A s min { η ( p ) , η ( q ) } , k * - k 2 } .

Then by Equation (8), we have

( η 2 ( ∘ ) k k * S ( ∘ ) k k * η 2 ) ( a )

= min { { ⋁ ( r , s ) ∈ A a min { ( η 2 ( ∘ ) k k * S ) ( r ) } , η 2 ( s ) } , k * - k 2 }

= min  { {⋁ _{(r,s)∈A a}  min  {⋁ _{(u2v2,v1)∈A r}  min {η (u₂) ,

η ( v 2 ) , k * - k 2 } , { ⋁ ( p , q ) ∈ A s min { η ( p ) , η ( q ) } , k * - k 2 } } ,

k * - k 2 } }

= ⋁ ( u 2 v 2 v 1 , pq ) ∈ A a min { η ( u 2 ) , η ( v 2 ) , η ( p ) , η ( q ) , k * - k 2 }

≤ ⋁ ( u 2 v 2 v 1 , pq ) ∈ A a min { η ( u 2 v 2 v 1 pq ) , k * - k 2 }

≤ min { η ( a ) , k * - k 2 }

= η k k * _ ( a ) .

Thus η 2 ( ∘ ) k k * 1 ( ∘ ) k k * η 2 ⪯ η k k * _ . □

Corollary 4.12. In an ordered semigroup S, the following are equivalent:

S is completely regular;

Finally, we consider the following particular cases of the present work:

(1). If we take k^* = 1, then the definition of an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy subset reduces to a new concept what we call as an (∈ , ∈ ∨ q _k )-fuzzy subset and defined as: A fuzzy subset η of S is said to be an (∈ , ∈ ∨ q _k )-fuzzy semiprime if x u 2 ∈ η ⇒ x u ∈ ∨ q k η for all u ∈ (0, 1] and x ∈ S. Thus we can apply all the results of this paper in the setting of (∈ , ∈ ∨ q _k )-fuzzy subsets.

(2). If we take k^* = 1 and k = 0, then the definition of an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy subset reduces to a new concept what we call as an (∈ , ∈ ∨ q)-fuzzy subset and defined as: A fuzzy subset η of S is said to be an (∈ , ∈ ∨ q)-fuzzy semiprime if x u 2 ∈ η ⇒ x u ∈ ∨ q η for all u ∈ (0, 1] and x ∈ S. Thus we can apply all the results of this paper in the setting of (∈ , ∈ ∨ q)-fuzzy subsets.

Using the notion of fuzzy set theory, we studied new types of fuzzy semiprime subsets in ordered semigroups and investigated several properties of them. In fact, we introduced the concept of an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subset in an ordered semigroup, and investigated the properties of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideals by concerning the (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime subsets. We provide characterizations of completely regular ordered semigroups in terms of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy semiprime generalized bi-ideals and their (k^*, k)-lower parts. We hope that this work will provide a deep impact on the upcoming research in this field and other related algebraic structures studies to open up new horizons of interest and innovations. As future directions, one can further study these new types of fuzzy semiprime subsets in different algebras such as BCK/BCI-algebras, MTL-algerbas, BL-algebras, EQ-algebras, B-algebras, MV-algebras, d-algebras, Q-algebras, etc. One may also apply these concepts to study some applications in many fields like decision making, knowledge base systems, medical diagnosis, data analysis, graph theory, etc.