Ordered semigroups characterized in terms of generalized fuzzy ideals

1 Introduction

In 1965, Lofti Zadeh [36] introduced the notion of a fuzzy subset of a set. This seminal paper has opened up new insights and application in a wide range of scientific fields. Rosenfeld [31] used the notion of a fuzzy subset to forth cornerstone papers in several areas of mathematics, among other disciplines. Kuroki initiated the theory of fuzzy semigroups in his paper [25]. The monograph by Mordeson et al. [28] deals with the theory of fuzzy semigroups and their use in fuzzy codes, fuzzy finite state machines and fuzzy languages. Because of this, semigroups and related structures are presently extensively investigated in fuzzy settings. Kehayopulu [17, 20] characterized regular, left regular and right regular ordered semigroups by means of fuzzy left, fuzzy right and fuzzy bi-ideals. Murali [29] defined the concept of belongingness of a fuzzy point to a fuzzy subset under a natural equivalence on a fuzzy subset. The idea of quasi-coincidence of a fuzzy point with a fuzzy set played a vital role in generating different types of fuzzy subgroups. Using these ideas, Bhakat and Das [4–8] introduced the concept of (α, β)-fuzzy subgroups by using the ’belong to’ (∈) relation and ’quasi coincident with’ (q) relation between a fuzzy point and a fuzzy subgroup and introduced the concept of (∈ , ∈ ∨ q)-fuzzy subgroup. Davvaz defined (∈ , ∈ ∨ q)-fuzzy subnearrings and ideals of a near ring in [9]. Kazanci and Yamak [16] studied (∈ , ∈ ∨ q)-fuzzy bi-ideals of a semigroup. In [34], Tang characterized ordered semigroups by (∈ , ∈ ∨ q)-fuzzy ideals. Generalizing the concept of the quasi-coincident of a fuzzy point with a fuzzy subset, Jun [15] defined (∈ , ∈ ∨ q _k )- fuzzy subalgebras in BCK/BCI-algebras, respectively. In [33] Shabir et al. characterized the regular semigroups by (∈ , ∈ ∨ q _k )-fuzzy ideals. In [21–23] Khan et al. characterized ordered semigroups in terms of fuzzy bi-ideals and intuitionistics fuzzy bi-ideals. In fact fuzzy ideals of ordered semigroups with special properties have always played an important role in the study of structure of ordered semigroups. Following the terminology given by Zadeh, fuzzy sets in an ordered semigroup S were first considered by Kehayopulu and Tsingelis in [17–20]. They defined fuzzy analogous for several notions, which have proved very useful in the theory of ordered semigroups. To mention [18], they did prove that each ordered groupoid can be embedded into an ordered groupoid having the greatest element (poe-groupoid) in terms of fuzzy sets. Xie and Tang [35] studied regular and intra-regular ordered semigroups in terms of fuzzy subsets. In [1–3] Akram et al., studied generalized fuzzy ideal of fuzzy K-algebras. In this paper, notions of (k^*, q)-quasi-coincident, (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left ideal, (∈ , ∈ ∨ (k^*, q _k ))-fuzzy right ideal and (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of an ordered semigroup are introduced. (k^*, k)-lower parts of an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left ideal, (∈ , ∈ ∨ (k^*, q _k ))-fuzzy right ideal and (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal are also introduced. We, then, have characterized these various fuzzy ideals. Finally, using properties of these fuzzy ideals, we have characterized regular, left regular, right regular, completely regular and left weakly regular ordered semigroups.

2 Preliminaries

An ordered semigroup is a pair (S, .) comprising of a semigroup S and an order relation ≤ (on S) that is compatible with the binary operation; i.e. for all a, b, s ∈ S, a ≤ b implies that sa ≤ sb and as ≤ bs.

For any non-empty subset A of an ordered semigroup S, define [17] by ( A ] = { a ∈ S | a ≤ b for some b ∈ A } .

Then for any non-empty subsets A, B of S[34], we have (1) A ⊆ (A]; (2) If A ⊆ B, then (A] ⊆ (B]; (3) (A] (B] ⊆ (AB]; (4) ((A]] = (A]; (5) ((A] (B]] = (AB].

A non-empty subset A of an ordered semigroup S is called a subsemigroup of S if A² ⊆ A.

A non-empty subset A of an ordered semigroup S is called a left(resp. right) ideal of S if (1) for any a ∈ S and b ∈ A, if a ≤ b, then a ∈ A and (2) SA ⊆ A(resp. AS ⊆ A). We denote by L (a) (resp. R (a)), the left (resp. right) ideal of S generated by a (a ∈ S). Then L (a) = (a ∪ Sa] (resp. R (a) = (a ∪ aS]) [35]. If A is both a left and right ideal of S, then A is called an ideal of S. We denote by I (a), the ideal of S generated by a (a ∈ S). Then I (a) = (a ∪ Sa ∪ aS ∪ SaS] [35].

A non-empty subset A of an ordered semigroup S is called a bi-ideal of S if (1) for any a ∈ S and b ∈ A, if a ≤ b, then a ∈ A; (2) for all a, b ∈ A ⇒ab ∈ A and (3) ASA ⊆ A. We denote by B (a), the bi-ideal of S generated by a (a ∈ S). Then B (a) = (a ∪ a² ∪ aSa] [35].

A non-empty subset A of an ordered semigroup S is called a generalized bi-ideal of S if (1) for any a ∈ S and b ∈ A, if a ≤ b, then a ∈ A and (2) ASA ⊆ A. We denote by B ˘ ( a ), the generalized bi-ideal of S generated by a (a ∈ S). Then B ˘ ( a ) = ( a ∪ aSa ].

Any function f from S to the closed interval [0, 1] is called a fuzzy subset [36] of S. The ordered semigroup S itself is regarded as a fuzzy subset of S by defining S (x) ≡1 for all x ∈ S.

Let f and g be two fuzzy subsets of S. Then, the inclusion relation f ⊆ g is defined [12] by f (x) ≤ g (x) for all x ∈ S. Also ( f ∩ g ) ( x ) = min { f ( x ) , g ( x ) } ( f ∪ g ) ( x ) = max { f ( x ) , g ( x ) } for all x ∈ S.

For any ordered semigroup (S, . , ≤) and x ∈ S, define [12] by A x = { ( y , z ) ∈ S × S | x ≤ yz } .

Then product f ∘ g of any fuzzy subsets f and g of S is defined [18] by ( f ∘ g ) ( x ) = { ⋁ ( y , z ) ∈ A x min { f ( y ) , g ( z ) } , A x ≠ ∅ , 0 , A x = ∅ ; for all x ∈ S. This operation “∘” defined above is associative and (F (S) , ∘ , ⊆) is a poe-semigroup, where F (S) denotes the set of all fuzzy subsets of S.

Let A be a non-empty 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 .

Clearly f _A is a fuzzy subset of S.

Let S be an ordered semigroup of S. A fuzzy subset f of S is called a fuzzy subsemigroup [17] of S if f ( xy ) ≥ min { f ( x ) , f ( y ) } for all x, y ∈ S.

A fuzzy subset f of S is called a fuzzy left(resp. right) ideal [17] of S if (1) x ≤ y ⇒ f (x) ≥ f (y) and (2) f (xy)   ≥  f (y) (resp. f (xy)   ≥  f (x)) for all x, y ∈ S; or equivalently S ∘ f ⊆ f (resp. f ∘ S ⊆ f). A fuzzy subset f of S is called a fuzzy ideal of S if it is both a fuzzy left and fuzzy right ideal of S.

A fuzzy subsemigroup f of S is called a fuzzy bi-ideal [17] of S if (1) x ≤ y ⇒ f (x) ≥ f (y) and (2) f (xyz) ≥ min {f (x) , f (z)} for all x, y, z ∈ S.

A fuzzy subset f of S is called a fuzzy generalized bi-ideal of S if (1) x ≤ y ⇒ f (x) ≥ f (y) and (2) f (xyz) ≥ min {f (x) , f (z)} for all x, y, z ∈ S.

Definition 2.1. [34] Let S be an ordered semigroup, a ∈ S and u ∈ (0, 1]. An ordered fuzzy point a _u of S is defined by a u ( x ) = { u , if x ∈ ( a ] , 0 , if x ∉ ( a ] .

Then a _u is a mapping from S into [0,1]. Thus an ordered fuzzy point of S is a fuzzy subset of S. For any fuzzy subset f of S, we shall also denote a _u  ⊆ f by a _u  ∈ f in the sequel. Then a _u  ∈ f if and only if f (a) ≥ u.

Definition 2.2. [34] An ordered fuzzy point a _u of an ordered semigroup S is said to be quasi-coincident with a fuzzy subset f of S, written as a _u qf, if f (a) + u > 1.

Definition 2.3. [34] Let f be any fuzzy subset of an ordered semigroup S. For any u ∈ (0, 1], the set U ( f ; u ) = { x ∈ S | f ( x ) ≥ u } , is called a level subset of f.

Theorem 2.4. [17] A fuzzy subset f of an ordered semigroup S is a fuzzy left (resp. right) ideal of S if and only if U (f ; u) (≠ ∅), where u ∈ (0, 1], is a left (resp. right) ideal of S.

Theorem 2.5. [12] A fuzzy subset f of an ordered semigroup S is a fuzzy generalized bi-ideal of S if and only if U (f ; u) (≠ ∅), where u ∈ (0, 1], is a generalized bi-ideal of S.

Theorem 2.6. [17] A non-empty subset A of an ordered semigroup S is a left (resp. right) ideal of S if and only if for any fuzzy subset f of S, the characteristic function f _A is a fuzzy left (resp. right) ideal of S.

Theorem 2.7. [12] A non-empty subset A of an ordered semigroup S is a generalized bi-ideal of S if and only if for any fuzzy subset f of S, the characteristic function f _A is a fuzzy generalized bi-ideal of S.

3 (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideals of ordered semigroups

In this section, first we define (k^*, q)-quasi-coincident and investigate mainly the properties of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left (resp. right) ideal, and (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of an ordered semigroup S. Some results about regular and left regular ordered semigroups, in terms of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideals, are also proved.

Definition 3.1. An ordered fuzzy point a _u of an ordered semigroup S, for any k^* ∈ (0, 1], is said to be (k^*, q)-quasi-coincident with a fuzzy subset f of S, written as a _u  (k^*, q) f, if f ( a ) + u > k * .

Let (S, . , ≤) be an ordered semigroup and 0 ≤ k < k^* ≤ 1. For an ordered fuzzy point x _u , we say that

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

Definition 3.2. A fuzzy subset f of an ordered semigroup S is called an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy subsemigroup of S if x _u  ∈ f and y _v  ∈ f imply (xy) _min{u,v} ∈ ∨ (k^*, q _k ) f for all u, v ∈ (0, 1] and x, y ∈ S.

Definition 3.3. A fuzzy subset f of an ordered semigroup S is called an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left (resp. right) ideal of S if for all u ∈ (0, 1] and x, y ∈ S, the following conditions hold:

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

A fuzzy subset f of an ordered semigroup S is called an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy ideal if it is both an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left ideal and (∈ , ∈ ∨ (k^*, q _k ))-fuzzy right ideal of S.

Definition 3.4. A fuzzy subset f of an ordered semigroup S is called an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy bi-ideal of S if for all u, v ∈ (0, 1] and x, y, z ∈ S, the following conditions hold:

x ≤ y, y _u  ∈ f ⇒ x _u  ∈ ∨ (k^*, q _k ) f;

Definition 3.5. A fuzzy subset f of an ordered semigroup S is called an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S if for all u, v ∈ (0, 1] and x, y, z ∈ S, the following conditions hold:

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

Theorem 3.6. Let S be an ordered semigroup and B be a generalized bi-ideal of S. Let f be a fuzzy subset of S defined by f ( x ) = { k * - k 2 , if x ∈ B ; 0 , if x ∉ B .

Then

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

Proof. (1) Let x, y ∈ S, x ≤ y and u ∈ (0, 1] be such that y _u  (k^*, q) f. Then y ∈ B, f (y) + u > k^*. Since B is a generalized bi-ideal of S and x ≤ y ∈ B, we have x ∈ B. Thus f ( x ) ≥ k * - k 2. If u ≤ k * - k 2, then f (x) ≥ u. So x _u  ∈ f. If u > k * - k 2, then f ( x ) + u > k * - k 2 + k * - k 2 = k * - k. Therefore x _u  (k^*, q _k ) f. Let x, y, z ∈ S and u, v ∈ (0, 1] be such that x _u  (k^*, q) f and z _v  (k^*, q) f. Then x, z ∈ B, f (x) + u > k^* and f (z) + v > k^*. Since B is a generalized bi-ideal of S, we have xyz ∈ B. Thus f ( xyz ) ≥ k * - k 2. If min { u , v } ≤ k * - k 2, then f (xyz) ≥ min {u, v} and so (xyz) _min{u,v} ∈ f. Again if min { u , v } > k * - k 2, then f ( xyz ) + min { u , v } > k * - k 2 + k * - k 2 = k * - k.

So (xyz) _min{u,v} (k^*, q _k ) f.

Therefore (xyz) _min{u,v} ∈ ∨ (k^*, q _k ) f.

(2) Let x, y ∈ S, x ≤ y and u ∈ (0, 1] be such that y _u  ∈ f. Then y ∈ B, f (y) ≥ u. Since B is a generalized bi-ideal of S and x ≤ y ∈ B, we have x ∈ B. Thus f ( x ) ≥ k * - k 2. If u ≤ k * - k 2, then f (x) ≥ u, so we have x _u  ∈ f. If u > k * - k 2, then f ( x ) + u > k * - k 2 + k * - k 2 = k * - k. So x _u  (k^*, q _k ) f. Therefore x _u  ∈ ∨ (k^*, q _k ) f. Let x, y, z ∈ S and u, v ∈ (0, 1] be such that x _u  ∈ f and z _v  ∈ f. Then x, z ∈ B, f (x) ≥ u and f (z) ≥ v. Since B is a generalized bi-ideal of S, we have xyz ∈ B. Thus f ( xyz ) ≥ k * - k 2. If min { u , v } ≤ k * - k 2, then f (xyz) ≥ min {u, v}. Therefore (xyz) _min{u,v} ∈ f. Again if min { u , v } > k * - k 2, then f ( xyz ) + min { u , v } > k * - k 2 + k * - k 2 = k * - k. So (xyz) _min{u,v} (k^*, q _k ) f.

Therefore (xyz) _min{u,v} ∈ ∨ (k^*, q _k ) f.

Theorem 3.7. Let f be a fuzzy subset of an ordered semigroup S. Then f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S if and only if

x ≤ y ⇒ f ( x ) ≥ min { f ( y ) , k * - k 2 };

Proof. Let f be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S and x, y ∈ S. Suppose to the contrary that f ( x ) < min { f ( y ) , k * - k 2 } for some x, y ∈ S. Choose u ∈ (0, 1] such that f ( x ) < u ≤ min { f ( y ) , k * - k 2 }. Then y _u  ∈ f, but ( x ) u ∈ ∨ ( k * , q k ) ¯ f, which is a contradiction. Hence f ( x ) ≥ min { f ( y ) , k * - k 2 }.

Suppose, again, to the contrary that f ( xyz ) < min { f ( x ) , f ( z ) , k * - k 2 } for some x, y, z ∈ S. Choose u ∈ (0, 1] such that f ( xyz ) < u ≤ min { f ( x ) , f ( z ) , k * - k 2 }. Then x _u  ∈ f and z _u  ∈ f, but ( xyz ) u ∈ ∨ ( k * , q k ) ¯ f, which is a contradiction. Hence f ( xyz ) ≥ min { f ( x ) , f ( z ) , k * - k 2 }.

Conversely, suppose that conditions (1) and (2) hold. Let f ( x ) ≥ min { f ( y ) , k * - k 2 } for all x, y ∈ S. Let y _u  ∈ f (u ∈ (0, 1]). Then f (y) ≥ u. So f ( x ) ≥ min { f ( y ) , k * - k 2 } ≥ min { u , k * - k 2 }. If u ≤ k * - k 2, then f (x) ≥ u implying that x _u  ∈ f. Again, if u > k * - k 2, then f ( x ) ≥ k * - k 2. So f ( x ) + u > k * - k 2 + k * - k 2 = k * - k, which implies that x _u  (k^*, q _k ) f. Hence x _u  ∈ ∨ (k^*, q _k ) f. Let f ( xyz ) ≥ min { f ( x ) , f ( z ) , k * - k 2 } for all x, y, z ∈ S. Let x _u  ∈ f and z _v  ∈ f (u, v ∈ (0, 1]). Then f (x) ≥ u and f (z) ≥ v. So f ( xyz ) ≥ min { f ( x ) , f ( z ) , k * - k 2 } ≥ min { u , v , k * - k 2 }. If min { u , v } ≤ k * - k 2, then f (xyz) ≥ min {u, v} implying that (xyz) _min{u,v} ∈ f. If min { u , v } > k * - k 2, then f ( xyz ) ≥ k * - k 2. So f ( xyz ) + min { u , v } > k * - k 2 + k * - k 2 = k * - k, which imply that (xyz) _min{u,v} (k^*, q _k ) f.

Hence (xyz) _min{u,v} ∈ ∨ (k^*, q _k ) f. Therefore, f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-idealof S.

Every (∈ , ∈ ∨ (k^*, q _k ))-fuzzy bi-ideal of S is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. However, the converse is not true, in general, as shown in the following example:

Example 3.8. Let S = {a, b, c} be an ordered semigroup with respect to the order relation a ≤ b, a ≤ c and the operation ^′ . ′ defined by the following Cayley table: . a b c a a a a b a a a c a a b

Define f on S by f (a) =0.55, f (b) =0 and f (c) =0.40. Then f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. Since f ( cc ) = f ( b ) = 0 < min { 0.40 , 0.40 , k * - k 2 } = min { f ( c ) , f ( c ) , k * - k 2 }, f is not an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy subsemigroup of S, and so f is not a (∈ , ∈ ∨ (k^*, q _k ))-fuzzy bi-ideal of S.

Theorem 3.9. Let f be a fuzzy subset of S. Then f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S if and only if U ( f ; u ) ( ≠ ∅ ) ( u ∈ ( 0 , k * - k 2 ] ) is a generalized bi-ideal of S.

Proof. Suppose that f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. Let x, y ∈ S be such that x ≤ y ∈ U (f ; u), where u ∈ ( 0 , k * - k 2 ]. Then f (y) ≥ u. By Theorem 3.7, f ( x ) ≥ min { f ( y ) , k * - k 2 } ≥ min { u , k * - k 2 } = u. Therefore x ∈ U (f ; u). Let x, z ∈ U (f ; u) and y ∈ S. Then f (x) ≥ u and f (z) ≥ u. So, by Theorem 3.7, f ( xyz ) ≥ min { f ( x ) , f ( z ) , k * - k 2 } ≥ min { u , k * - k 2 } = u. Thus f (xyz) ≥ u. Therefore xyz ∈ U (f ; u). Hence U (f ; u) is a generalized bi-ideal.

Conversely, suppose that U (f ; u) (≠ ∅) is a generalized bi-ideal of S for all u ∈ ( 0 , k * - k 2 ]. Take any x, y ∈ S with x ≤ y and suppose to the contrary that f ( x ) < min { f ( y ) , k * - k 2 }. Then f ( x ) < u ≤ min { f ( y ) , k * - k 2 }, for some u ∈ ( 0 , k * - k 2 ]. This implies that y ∈ U (f ; u), but x ∉ U (f ; u). This is a contradiction. Thus f ( x ) ≥ min { f ( y ) , k * - k 2 } for all x, y ∈ S with x ≤ y. Again suppose to the contrary that f ( xyz ) < min { f ( x ) , f ( z ) , k * - k 2 } for some x, y, z ∈ S. Then there exist u ∈ ( 0 , k * - k 2 ] such that f ( xyz ) < u ≤ min { f ( x ) , f ( z ) , k * - k 2 }. This implies that x _u  ∈ U (f ; u) and z _u  ∈ U (f ; u), but (xyz)  _u  ∉ U (f ; u). We get a contradiction. Therefore f ( xyz ) ≥ min { f ( x ) , f ( z ) , k * - k 2 } for all x, y, z ∈ S. Hence, by Theorem 3.7, f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S.

Example 3.10. Let S = {a, b, c, d} be an ordered semigroup with respect to the order relation a ≤ b and the operation ^′ . ′ defined by the following Cayley table: . a b c d a a a a a b a a a a c a a b a d a a b b

Then {a} , {a, b} , {a, c} , {a, d} , {a, b, c} , {a, c, d} and {a, b, c, d} are generalized bi-ideals of S. Define f on S by f (a) =0.9, f (b) =0.4, f (c) =0.45 and f (d) =0.8. Then U ( f ; u ) = { S , if 0 < u ≤ 0.4 ; { a , c , d } , if 0.4 < u ≤ 0.45 ; { a , d } , if 0.45 < u ≤ 0.8 ; { a } , if 0.8 < u ≤ 0.9 ; ∅ , if 0.9 < u ≤ 1 .

Theorem 3.11. If f is a nonzero (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S, then the set f₀ = {x ∈ S | f (x) >0} is a generalized bi-ideal of S.

Proof. Let f be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. Let x, y ∈ S, x ≤ y and y ∈ f₀. Then f (y) >0. Since f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S, we have f ( x ) ≥ min { f ( y ) , k * - k 2 } > 0, since f (y) >0. Thus f (x) >0 and so x ∈ f₀. Let x, z ∈ f₀ and y ∈ S. Then, f (x) >0 and f (z) >0. Therefore f ( xyz ) ≥ min { f ( x ) , f ( z ) , k * - k 2 } > 0. Thus xyz ∈ f₀ and, consequently, f₀ is a generalized bi-ideal of S.

Theorem 3.12. A non-empty subset B of an ordered semigroup S is a generalized bi-ideal of S if and only if the characteristic function f _B is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S.

Proof. The proof is straightforward, so we omit it. Similar to the proof of Theorem 3.6, we may prove the following theorems:

Theorem 3.13. Let S be an ordered semigroup and L be a left ideal of S. Let f be a fuzzy subset of S defined by f ( x ) = { k * - k 2 , if x ∈ L ; 0 , if x ∉ L .

Then

f is a ((k^*, q) , ∈ ∨ (k^*, q _k ))-fuzzy left ideal of S.

Theorem 3.14. Let S be an ordered semigroup and R be a right ideal of S. Let f be a fuzzy subset of S defined by f ( x ) = { k * - k 2 , if x ∈ R ; 0 , if x ∉ R .

Then

f is a ((k^*, q) , ∈ ∨ (k^*, q _k ))-fuzzy right ideal of S.

Theorem 3.15. Let f be a fuzzy subset of an ordered semigroup S. Then f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left ideal of S if and only if

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

Proof. Let f be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left ideal of S and x, y ∈ S. Suppose to the contrary that f ( x ) < min { f ( y ) , k * - k 2 } for some x, y ∈ S. Choose u ∈ (0, 1] such that f ( x ) < u ≤ min { f ( y ) , k * - k 2 }. Then y _u  ∈ f, but ( x ) u ∈ ∨ ( k * , q k ) ¯ f. This is a contradiction. Hence f ( x ) ≥ min { f ( y ) , k * - k 2 }. Again suppose to the contrary that f ( xy ) < min { f ( y ) , k * - k 2 } for some x, y ∈ S. Choose u ∈ (0, 1] such that f ( xy ) < u ≤ min { f ( y ) , k * - k 2 }. Then x ∈ S and y _u  ∈ f, but ( xy ) u ∈ ∨ ( k * , q k ) ¯ f. So, we get a contradiction. Hence f ( xy ) ≥ min { f ( y ) , k * - k 2 }.

Conversely, suppose that conditions (1) and (2) hold. Let f ( x ) ≥ min { f ( y ) , k * - k 2 } for all x, y ∈ S and let y _u  ∈ f (u ∈ (0, 1]). Then f (y) ≥ u. So f ( x ) ≥ min { f ( y ) , k * - k 2 } ≥ min { u , k * - k 2 }. Now, if u ≤ k * - k 2, then f (x) ≥ u imply that x _u  ∈ f. Again, if u > k * - k 2, then f ( x ) ≥ k * - k 2. So f ( x ) + u > k * - k 2 + k * - k 2 = k * - k, which implies that x _u  (k^*, q _k ) f. Hence x _u  ∈ ∨ (k^*, q _k ) f. Suppose next that f ( xy ) ≥ min { f ( y ) , k * - k 2 } for all x, y ∈ S. Let x ∈ S and y _u  ∈ f for all u ∈ (0, 1]. Then f (y) ≥ u. So f ( xy ) ≥ min { f ( y ) , k * - k 2 } ≥ min { u , k * - k 2 }. Now if u ≤ k * - k 2, then f (xy) ≥ u imply that (xy)  _u  ∈ f. If u > k * - k 2, then f ( xy ) ≥ k * - k 2. So f ( xy ) + u > k * - k 2 + k * - k 2 = k * - k, which implies that (xy)  _u  (k^*, q _k ) f. Hence (xy)  _u  ∈ ∨ (k^*, q _k ) f. Therefore, f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left ideal of S.

Clearly, every fuzzy left ideal of an ordered semigroup S is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left ideal of S, but the converse is not necessarily true, in general, as shown by the following example:

Example 3.16. Let S = {a, b, c, d} be an ordered semigroup with respect to the order relation b ≤ a, c ≤ a, c ≤ b and the operation ^′ . ′ defined by the following Cayley table: . a b c d a a a a a b a a a a c a a b a d a a b b

Define f on S by f (a) =0.7, f (b) =0.8, f (c) =0.9 and f (d) =0.4. Then f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left ideal of S. But, f is not a fuzzy left ideal of S, since f (bb) = f (a) =0.7 < 0.8 = f (b).

Similar to the proof of Theorem 3.15, we may prove the following theorem:

Theorem 3.17. Let f be a fuzzy subset of an ordered semigroup S. Then f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy right ideal of S if and only if

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

Theorem 3.18. Let f be a fuzzy subset of S. Then f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left (resp. right)ideal of S if and only if U (f ; u) (≠ ∅) (u ∈ ( 0 , k * - k 2 ]) is a left (resp. right)ideal of S.

Definition 3.19. [35] An ordered semigroup S is called regular ordered semigroup if, for each a ∈ S, there exist an element x ∈ S such that a≤ axa ; or equivalently A ⊆ (ASA], ∀A ⊆ S.

Definition 3.20. [32] An ordered semigroup S is called left weakly regular ordered semigroup if, for each a ∈ S, there exist an element x, y ∈ S such that a≤ xaya ; or equivalently A ⊆ ((SA) ²], ∀A ⊆ S.

Proposition 3.21. Let f be a fuzzy subset of S. Then every (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of a regular ordered semigroup S is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy bi-ideal of S.

Proof. Let x, y ∈ S and f be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. Since S is regular ordered semigroup, there exist a ∈ S such that x ≤ xax. Then f ( xy ) ≥ f ( ( xax ) y ) = f ( x ( ax ) y ) ≥ min { f ( x ) , f ( y ) , k * - k 2 }, This implies that f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy subsemigroup of S. Hence, f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy bi-ideal of S.

Proposition 3.22. Let f be a fuzzy subset of S. Then every (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of a left weakly regular ordered semigroup S is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy bi-ideal of S.

Proof. Let x, y ∈ S and f be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. Since S is left weakly regular ordered semigroup, there exist a, b ∈ S such that y ≤ ayby. Then f ( xy ) ≥ f ( x ( ayby ) ) = f ( x ( ayb ) y ) ≥ min { f ( x ) , f ( y ) , k * - k 2 }. Therefore f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy subsemigroup of S. Hence, f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy bi-ideal of S.

4 (k^*, k)-lower parts of (∈ , ∈ ∨ (k^*, q _k ))- fuzzy generalized bi-ideals

In this section we investigate properties of (k^*, k)-lower parts of (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideals of ordered semigroup. Then, using properties of generalized bi-ideals, we characterize left simple, right simple, regular, completely regular and left weakly regular ordered semigroups.

Definition 4.1. Let f be a fuzzy subset of an ordered semigroup S. Then we define the (k^*, k)-lower part f k k * _ of f as follows: f k k * _ ( x ) = min { f ( x ) , k * - k 2 } for all x ∈ S and 0 ≤ k < k^* ≤ 1.

Clearly, f k k * _ is a fuzzy subset of S.

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

Definition 4.2. Let f and g be any fuzzy subsets of S. Define f ( ∩ ) k k * g, f ( ∪ ) k k * g and f ( ∘ ) k k * g as follows: ( f ( ∩ ) k k * g ) ( x ) = min { ( f ∩ g ) ( x ) , k * - k 2 } ( f ( ∪ ) k k * g ) ( x ) = min { ( f ∪ g ) ( x ) , k * - k 2 } ( f ( ∘ ) k k * g ) ( x ) = min { ( f ∘ g ) ( x ) , k * - k 2 } for all x ∈ S and 0 ≤ k < k^* ≤ 1.

Then, clearly, f ( ∩ ) k k * g, f ( ∪ ) k k * g and f ( ∘ ) k k * g are all fuzzy subsets of S.

The following Lemma easily follows:

Lemma 4.3. Let f and g be any fuzzy subsets of an ordered semigroup S. Then

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

Lemma 4.4. Let A and B be any non-empty subsets of an ordered semigroup S. Then

f A ( ∩ ) k k * f B = ( f k k * _ ) A ∩ B;

Proof. (1) f A ( ∩ ) k k * f B = ( f k k * _ ) A ∩ B for all x ∈ S. Indeed: If x ∈ A ∩ B, then ( f k k * _ ) A ∩ B ( x ) = k * - k 2. Since x ∈ A and x ∈ B, we have f _A  (x) = f _B  (x) =1, so ( f A ( ∩ ) k k * f B ) ( x ) = min { f A ( x ) , f B ( x ) , k * - k 2 } = min { 1 , 1 , k * - k 2 } = k * - k 2 .

If x ∉ A ∩ B, then ( f k k * _ ) A ∩ B ( x ) = 0. Suppose that x ∉ A. Then, by Lemma 4.2(2), ( f A ( ∩ ) k k * f B ) ( x ) = min { ( f k k * _ ) A ( x ) , ( f k k * _ ) B ( x ) } ≤ ( f k k * _ ) A ( x ) = 0. On the other hand, since f A ( ∩ ) k k * f B is a fuzzy subset of S, we have ( f A ( ∩ ) k k * f B ) ( x ) ≥ 0. Thus f A ( ∩ ) k k * f B = ( f k k * _ ) A ∩ B.

(2) The proof is similar to the proof of (1).

(3) Let x ∈ S. If x ∈ (AB], then ( f k k * _ ) ( AB ] ( x ) = k * - k 2, and x ≤ ab for some a ∈ A and b ∈ B. Thus (a, b) ∈ A _x , and ( f A ( ∘ ) k k * f B ) ( x ) = min { ( f A ∘ f B ) ( x ) , k * - k 2 } = ⋁ ( y , z ) ∈ A x min { f A ( y ) , f B ( z ) , k * - k 2 } ≤ min { f A ( a ) , f B ( b ) , k * - k 2 } = min { 1 , 1 , k * - k 2 } = k * - k 2 = ( f k k * _ ) ( AB ] ( x ) .

On the other hand, since (f _A  ∘ f _B ) (x) ≤1 for all x ∈ S, we have ( f A ( ∘ ) k k * f B ) ( x ) = min { ( f A ∘ f B ) ( x ) , k * - k 2 } ≤ k * - k 2 = ( f k k * _ ) ( AB ] ( x ). Therefore f A ( ∘ ) k k * f B = ( f k k * _ ) ( AB ]. Similarly we may prove for the case when x ∉ (AB]. Thus f A ( ∘ ) k k * f B = ( f k k * _ ) ( AB ]. Hence f A ( ∘ ) k k * f B = ( f k k * _ ) ( AB ].

Lemma 4.5. Let S be an ordered semigroup and B be any non-empty subset of S. Then the (k^*, k)-lower part ( f k k * _ ) B of the characteristic function f _B of B is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S if and only if B is a generalized bi-ideal of S.

Proof. Let B be a generalized bi-ideal of S. Then, by Theorems 2.7 and 3.12, ( f k k * _ ) B is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S.

Conversely, suppose that ( f k k * _ ) B is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. Let x, y ∈ S, x ≤ y. If y ∈ B, then ( f k k * _ ) B ( y ) = k * - k 2. Since ( f k k * _ ) B is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S, and x ≤ y, we have ( f k k * _ ) B ( x ) ≥ min { ( f k k * _ ) B ( y ) , k * - k 2 } = k * - k 2. It follows that ( f k k * _ ) B ( x ) = k * - k 2 and so x ∈ B. Let x, z ∈ B and y ∈ S. ( f k k * _ ) B ( x ) = k * - k 2 and ( f k k * _ ) B ( z ) = k * - k 2. Now, ( f k k * _ ) B ( xyz ) ≥ min { ( f k k * _ ) B ( x ) , ( f k k * _ ) B ( z ) , k * - k 2 } = k * - k 2 .

Hence ( f k k * _ ) B ( xyz ) = k * - k 2 and so xyz ∈ B. Therefore, B is a generalized bi-ideal of S.

Similarly, we may prove the following:

Lemma 4.6. Let S be an ordered semigroup and B be any non-empty subset of S. Then the (k^*, k)-lower part ( f k k * _ ) B of the characteristic function f _B of B is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy left (resp. right) ideal of S if and only if B is a left (resp. right)ideal of S.

Proposition 4.7. If f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S, then f k k * _ is a fuzzy generalized bi-ideal of S.

Proof. Let x, y ∈ S, x ≤ y. Since f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S and x ≤ y, we have f ( x ) ≥ min { f ( y ) , k * - k 2 }. It follows that min { f ( x ) , k * - k 2 } ≥ min { f ( y ) , k * - k 2 } and, hence, ( f k k * _ ) ( x ) ≥ ( f k k * _ ) ( y ). Let x, y, z ∈ S, we have f ( xyz ) ≥ min { f ( x ) , f ( y ) , k * - k 2 }. Then min { f ( xyz ) , k * - k 2 } ≥ min { f ( x ) , f ( z ) , k * - k 2 }, and so ( f k k * _ ) ( xyz ) ≥ min { ( f k k * _ ) ( x ) , ( f k k * _ ) ( z ) }. Therefore, f k k * _ is a fuzzy generalized bi-ideal of S.

Definition 4.8. [23] An ordered semigroup S is called left (resp. right) regular if, for each a ∈ S, there exist an element x ∈ S such that a ≤ xa²(resp. a ≤ a²x); or equivalently A ⊆ (SA²] (resp. (A²S]), ∀A ⊆ S.

Definition 4.9. [27] An ordered semigroup S is called left (resp. right) simple if for every left (resp. right) ideal A of S, A = S. Further S is called simple if it is both left and right simple.

Definition 4.10. An ordered semigroup S is called completely regular if it is left regular, right regular and regular.

Lemma 4.11. [32] An ordered semigroup S is left (resp. right) simple if and only if (Sa] = S (resp. (aS] = S) for every a ∈ S.

Theorem 4.12. If S is regular, left and right simple ordered semigroup, then ( f k k * _ ) ( a ) = ( f k k * _ ) ( b ), for each (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal f of S and for all a, b ∈ S.

Proof. Suppose that S is regular, left and right simple and f be an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S. Let E _S  = {e ∈ S | e ≤ e²}. Then E _S ≠ ∅, as for any a, x ∈ S, such that a ≤ axa, ax, xa ∈ E _S . Let b, c ∈ E _S . Since S is left and right simple, by Lemma 4.11, it follows that S = (Sb] = (bS]. Then c ≤ xb and c ≤ by for some x, y ∈ S. So c² = (by) (xb) ≤ b (yx) b. Since f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S, we have f ( c 2 ) ≥ min { f ( b ( yx ) b ) , k * - k 2 } = min { f ( b ) , k * - k 2 } .

Thus min { f ( c 2 ) , k * - k 2 } ≥ min { f ( b ) , k * - k 2 }. So, ( f k k * _ ) ( c 2 ) ≥ ( f k k * _ ) ( b ). (1) Since c ∈ E _S , we have c ≤ c². So, f (c) ≥ f (c²). This implies min { f ( c ) , k * - k 2 } ≥ min { f ( c 2 ) , k * - k 2 }. Therefore ( f k k * _ ) ( c ) ≥ ( f k k * _ ) ( c 2 ). Thus, by(1), we have ( f k k * _ ) ( c ) ≥ ( f k k * _ ) ( b ). On the other hand, by Lemma 4.11, we also have (Sc] = S = (cS]. So b ≤ sc and b ≤ ct for some s, t ∈ S. Thus by the argument similar to the above, we get ( f k k * _ ) ( b ) ≥ ( f k k * _ ) ( c ). Therefore ( f k k * _ ) ( b ) = ( f k k * _ ) ( c ). Hence f k k * _ is a constant on E _S .

Now, take any a ∈ S. Then there exist x ∈ S such that a ≤ axa and ax, xa ∈ E _S . So ( f k k * _ ) ( ax ) = ( f k k * _ ) ( b ) = ( f k k * _ ) ( xa ). Since (ax) a (xa) = (axa) xa ≥ axa ≥ a, we have f ( a ) ≥ min { f ( ( ax ) a ( xa ) ) , k * - k 2 } ≥ min { f ( ax ) , f ( xa ) ) , k * - k 2 } .

Thus ( f k k * _ ) ( a ) ≥ ( f k k * _ ) ( b ). Since S = (Sa] = (aS], we have b ≤ ua and b ≤ av for some u, v ∈ S. Then b² ≤ (av) (ua) = a (vu) a. So f ( b 2 ) ≥ min { f ( a ( vu ) a ) , k * - k 2 } ≥ min { f ( a ) , f ( a ) , k * - k 2 } = min { f ( a ) , k * - k 2 } .

Therefore, min { f ( b 2 ) , k * - k 2 } ≥ min { f ( a ) , k * - k 2 } and we have ( f k k * _ ) ( b 2 ) ≥ ( f k k * _ ) ( a ). Since b ∈ E _S , we have, b² ≥ b. Then f ( b ) ≥ min { f ( b 2 ) , k * - k 2 }. So min { f ( b ) , k * - k 2 } ≥ min { f ( b 2 ) , k * - k 2 }. Therefore, ( f k k * _ ) ( b ) ≥ ( f k k * _ ) ( b 2 ) and, so, ( f k k * _ ) ( b ) ≥ ( f k k * _ ) ( a ). Hence ( f k k * _ ) ( b ) = ( f k k * _ ) ( a ). Thus f k k * _ is a constant function on S.

Lemma 4.13. [32] An ordered semigroup S is completely regular if and only if A ⊆ (A²SA²], for every A ⊆ S.

Theorem 4.14. Let f be a fuzzy subset of an ordered semigroup S. Then S is completely regular if and only if ( f k k * _ ) ( a ) = ( f k k * _ ) ( a 2 ), for each (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal f of S and for each a ∈ S.

Proof. Let a ∈ S. Since S is completely regular, by Lemma 4.13, a ∈ (a²Sa²]. Then, there exist x ∈ S Such that a ≤ a²xa². Since f is an (∈ , ∈ ∨ (k^*, q _k ))-fuzzy generalized bi-ideal of S, f ( a ) ≥ min { f ( a 2 xa 2 ) , k * - k 2 } ≥ min { f ( a 2 ) , f ( a 2 ) , k * - k 2 } ≥ min { f ( a ) , f ( a ) , k * - k 2 } = min { f ( a ) , k * - k 2 } .