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IOS Press. 239. Characterizations of regular ordered semigroups by generalized fuzzy ideals. Jian Tanga,∗ and Xiangyun Xieb. aSchool of Mathematics and ...
239

Journal of Intelligent & Fuzzy Systems 26 (2014) 239–252 DOI:10.3233/IFS-120731 IOS Press

Characterizations of regular ordered semigroups by generalized fuzzy ideals Jian Tanga,∗ and Xiangyun Xieb a School b School

of Mathematics and Computational Science, Fuyang Normal College, Fuyang, Anhui, P.R. China of Mathematics and Computational Science, Wuyi University, Jiangmen, Guangdong, P.R. China

Abstract. Let S be an ordered semigroup. In this paper we first introduce the concepts of (∈, ∈ ∨qk )-fuzzy ideals, (∈, ∈ ∨qk )-fuzzy bi-ideals and (∈, ∈ ∨qk )-fuzzy generalized bi-ideals of an ordered semigroup S by the ordered fuzzy points of S, and investigate their related properties. Furthermore, characterizations of regular ordered semigroups by the properties of (∈, ∈ ∨qk )-fuzzy left ideals, (∈, ∈ ∨qk )-fuzzy right ideals and (∈, ∈ ∨qk )-fuzzy (generalized) bi-ideals are given. Keywords: Regular ordered semigroup, intra-regular ordered semigroup, strongly convex fuzzy subset, (∈, ∈ ∨qk )-fuzzy left (right) ideal, (∈, ∈ ∨qk )-fuzzy (generalized) bi-ideal

1. Introduction The theory of fuzzy sets, proposed by Zadeh in 1965, has provided a useful mathematical tool for describing the behavior of systems that are too complex or ill defined to admit precise mathematical analysis by classical methods and tools. Extensive applications of fuzzy set theory have been found in various fields such as computer science, artificial intelligence, expert systems, management science, operations research, pattern recognition, and others. It soon invoked a natural question concerning a possible connection between fuzzy sets and algebraic systems. Rosenfeld [18] inspired the fuzzification of algebraic structures and introduced the notion of fuzzy subgroups. Since then, Kuroki initiated the theory of fuzzy semigroups in his paper [15]. In [15–17], Kuroki introduced the concepts of fuzzy ideals and fuzzy bi-ideals and fuzzy generalized bi-ideals of semigroups and investigated their related properties. Following the works by Kuroki [15–17], the concept of (α, β)-fuzzy subgroups was introduced by Bhakat and Das [1, 2], based on the “belongs to” relation ∗ Corresponding author. Jian Tang, School of Mathematics and Computational Science, Fuyang Normal College, Fuyang, Anhui 236037, P.R. China. Tel.: +86 0558 2591133. E-mails: [email protected] (J. Tang); [email protected] (X.Y. Xie).

(∈) and “quasi-coincident with” relation (q) between a fuzzy point and a fuzzy subgroup. In particular, the concept of an (∈, ∈ ∨q)-fuzzy subgroup is a useful generalization of Rosenfeld’s fuzzy subgroup [18]. In [3] Davvaz defined (∈, ∈ ∨q)-fuzzy subnearring and ideals of a nearring. In [5] Jun and Song initiated the study of (α, β)-fuzzy interior ideals of a semigroup. Ma and Zhan [11] studied (∈, ∈ ∨q)-fuzzy h-bi-ideals and h-quasi-ideals of a hemiring (See also [30–32]). In [29] Zhan and Yin redefined fuzzy k-ideals and fuzzy k-interior ideals of hemirings and gave some characterizations of semiregular hemirings and semisimple hemirings. In [4] Davvaz and Khan studied (∈, ∈ ∨q)fuzzy generalized bi-ideals of an ordered semigroup and gave some characterizations of regular ordered semigroups. Generalizing the concept of the quasicoincident of a fuzzy point with a fuzzy subset Jun [6, 7] defined (∈, ∈ ∨qk )-fuzzy subgroups and (∈, ∈ ∨qk )-fuzzy subalgebras in BCK/BCI-algebras, respectively. In [8] (∈, ∈ ∨qk )-fuzzy ideals of hemirings are defined and investigated. Following the terminology given by Zadeh, fuzzy sets in an ordered semigroup S were first considered by Kehayopulu and Tsingelis in [22], then they defined “ fuzzy” analogous for several notations, which have proven useful in the theory of ordered semigroups.

1064-1246/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved

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J. Tang and X. Xie / Characterizations of regular ordered semigroups by generalized fuzzy ideals

A theory of fuzzy sets on ordered semigroups has been recently developed (see [10, 13, 24–27, 33, 34, 37–40]). The concept of ordered fuzzy points of an ordered semigroup was introduced by Xie and Tang [37], and prime fuzzy ideals of an ordered semigroup were studied in [39]. As we know, regular semigroups play an important role in the abstract algebra. In 1956, Is´eki studied some properties of regular semigroups, and proved that a commutative semigroup S is regular if and only if every ideal of S is idempotent [35]. In 1961, Calais proved that a semigroup S is regular if and only if the right and the left ideals of S are idempotent and for every right ideal A and every left ideal B of S, the product AB is a quasiideal of S [36]. Then Kehayopulu extended those results to ordered semigroups [19, 20]. Recently, Kehayopulu also extended those similar “fuzzy” results to ordered semigroups, and characterized the regular, the left regular and the right regular ordered semigroups by means of fuzzy left ideals, fuzzy right ideals, fuzzy semiprime subsets and fuzzy quasi-ideals (see [24–26]). In [38], the authors also characterized regular ordered semigroups and intra-regular ordered semigroups in terms of fuzzy left ideals, fuzzy right ideals, fuzzy (generalized) bi-ideals and fuzzy quasi-ideals. As a further study, the concepts of (∈, ∈ ∨qk )-fuzzy ideals, (∈, ∈ ∨qk )-fuzzy bi-ideals and (∈, ∈ ∨qk )fuzzy generalized bi-ideals in an ordered semigroup are introduced, and related properties are investigated. Furthermore, regular ordered semigroups are characterized by the properties of (∈, ∈ ∨qk )-fuzzy left ideals, (∈, ∈ ∨qk )-fuzzy right ideals and (∈, ∈ ∨qk )-fuzzy (generalized) bi-ideals. The paper illustrates that one can pass from the theory of semigroups or ordered semigroups to the theory of “fuzzy” ordered semigroups. As an application of the results of this paper, the corresponding results of semigroups (without order) are also obtained.

2. Preliminaries and some notations Throughout this paper, we denote by S an ordered semigroup, i.e., a semigroup S with an order relation “≤” such that a ≤ b implies xa ≤ xb and ax ≤ bx for any x ∈ S. A function f from S to the real closed interval [0, 1] is a fuzzy subset of S. The ordered semigroup S itself is a fuzzy subset of S such that S(x) ≡ 1 for all x ∈ S (the fuzzy subset S is also denoted by 1 in [16]). Let f and g be two fuzzy subsets of S. Then the inclusion relation f ⊆ g is defined by f (x) ≤ g(x) for all x ∈ S, and f ∩ g, 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) for all x ∈ S, respectively. The set of all fuzzy subsets of S is denoted by F (S). (F (S), ⊆, ∩, ∪) forms a complete lattice [26]. Let (S, ·, ≤) be an ordered semigroup. For x ∈ S, we define Ax := {(y, z) ∈ S × S| x ≤ yz}. The product f ◦ g of f and g is defined by ⎧  ⎨ [f (y) ∧ g(z)], if Ax = / ∅, (f ◦ g)(x) = (y,z)∈Ax ⎩ 0 , if Ax = ∅, for all x ∈ S. It is well known (see Theorem of [23]) that this operation “◦” is associative and (F (S), ◦, ⊆) is a poe-semigroup. Let S be an ordered semigroup. For H ⊆ S, we define (H] := {t ∈ S | t ≤ h for some h ∈ H}. For H = {a}, we write (a] instead of ({a}]. For two subsets A, B of S, we have: (1) A ⊆ (A]; (2) If A ⊆ B, then (A] ⊆ (B]; (3) (A](B] ⊆ (AB]; (4) ((A]] = (A]; (5) ((A](B]] = (AB] (cf. [21]). By a subsemigroup of S we mean a nonempty subset A of S such that A2 ⊆ A. A nonempty subset A of an ordered semigroup S is called a left (resp. right) ideal of S if (1) SA ⊆ A (resp. AS ⊆ A) and (2) If a ∈ A and S b ≤ a, then b ∈ A. If A is both a left and a right ideal of S, then it is called an (two-sided) ideal of S [21]. We denote by I(a) the (two-sided) ideal of S generated by a (a ∈ S). Then I(a) = (a ∪ Sa ∪ aS ∪ SaS] [21]. A subsemigroup B of an ordered semigroup S is called a bi-ideal of S if (1) BSB ⊆ B and (2) If a ∈ B and S b ≤ a, then b ∈ B [28]. A non-empty subset B of an ordered semigroup S is called a generalized bi-ideal of S if (1) BSB ⊆ B. (2) If a ∈ B and S b ≤ a, then b ∈ B [38]. Let A be a nonempty subset of S. We denote by fA the characteristic mapping of A, that is the mapping of S into [0, 1] defined by  1, if x ∈ A, fA (x) = 0, if x ∈ / A. Then fA is a fuzzy subset of S.

J. Tang and X. Xie / Characterizations of regular ordered semigroups by generalized fuzzy ideals

Let S be an ordered semigroup. A fuzzy subset f of S is called a fuzzy subsemigroup of S if f (xy) ≥ f (x) ∧ f (y) for all x, y ∈ S [22]. A fuzzy subset f of S is called a fuzzy left (resp. right) ideal of S if (1) x ≤ y ⇒ f (x) ≥ f (y), and (2) f (xy) ≥ f (y) (resp. f (xy) ≥ f (x)) ∀x, y ∈ S. 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 a fuzzy right ideal of S [22, 25]. A fuzzy subsemigroup f of S is called a fuzzy bi-ideal of S if (1) x ≤ y ⇒ f (x) ≥ f (y), and (2) f (xyz) ≥ min{f (x), f (z)} = f (x) ∧ f (z), ∀x, y, z ∈ S [24]. We denote by aλ an ordered fuzzy point of an ordered semigroup S, where  λ, if x ∈ (a], aλ (x) = 0, if x ∈ / (a]. It is easy to see that an ordered fuzzy point of an ordered semigroup S is a fuzzy subset of S. For any fuzzy subset f of S, we also denote aλ ⊆ f by aλ ∈ f in the sequel [37]. Definition 2.1. (cf. [39, Definition 3]). Let f be a fuzzy subset of S, we define (f ] by the rule that  (f ](x) = f (y) y≥x

for all x ∈ S. A fuzzy subset of S is called strongly convex if f = (f ]. By Definition 2.1, for an ordered fuzzy point aλ and a strongly convex fuzzy subset f of S we have aλ ∈ f if and only if f (a) ≥ λ. Lemma 2.2. (cf. [39, Theorem 2]). Let f be a strongly convex fuzzy subset of an ordered semigroup S. Then  f = ys . ys ∈f

Lemma 2.3. [39]. Let f be a fuzzy subset of an ordered semigroup S. Then f is a strongly convex fuzzy subset of S if and only if x ≤ y ⇒ f (x) ≥ f (y), for all x, y ∈ S. Definition 2.4. An ordered fuzzy point aλ of an ordered semigroup S is said to be quasi-coincident with a fuzzy subset f of S, written as aλ qf if f (a) + λ > 1. If aλ ∈ f or aλ qf , then we write aλ ∈ ∨qf. Lemma 2.5. [37]. Let aλ , bµ (λ = / 0, µ = / 0) be ordered fuzzy points of S, and f, g fuzzy subsets of S. Then the following statements are true:

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(1) aλ ◦ bµ = (ab)λ∧µ for all ordered fuzzy points aλ and bµ of S. In particular, aλ ◦ aλ = (a2 )λ . (2) If f, g are fuzzy ideals of S, then f ◦ g, f ∪ g are fuzzy ideals of S. (3) If f ⊆ g, and h ∈ F (S), then f ◦ h ⊆ g ◦ h, h ◦ f ⊆ h ◦ g. The reader is referred to [37,41] for notation and terminology not defined in this paper. 3. (∈, ∈ ∨qk )-fuzzy (generalized) bi-ideals of ordered semigroups In what follows, let S denote an ordered semigroup and k an arbitrary element of [0, 1) unless otherwise specified. Generalizing the concept of aλ qf, we defined aλ qk f, where k ∈ [0, 1) as aλ qk f if f (a) + λ + k > 1. If aλ ∈ f or aλ qk f , then we write aλ ∈ ∨qk f. The symbol ∈ ∨qk means that ∈ ∨qk does not hold. In this section, we study mainly the (∈, ∈ ∨qk )-fuzzy biideals and (∈, ∈ ∨qk )-fuzzy generalized bi-ideals of an ordered semigroup S by the ordered fuzzy points of S, respectively. Definition 3.1. [40]. A fuzzy subset f of an ordered semigroup S is called an (∈, ∈ ∨qk )-fuzzy subsemigroup of S if xt ∈ f and yr ∈ f imply (xy)t∧r ∈ ∨qk f for all t, r ∈ (0, 1] and x, y ∈ S. Definition 3.2. [40]. A fuzzy subset f of an ordered semigroup S is called an (∈, ∈ ∨qk )-fuzzy left (resp. right) ideal of S, if for all t ∈ (0, 1] and x, y ∈ S, the following conditions hold: (1) x ≤ y ⇒ f (x) ≥ f (y), and (2) yt ∈ f, x ∈ S ⇒ (xy)t ∈ ∨qk f ∨qk f ).

(resp.

(yx)t ∈

A fuzzy subset f of S is called an (∈, ∈ ∨qk )-fuzzy ideal if it is both an (∈, ∈ ∨qk )-fuzzy left ideal and an (∈, ∈ ∨qk )-fuzzy right ideal of S. Lemma 3.3. [40]. Let f be a strongly convex fuzzy subset of an ordered semigroup S. Then f is an (∈, ∈ ∨qk )-fuzzy subsemigroup of S if and only if f (xy) ≥ f (x) ∧ f (y) ∧

1−k 2

for all x, y ∈ S. Lemma 3.4. [40]. Let S be an ordered semigroup and f a fuzzy subset of S. Then f is an (∈, ∈ ∨qk )-fuzzy left (resp. right) ideal of S if and only if f satisfies

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J. Tang and X. Xie / Characterizations of regular ordered semigroups by generalized fuzzy ideals

(1) x ≤ y ⇒ f (x) ≥ f (y),  and (2) f (xy) ≥ f (y) ∧ 1−k resp. f (xy) ≥ f (x) ∧ 2 for all x, y ∈ S.

1−k 2



Definition 3.5. A fuzzy subset f of an ordered semigroup S is called an (∈, ∈ ∨qk )-fuzzy bi-ideal of S, if for all t, r ∈ (0, 1] and x, y, z ∈ S, the following conditions hold: (1) x ≤ y ⇒ f (x) ≥ f (y). (2) xt ∈ f, yr ∈ f ⇒ (xy)t∧r ∈ ∨qk f. (3) xt ∈ f, zr ∈ f and y ∈ S ⇒ (xyz)t∧r ∈ ∨qk f. Clearly, every (∈, ∈ ∨qk )-fuzzy bi-ideal of an ordered semigroup S is an (∈,∈ ∨qk )-fuzzy subsemigroup of S. If we take k = 0 in Definition 3.5, then we have the concept of (∈, ∈ ∨q)-fuzzy bi-ideals of S given in [9]. So the concept of (∈, ∈ ∨qk )-fuzzy bi-ideals is a generalization of (∈, ∈ ∨q)-fuzzy bi-ideals in an ordered semigroup S. Theorem 3.6. Let S be an ordered semigroup and f a fuzzy subset of S. Then f is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S if and only if f satisfies that (1) x ≤ y ⇒ f (x) ≥ f (y). (2) f (xy) ≥ f (x) ∧ f (y) ∧ 1−k 2 (3) f (xyz) ≥ f (x) ∧ f (z) ∧ 1−k 2

for all x, y ∈ S. for all x, y, z ∈ S.

Proof. Let f be an (∈, ∈ ∨qk )-fuzzy bi-ideal of S and x, y, z ∈ S. Then, by Lemma 3.3 and Definition 3.5, the conditions (1) and (2) hold. Furthermore, f (xyz) ≥ f (x) ∧ f (z) ∧ 1−k 2 . Indeed, if f (xyz) < f (x) ∧ f (z) ∧ 1−k for some x, y, z ∈ S, then there exists t ∈ (0, 1] 2 such that f (xyz) < t < f (x) ∧ f (z) ∧ 1−k 2 . Then xt ∈ f, zt ∈ f and (xyz)t ∈ ∨qk f, which is a contradiction. Hence f (xyz) ≥ f (x) ∧ f (z) ∧ 1−k 2 . Conversely, assume that the conditions (1), (2) and (3) hold. Let x, y ∈ S and t, r ∈ (0, 1] be such that xt ∈ f, yr ∈ f. Then, by Definition 3.1 and Lemma 3.3, (xy)t∧r ∈ ∨qk f. Furthermore, let x, y, z ∈ S and t, r ∈ (0, 1] be such that xt ∈ f, zr ∈ f. Then f (x) ≥ t, f (z) ≥ r and f (xyz) ≥ f (x) ∧ f (z) ∧

1−k 1−k ≥t∧r∧ . 2 2

then f (xyz) ≥ 1−k If t ∧ r > 2 , which implies that f (xyz) + t ∧ r + k > 1. If t ∧ r ≤ 1−k 2 , then f (xyz) ≥ t ∧ r. Therefore, (xyz)t∧r ∈ ∨qk f. 1−k 2 ,

One can easily observe that every fuzzy bi-ideal of an ordered semigroup S is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S, but not conversely. We can illustrate it by the following example:

Example 3.7. We consider the ordered semigroup S := {a, b, c, d, e} defined by the following multiplication “·” and the order “≤ ”: · a b c d e

a a a a a a

b a a a a a

c a a c c c

d a a c d c

e a a e e e

≤ := {(a, a), (a, c), (a, e), (b, b), (b, c), (b, e), (c, c), (c, e), (d, c), (d, e), (d, d), (e, e)}. Let f be a fuzzy subset of S such that f (a) = f (b) = f (d) = 0.8, f (c) = 0.7, f (e) = 0.6. Then we can easily show that f is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S by Theorem 3.6. But f is not a fuzzy bi-ideal of S, since f (dcd) = f (c) = 0.7 < 0.8 = min{f (d), f (d)}. Theorem 3.8. Let {fi | i ∈ I} be a family of (∈, ∈ ∨qk )-fuzzy bi-ideals of an ordered semigroup S. Then f := fi is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S, where i∈I

( fi )(x) = (fi (x)). i∈I

i∈I

Proof. Let x, y, z ∈ S. Then, since each fi (i ∈ I) is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S, we have

 (fi (xy)) f (xy) = fi (xy) = i∈I





i∈I

fi (x) ∧ fi (y) ∧

i∈I

 =





(fi (x) ∧ fi (y)) ∧

i∈I





=

1−k 2

fi

(x) ∧



i∈I



1−k 2

 (y) ∧

fi

i∈I

1−k 2

1−k = f (x) ∧ f (y) ∧ , 2 and

 f (xyz) = fi (xyz) = (fi (xyz)) i∈I





i∈I

fi (x) ∧ fi (z) ∧

i∈I

 =





(fi (x) ∧ fi (z)) ∧

i∈I

 =

1−k 2

i∈I

fi

(x) ∧

i∈I

1−k = f (x) ∧ f (z) ∧ . 2



1−k 2



fi

(z) ∧

1−k 2

J. Tang and X. Xie / Characterizations of regular ordered semigroups by generalized fuzzy ideals

Furthermore, if x ≤ y, then f (x) ≥ f (y). Indeed, since each fi (i ∈ I) is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S, we have fi (x) ≥ fi (y) for all i ∈ I. Thus

 fi (x) = (fi (x)) f (x) = i∈I





i∈I

(fi (y)) = fi (y) = f (y).

i∈I

 is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S, where ( fi )(x) = i∈I  (fi (x)). i∈I

Proof. (1) For all x, y ∈ S, we have

  f (xy) = fi (xy) = (fi (xy)) i∈I

i∈I



Therefore, f is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S by Theorem 3.6. Suppose {fi | i ∈ I} is a family of (∈, ∈ ∨qk )fuzzy bi-ideals of an ordered semigroup S. Is it true  that fi is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S? where i∈I   ( fi )(x) = (fi (x)). The following example gives a i∈I

i∈I

negative answer to the above question. Example 3.9. We consider the ordered semigroup S := {a, b, c, d} defined by the following multiplication “·” and the order “≤”: · a b c d

a a a a a

b a a a a

c a d a a

d a a a a

≤:= {(a, a), (a, d), (b, b), (c, c), (d, d)}. Let f1 and f2 be two fuzzy subsets of S such that f1 (a) = 0.4, f1 (b) = 0.4, f1 (c) = 0, f1 (d) = 0; f2 (a) = 0.4, f2 (b) = 0, f2 (c) = 0.4, f2 (d) = 0. Then f1 and f2 are both (∈, ∈ ∨qk )-fuzzy bi-ideals of S for any k ∈ [0, 1), but f1 ∪ f2 is not an (∈, ∈ ∨qk )fuzzy bi-ideal of S, since (f1 ∪ f2 )(bc) = (f1 ∪ f2 )(d) = f1 (d) ∨ f2 (d) = 0 < 0.4 ∧ 0.4 ∧

1−k 2



i∈I

fi (x) ∧ fi (y) ∧

i∈I

Theorem 3.10. Let {fi | i ∈ I} be a family of (∈, ∈ ∨qk )-fuzzy bi-ideals of an ordered semigroup S such  that fi ⊆ fj or fj ⊆ fi for all i, j ∈ I. Then f := fi i∈I

1−k 2



(Since fi is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S)



  1−k = fi (x) ∧ fi (y) ∧ (∗) 2 i∈I

i∈I

1−k = f (x) ∧ f (y) ∧ . 2 In the following we show that Equation (∗) holds. It   is obvious that (fi (x) ∧ fi (y) ∧ 1−k fi )(x)∧ 2 )≤( i∈I i∈I   ( fi )(y) ∧ 1−k (fi (x) ∧ fi (y) ∧ 2 . Assume that i∈I i∈I   1−k / ( fi )(x) ∧ ( fi )(y) ∧ 1−k Then there 2 )= 2 . i∈I i∈I  exists r such that (fi (x) ∧ fi (y) ∧ 1−k 2 ) 1, and f (y) ≥ t or f (y) + t + k > 1. Since f is an (∈, ∈ ∨qk )fuzzy bi-ideal of S, we have   1−k . f (xy) ≥ min f (x), f (y), 2 Case 1. Let f (x) ≥ t and f (y) ≥ t. If t > 1−k 2 , then 1−k 1−k 1−k f (xy) ≥ f (x) ∧ f (y) ∧ 2 ≥ t ∧ t ∧ 2 = 2 , and thus (xy)t qk f. If t ≤ 1−k 2 , then f (xy) ≥ f (x) ∧ f (y) ∧ 1−k 1−k 2 ≥ t ∧ t ∧ 2 = t, and so (xy)t ∈ f. Hence (xy)t ∈ ∨qk f. Case 2. Let f (x) ≥ t and f (y) + t + k > 1. If t > 1−k 2 , then 1−k 1−k = f (y) ∧ 2 2 1−k > (1 − t − k) ∧ = 1 − t − k, 2

f (xy) ≥ f (x) ∧ f (y) ∧

i.e., f (xy) + t + k > 1, and thus (xy)t qk f. If t ≤ then 1−k f (xy) ≥ f (x) ∧ f (y) ∧ 2 1−k ≥ t ∧ (1 − t − k) ∧ = t, 2 and so (xy)t ∈ f. Hence (xy)t ∈ ∨qk f.

1−k 2 ,

Case 3. Let f (x) + t + k > 1 and f (y) ≥ t. Similar to the proof of Case 2, we have (xy)t ∈ ∨qk f. Case 4. Let f (x) + t + k > 1 and f (y) + t + k > 1. If t > 1−k 2 , then 1−k 2 1−k > (1 − t − k) ∧ = 1 − t − k, 2

f (xy) ≥ f (x) ∧ f (y) ∧

i.e., f (xy) + t + k > 1, and thus (xy)t qk f. If t ≤ then

1−k 2 ,

1−k 2 1−k 1−k = ≥ (1 − t − k) ∧ ≥ t, 2 2

f (xy) ≥ f (x) ∧ f (y) ∧

and so (xy)t ∈ f. Hence (xy)t ∈ ∨qk f. Thus, in any case, we have (xy)t ∈ ∨qk f, and so xy ∈ [f ]t . Now let x, z ∈ [f ]t for t ∈ (0, 1] and y ∈ S. Then xt ∈ ∨qk f and zt ∈ ∨qk f, that is, f (x) ≥ t or f (x) + t + k > 1, and f (z) ≥ t or f (z) + t + k > 1. Since f is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S, we have f (xyz) ≥ min{f (x), f (z), 1−k 2 }. In a similar way as in the previous proof process of this theorem, we can show that (xyz)t ∈ ∨qk f, and so xyz ∈ [f ]t . Furthermore, let x ∈ [f ]t , S y ≤ x. Then y ∈ [f ]t . Indeed, since x ∈ [f ]t , we have xt ∈ ∨qk f, that is f (x) ≥ t or f (x) + t + k > 1. Then, by Theorem 3.6, we have f (y) ≥ f (x) ≥ t or f (y) ≥ f (x) ≥ 1 − t − k, that is, yt ∈ ∨qk f, and so y ∈ [f ]t . Therefore, [f ]t is a bi-ideal of S. Conversely, let f be a strongly convex fuzzy subset of S and t ∈ (0, 1] be such that [f ]t is a bi-ideal of S. If possible, let f (xy) < f (x) ∧ f (y) ∧ 1−k 2 for 1−k some x, y ∈ S. Then we can take t ∈ (0, 2 ] such that f (xy) < t < f (x) ∧ f (y) ∧ 1−k 2 . Then, since f is strongly convex, x, y ∈ [f ]t , which implies that xy ∈ [f ]t . Hence f (xy) ≥ t or f (xy) + t + k > 1, which is impossible. Therefore, f (xy) ≥ f (x) ∧ f (y) ∧ 1−k 2 for all x, y ∈ S. In a similar way we can show that f (xyz) ≥ f (x) ∧ f (z) ∧ 1−k 2 for all x, y, z ∈ S. Using Lemma 2.3 and Theorem 3.6, we conclude that f is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S. Definition 3.13. A fuzzy subset f of an ordered semigroup S is called an (∈, ∈ ∨qk )-fuzzy generalized bi-ideal of S, if for all t, r ∈ (0, 1] and x, y, z ∈ S, the following conditions hold: (1) x ≤ y ⇒ f (x) ≥ f (y). (2) xt ∈ f, zr ∈ f and y ∈ S ⇒ (xyz)t∧r ∈ ∨qk f.

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245

The following Theorem 3.14 can be proved in a similar way as in the proof of Theorem 3.6.

4. k-lower parts of (∈, ∈ ∨qk )-fuzzy ideals of ordered semigroups

Theorem 3.14. Let S be an ordered semigroup and f a fuzzy subset of S. Then f is an (∈, ∈ ∨qk )-fuzzy generalized bi-ideal of S if and only if f satisfies that

In this section we investigate mainly the properties of k-lower parts of (∈, ∈ ∨qk )-fuzzy ideals of S.

(1) x ≤ y ⇒ f (x) ≥ f (y). (2) f (xyz) ≥ f (x) ∧ f (z) ∧

1−k 2

for all x, y, z ∈ S.

Theorem 3.15. Let S be an ordered semigroup and f a strongly convex fuzzy subset of S. Then f is an (∈, ∈ ∨qk )-fuzzy generalized bi-ideal of S if and only if the (∈ ∨qk )-level subset [f ]t of f is a generalized bi-ideal of S for all t ∈ (0, 1]. Proof. The proof is similar to that of Theorem 3.12, we omit it. One can easily observe that every (∈, ∈ ∨qk )-fuzzy bi-ideal of an ordered semigroup S is an (∈, ∈ ∨qk )fuzzy generalized bi-ideal of S, but not conversely. We can illustrate it by the following example: Example 3.16. We consider the ordered semigroup S := {a, b, c, d} defined by the following multiplication “·” and the order “≤”: · a b c d

a a a a a

b a a a a

c a a b b

d a a a b

≤:= {(a, a), (a, b), (a, c), (a, d), (b, b), (c, c), (d, d)}. Let f be a fuzzy subset of S such that f (a) = 0.6, f (b) = 0, f (c) = 0.3, f (d) = 0. Then, by Theorem 3.14, we can easily show that f is an (∈ , ∈ ∨qk )-fuzzy generalized bi-ideal of S for any k ∈ [0, 1). But f is not an (∈ ∨qk )-fuzzy bi-ideal of S, since f (cc) = f (b) = 0 < min{0.3, 0.3, 1−k 2 }= 1−k min{f (c), f (c), 2 }, that is f is not an (∈, ∈ ∨qk )fuzzy subsemigroup of S. An ordered semigroup (S, ·, ≤) is called regular if, for each element a of S, there exists an element x in S such that a ≤ axa (cf. [20]). Equivalent definition: (1) A ⊆ (ASA], ∀A ⊆ S. (2) a ∈ (aSa], ∀a ∈ S. Theorem 3.17. Every (∈, ∈ ∨qk )-fuzzy generalized biideal of a regular ordered semigroup S is an (∈, ∈ ∨qk )fuzzy bi-ideal of S. Proof. The proof is similar to that of Lemma 8 in [12], we omit it.

Definition 4.1. Let f be a fuzzy subset of an ordered semigroup S. Then we define the k-lower part fk of f as follows: fk (x) = f (x) ∧ 1−k 2 for all x ∈ S. Clearly, fk is still a fuzzy subset of S. Furthermore, let f, g be fuzzy subsets of S. We define the fuzzy subsets f ∩k g and f ◦k g of S as follows: 1−k , 2 1−k (f ◦k g)(x) = (f ◦ g)(x) ∧ 2 for all x ∈ S. (f ∩k g)(x) = (f ∩ g)(x) ∧

Lemma 4.2. Let f and g be fuzzy subsets of an ordered semigroup S. Then the following statements are true: (1) (fk )k = fk , fk ⊆ f. (2) If f ⊆ g, and h ∈ F (S), then f ◦k h ⊆ g ◦k h, h ◦k f ⊆ h ◦k g. (3) f ∩k g = fk ∩ gk . (4) f ◦k g = fk ◦ gk . (5) f ◦k S = fk ◦ S, S ◦k f = S ◦ fk , f ◦k S ◦k f = fk ◦ S ◦ fk and S ◦k f ◦k S = S ◦ fk ◦ S. Proof. The proof is straightforward, we omit it. In Example 3.7 of [40], the authors have illustrated that an (∈, ∈ ∨qk )-fuzzy ideal of an ordered semigroup S is not necessarily a fuzzy ideal of S. In the following proposition, we show that if f is an (∈, ∈ ∨qk )-fuzzy ideal of S, then the k-lower part fk of f is a fuzzy ideal of S. Proposition 4.3. If f is an (∈, ∈ ∨qk )-fuzzy ideal of an ordered semigroup S, then the k-lower part fk of f is a fuzzy ideal of S. Proof. Let f be an (∈, ∈ ∨qk )-fuzzy ideal of S and x, y ∈ S. Then, f (xy) ≥ f (x) ∧ 1−k 2 and we have 1−k fk (xy) = f (xy) ∧ 1−k ≥ f (x) ∧ = fk (x). More2 2 over, if x ≤ y, then fk (x) ≥ fk (y). Indeed, since f is an (∈, ∈ ∨qk )-fuzzy ideal of S, we have f (x) ≥ f (y), and 1−k so fk (x) = f (x) ∧ 1−k 2 ≥ f (y) ∧ 2 = fk (y). Therefore, fk is a fuzzy right ideal of S. In a similar way we can show that fk is also a fuzzy left ideal of S, and so fk is a fuzzy ideal of S.

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Lemma 4.4. Let A, B be any nonempty subsets of an ordered semigroup S. Then the following statements are true: (1) fA ∩k fB = (fA∩B )k . (2) fA ◦k fB = (f(AB] )k . (3) S ◦k fA = (f(SA] )k , fA ◦k S = (f(AS] )k , fA ◦k S◦k fA = (f(ASA] )k and S ◦k fA ◦k S = (f(SAS] )k . Proof. The proof is similar to that of Lemma 3.8 in [37] with a slight modification, we omit it. Lemma 4.5 [14]. Let (S, ·) be a semigroup and ∅ = / A⊆ S. Then A is a left (right) ideal of S if and only if the characteristic function fA of A is an (∈, ∈ ∨qk )-fuzzy left (right) ideal of S. Theorem 4.6. Let (S, ·, ≤) be an ordered semigroup and ∅ = / A ⊆ S. Then A is a left (right) ideal of S if and only if the characteristic function fA of A is an (∈, ∈ ∨qk )-fuzzy left (right) ideal of S. Proof. (=⇒) Let x, y ∈ S, x ≤ y. Then fA (x) ≥ fA (y). Indeed, if y ∈ A, then fA (y) = 1. Since S x ≤ y ∈ A, we have x ∈ A, then fA (x) = 1. Thus fA (x) ≥ fA (y). If y ∈/ A, then fA (y) = 0. Since x ∈ S, we have fA (x) ≥ 0. Thus fA (y) = 0 ≤ fA (x). (⇐=) Let x ∈ A, S y ≤ x. Then y ∈ A. Indeed: It is enough to prove that fA (y) = 1. Since x ∈ A, fA (x) = 1. Since fA is an (∈, ∈ ∨qk )-fuzzy left (right) ideal of S and y ≤ x, we have fA (y) ≥ fA (x) = 1. Since y ∈ S, we have fA (y) ≤ 1. The rest of the proof is a consequence of Lemma 4.5.

(⇐=) Assume that fA is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S. Then for every x, y ∈ A, we have 1−k 2 1−k 1−k = = 1∧1∧ > 0, 2 2 and so fA (xy) = 1, i.e., xy ∈ A. Similarly, for every x, z ∈ A, y ∈ S, we can prove that xyz ∈ A. The rest of the proof is similar to the proof process of Theorem 4.6, we omit it. fA (xy) ≥ fA (x) ∧ fA (y) ∧

Theorem 4.9. Let (S, ·, ≤) be an ordered semigroup and ∅ = / A ⊆ S. Then A is a generalized bi-ideal of S if and only if the characteristic function fA of A is an (∈, ∈ ∨qk )-fuzzy generalized bi-ideal of S. Proof. The proof is similar to that of Theorem 4.8. Lemma 4.10. Let (S, ·, ≤) be an ordered semigroup and f a strongly convex fuzzy subset of S. Then f is an (∈, ∈ ∨qk )-fuzzy subsemigroup of S if and only if f ◦k f ⊆ fk . Proof. Let f be an (∈, ∈ ∨qk )-fuzzy subsemigroup of S and x ∈ S. If Ax = ∅, then (f ◦k f )(x) = (fk ◦ fk )(x) = 0 ≤ fk (x). If Ax = / ∅, then we have (f ◦k f )(x) = (f ◦ f )(x) ∧



=

[f (y) ∧ f (z) ∧

(y,z)∈Ax





f (yz) ∧

(y,z)∈Ax





Proof. It is obvious by Theorem 4.6.

Proof. (=⇒) Let A be a bi-ideal of S and x, y ∈ S. Then fA (xy) ≥ fA (x) ∧ fA (y) ∧ 1−k 2 . Indeed: If x ∈ A and y ∈ A, then, since A is a bi-ideal of S, xy ∈ A2 ⊆ A, and we have fA (xy) = 1 ≥ fA (x) ∧ fA (y) ∧ 1−k 2 . If 1−k x∈ / A or y ∈ / A, then fA (x) ∧ fA (y) ∧ 2 = 0. Since xy ∈ S, we have fA (xy) ≥ 0. Thus fA (xy) ≥ fA (x) ∧ fA (y) ∧ 1−k 2 . Now, let x, y, z ∈ S. Then, in the above way, we have fA (xyz) ≥ fA (x) ∧ fA (z) ∧ 1−k 2 .

[f (y) ∧ f (z)] ∧

(y,z)∈Ax

Theorem 4.7. Let (S, ·, ≤) be an ordered semigroup and ∅= / A ⊆ S. Then A is an ideal of S if and only if the characteristic function fA of A is an (∈, ∈ ∨qk )-fuzzy ideal of S.

Theorem 4.8. Let (S, ·, ≤) be an ordered semigroup and ∅= / A ⊆ S. Then A is a bi-ideal of S if and only if the characteristic function fA of A is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S.



=

1−k 2

f (x) ∧

(y,z)∈Ax

1−k 2 1−k 1−k ]∧ 2 2

1−k 2

1−k (since x ≤ yz) 2

1−k = fk (x). 2 It thus implies that f ◦k f ⊆ fk . Conversely, if f ◦k f ⊆ fk , then for all x, y ∈ S = f (x) ∧

f (xy) ≥ fk (xy) ≥ (f ◦k f )(xy) = (fk ◦ fk )(xy)  [fk (p) ∧ fk (q)] ≥ fk (x) ∧ fk (y) = (p,q)∈Axy



1−k f (x) ∧ 2





1−k = ∧ f (y) ∧ 2 1−k = f (x) ∧ f (y) ∧ . 2



J. Tang and X. Xie / Characterizations of regular ordered semigroups by generalized fuzzy ideals

Therefore, f is an (∈, ∈ ∨qk )-fuzzy subsemigroup of S by Lemma 3.3. Lemma 4.11. Let (S, ·, ≤) be an ordered semigroup and f a fuzzy subset of S. Then f is an (∈, ∈ ∨qk )-fuzzy left ideal of S if and only if f satisfies that (1) x ≤ y ⇒ f (x) ≥ f (y), for all x, y ∈ S. (2) S ◦k f ⊆ fk .

=

 (y,z)∈Ax



1−k 2

1−k 2 (y,z)∈Ax    1−k 1−k = f (z) ∧ ∧ 2 2 =

[1 ∧ f (z)] ∧

(y,z)∈Ax



 (y,z)∈Ax



 (y,z)∈Ax

f (yz) ∧

Lemma 4.13. Let S be an ordered semigroup and f a fuzzy subset of S. Then f is an (∈, ∈ ∨qk )-fuzzy ideal of S if and only if f satisfies that

Proposition 4.14. If f is an (∈, ∈ ∨qk )-fuzzy generalized bi-ideal of an ordered semigroup S, then f ◦k S ◦k f ⊆ fk . Proof. Let f be an (∈, ∈ ∨qk )-fuzzy generalized bi-ideal of S. Then (f ◦k S ◦k f )(a) ≤ fk (a) for all a ∈ S. Indeed, since fk is a fuzzy subset of S, we have fk (a) ≥ 0 for all a ∈ S. If (f ◦k S ◦k f )(a) = 0, then (f ◦k S ◦k f )(a) ≤ fk (a). If (f ◦k S ◦k f )(a) = / 0, then, by Lemma 4.2(5), (fk ◦ S ◦ fk )(a) = (f ◦k S ◦k f )(a) = / 0, and there exist x, y, p, q ∈ S such that (x, y) ∈ Aa and (p, q) ∈ Ax , i.e., a ≤ xy and x ≤ pq. Since f is an (∈ ∨qk )-fuzzy generalized bi-ideal of S, we have f (pqy) ≥ f (p) ∧ f (y) ∧ 1−k 2 . Thus, by Lemma 4.2(5),

1−k 2

[S(y) ∧ f (z)] ∧

(1) x ≤ y ⇒ f (x) ≥ f (y), for all x, y ∈ S. (2) f ◦k S ⊆ fk .

(1) x ≤ y ⇒ f (x) ≥ f (y), for all x, y ∈ S. (2) S ◦k f ⊆ fk , f ◦k S ⊆ fk .

Proof. (=⇒) Let f be an (∈, ∈ ∨qk )-fuzzy left ideal of S and x ∈ S. Then (S ◦k f )(x) ≤ fk (x). Indeed, if Ax = ∅, then (S ◦k f )(x) = (S ◦ fk )(x) = 0 ≤ fk (x). If Ax = / ∅, then, by Lemma 3.4, (S ◦k f )(x) = (S ◦ f )(x) ∧

247

1−k 2

(f ◦k S ◦k f )(a)

1−k f (x) ∧ (since x ≤ yz) 2

1−k = f (x) ∧ = fk (x). 2

= (fk ◦ S ◦ fk )(a) =

=

(⇐=) Assume that (2) holds. Then f (xy) ≥ f (x) ∧ for all x, y ∈ S. Indeed, let x, y ∈ S. Put a = xy. Then, by S ◦k f ⊆ fk , 1−k 2

f (xy) = f (a) ≥ fk (a) ≥ (S ◦k f )(a) = (S ◦ fk )(a)  = [S(b) ∧ fk (c)] (b,c)∈Aa

≥ S(x) ∧ fk (y) = 1 ∧ fk (y) 1−k = fk (y) = f (y) ∧ . 2 By hypothesis (1) and Lemma 3.4, f is an (∈, ∈ ∨qk )fuzzy left ideal of S. Similar to Lemma 4.11, we have the following two lemmas. The proofs are similar to that of Lemma 4.11. Lemma 4.12. Let S be an ordered semigroup and f a fuzzy subset of S. Then f is an (∈, ∈ ∨qk )-fuzzy right ideal of S if and only if f satisfies that

 (x,y)∈Aa

=





[(fk ◦ S)(x) ∧ fk (y)]

(x,y)∈Aa

⎧ ⎨ 

[fk (p) ∧ S(q)] ∧ fk (y)



(p,q)∈Ax



(x,y)∈Aa (p,q)∈Ax

⎫ ⎬ ⎭

  1−k f (p) ∧ 2

 1−k ∧ f (y) ∧ 2     1−k 1−k = f (p) ∧ f (y) ∧ ∧ 2 2 

(x,y)∈Aa (p,q)∈Ax







f (pqy) ∧

(x,y)∈Aa (p,q)∈Ax



 (x,y)∈Aa

≤ f (a) ∧

f (xy) ∧

1−k 2

1−k (Since xy ≤ pqy) 2

1−k = fk (a) (Since a ≤ xy), 2

which means that f ◦k S ◦k f ⊆ fk .

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In a similar way as in the previous proposition, by Lemma 4.10, we can show the following proposition:

(f ◦k g)(y) = (fk ◦ gk )(y) =

Proposition 4.15. If f is an (∈, ∈ ∨qk )-fuzzy bi-ideal of an ordered semigroup S, then f ◦k f ⊆ fk and f ◦k S ◦k f ⊆ fk . Lemma 4.16. Let f be a fuzzy subset of an ordered semigroup S and f ◦k S ◦k f ⊆ fk . Then f (xyz) ≥ f (x) ∧ f (z) ∧ 1−k 2 for all x, y, z ∈ S. Proof. For all x, y, z ∈ S, put a = xyz. Since f ◦k S ◦k f ⊆ fk , we have f (xyz) = f (a) ≥ fk (a) ≥ (f ◦k S ◦k f )(a) = (fk ◦ S ◦ fk )(a)  = [(fk ◦ S)(p) ∧ fk (q)] (p,q)∈Aa

≥ (fk ◦ S)(xy) ∧ fk (z)  = [fk (u) ∧ S(v)] ∧ fk (z) (u,v)∈Axy

≥ fk (x) ∧ S(y) ∧ fk (z)     1−k 1−k = f (x) ∧ ∧ 1 ∧ f (z) ∧ 2 2 = f (x) ∧ f (z) ∧

1−k . 2

Proposition 4.17. Let f and g be two (∈, ∈ ∨qk )-fuzzy generalized bi-ideals of an ordered semigroup S. Then f ◦k g is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S. Proof. Let f, g be two (∈, ∈ ∨qk )-fuzzy generalized biideals of S. Then, by Proposition 4.14, f ◦k S ◦k f ⊆ fk and g ◦k S ◦k g ⊆ gk . Thus, by Lemma 4.2(2), we have (f ◦k g) ◦k (f ◦k g) ⊆ f ◦k (g ◦k S ◦k g) ⊆ f ◦k g. Clearly, f ◦k g is a strongly convex fuzzy subset of S. Therefore, by Lemma 4.10, f ◦k g is an (∈, ∈ ∨qk )-fuzzy subsemigroup of S. Furthermore, by Lemma 4.2(2) we have (f ◦k g) ◦k S ◦k (f ◦k g) = f ◦k g ◦k (S ◦k f ) ◦k g ⊆ f ◦k (g ◦k S ◦k g) ⊆ f ◦k g. By Lemma 4.16, (f ◦k g)(xyz) ≥ (f ◦k g)(x) ∧ (f ◦k g)(z) ∧ 1−k 2 for all x, y, z ∈ S. Furthermore, if x ≤ y, then (f ◦k g)(x) ≥ (f ◦k g)(y). Indeed, if Ay = ∅, then (f ◦k g)(y) = (fk ◦ gk )(y) = 0. Since f ◦k g is a fuzzy subset of S, we have (f ◦k g)(x) ≥ 0 = (f ◦k g)(y). If Ay = / ∅, then, since x ≤ y, we have Ay ⊆ Ax . Thus, by Lemma 4.2(4), we have







[fk (u) ∧ gk (v)]

(u,v)∈Ay

[fk (u) ∧ gk (v)] = (fk ◦ gk )(x)

(u,v)∈Ax

= (f ◦k g)(x). Therefore, f ◦k g is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S by Theorem 3.6.

5. Characterizations of regular and intraregular ordered semigroups In this section we give some characterizations of regular ordered semigroups and intra-regular ordered semigroups by the properties of their (∈, ∈ ∨qk )-fuzzy left ideals, (∈, ∈ ∨qk )-fuzzy right ideals and (∈, ∈ ∨qk )-fuzzy (generalized) bi-ideals. Lemma 5.1. [38]. An ordered semigroup S is regular if and only if (BSB] = B for any bi-ideal B of S. Now we give characterizations of regular ordered semigroups by (∈, ∈ ∨qk )-fuzzy (generalized) biideals. Theorem 5.2. Let S be an ordered semigroup. Then the following statements are equivalent: (1) S is regular. (2) fk = f ◦k S ◦k f for any (∈, ∈ ∨qk )-fuzzy biideal f of S. Proof. (1) ⇒ (2). Let f be any (∈, ∈ ∨qk )-fuzzy biideal of S and a ∈ S. Then, since S is regular, there exists x ∈ S such that a ≤ axa, and (ax, a) ∈ Aa . Thus (f ◦k S ◦k f )(a) = (fk ◦ S ◦ fk )(a)  [(fk ◦ S)(y) ∧ fk (z)] ≥ (fk ◦ S)(ax) ∧ fk (a) = (y,z)∈Aa

=



[fk (p) ∧ S(q)] ∧ fk (a)

(p,q)∈Aax

≥ [fk (a)∧S(x)]∧fk (a) = fk (a)∧1∧fk (a) = fk (a), and so we have f ◦k S ◦k f ⊇ fk . Since f is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S, by Proposition 4.15, we have f ◦k S ◦k f ⊆ fk . Thus f ◦k S ◦k f = fk . (2) ⇒ (1). Let B be any bi-ideal of S. Then, by Lemma 4.8, fB is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S, and for each a ∈ B, by hypothesis and Lemma 4.4(3), we have

J. Tang and X. Xie / Characterizations of regular ordered semigroups by generalized fuzzy ideals

(f(BSB] )k = fB ◦k S ◦k fB = (fB )k ,

249

which implies that (BSB] = B. By Lemma 5.1, S is regular. In a similar way, the following theorem can be proved.

(2) ⇒ (1). Since S itself is an (∈, ∈ ∨qk )-fuzzy ideal of S, by hypothesis we have fk = fk ∩ S = f ∩k S = f ◦k S ◦k f. By Theorem 5.2, S is regular. In a similar way we can show the following two theorems:

Theorem 5.3. Let S be an ordered semigroup. Then the following statements are equivalent:

Theorem 5.5. Let S be an ordered semigroup. Then the following statements are equivalent:

(1) S is regular. (2) fk = f ◦k S ◦k f for any (∈, ∈ ∨qk )-fuzzy generalized bi-ideal f of S.

(1) S is regular. (2) f ◦k g ◦k f = f ∩k g for every (∈, ∈ ∨qk )-fuzzy generalized bi-ideal f and every (∈, ∈ ∨qk )-fuzzy ideal g of S.

Theorem 5.4. Let S be an ordered semigroup. Then the following statements are equivalent: (1) S is regular. (2) f ◦k g ◦k f = f ∩k g for any (∈, ∈ ∨qk )-fuzzy biideal f and any (∈, ∈ ∨qk )-fuzzy ideal g of S. Proof. (1) ⇒ (2). Let f, g be an (∈, ∈ ∨qk )-fuzzy bi-ideal and an (∈, ∈ ∨qk )-fuzzy ideal of S, respectively. Then, by Lemma 4.2(2) and Proposition 4.15, we have f ◦k g ◦k f ⊆ f ◦k S ◦k f ⊆ fk . Since g is an (∈, ∈ ∨qk )-fuzzy ideal of S, by Proposition 4.3 we have f ◦k g ◦k f = fk ◦ gk ◦ fk ⊆ S ◦ gk ◦ S ⊆ S ◦ gk ⊆ gk . Thus f ◦k g ◦k f ⊆ fk ∩ gk = f ∩k g. On the other hand, let a ∈ S. Then, since S is regular, there exists x ∈ S such that a ≤ axa ≤ (axa)xa, so (a, xaxa) ∈ Aa . Since g is an (∈, ∈ ∨qk )-fuzzy ideal of 1−k S, we have g(xax) ≥ g(ax) ∧ 1−k 2 ≥ g(a) ∧ 2 , and 1−k 1−k so gk (xax) = g(xax) ∧ 2 ≥ g(a) ∧ 2 ∧ 1−k 2 = g(a) ∧ 1−k = g (a). Thus k 2 (f ◦k g ◦k f )(a) = (fk ◦ gk ◦ fk )(a)  = [fk (y) ∧ (gk ◦ fk )(z)] (y,z)∈Aa

≥ fk (a) ∧ (gk ◦ fk )(xaxa)}    = fk (a) ∧ [gk (p) ∧ fk (q)] (p,q)∈Axaxa

≥ fk (a) ∧ [gk (xax) ∧ fk (a)] ≥ fk (a) ∧ gk (a) ∧ fk (a) = fk (a) ∧ gk (a) = (fk ∩ gk )(a) = (f ∩k g)(a), which means that f ◦k g ◦k f ⊇ f ∩k g. Therefore f ◦k g ◦k f = f ∩k g.

Theorem 5.6. Let S be an ordered semigroup. Then the following statements are equivalent: (1) S is regular. (2) f ◦k g = f ∩k g for every (∈, ∈ ∨qk )-fuzzy right ideal f and every (∈, ∈ ∨qk )-fuzzy left ideal g of S. Lemma 5.7. [38]. Let S be an ordered semigroup. Then the following statements are equivalent: (1) S is regular. (2) B ∩ L ⊆ (BL] for every bi-ideal B and every left ideal L of S. (3) R ∩ B ∩ L ⊆ (RBL] for every bi-ideal B, every right ideal R and every left ideal L of S. Now we give characterizations of regular ordered semigroups by (∈, ∈ ∨qk )-fuzzy left ideals, (∈, ∈ ∨qk )fuzzy right ideals and (∈, ∈ ∨qk )-fuzzy (generalized) bi-ideals. Theorem 5.8. Let S be an ordered semigroup. Then the following statements are equivalent: (1) S is regular. (2) f ∩k g ⊆ f ◦k g for every (∈, ∈ ∨qk )-fuzzy generalized bi-ideal f and every (∈, ∈ ∨qk )-fuzzy left ideal g of S. (3) f ∩k g ⊆ f ◦k g for every (∈, ∈ ∨qk )-fuzzy biideal f and every (∈, ∈ ∨qk )-fuzzy left ideal g of S. (4) h ∩k f ∩k g ⊆ h ◦k f ◦k g for every (∈, ∈ ∨qk )fuzzy generalized bi-ideal f , every (∈, ∈ ∨qk )fuzzy left ideal g and every (∈, ∈ ∨qk )-fuzzy right ideal h of S. (5) h ∩k f ∩k g ⊆ h ◦k f ◦k g for every (∈, ∈ ∨qk )fuzzy bi-ideal f , every (∈, ∈ ∨qk )-fuzzy left ideal g and every (∈, ∈ ∨qk )-fuzzy right ideal h of S.

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Proof. (1) ⇒ (2). Let f and g be an (∈, ∈ ∨qk )-fuzzy generalized bi-ideal and an (∈, ∈ ∨qk )-fuzzy left ideal of S, respectively. If a ∈ S, then, since S is regular, there exists x ∈ S such that a ≤ axa. Since (a, xa) ∈ Aa , we have (f ◦k g)(a) = (fk ◦ gk )(a)  = [fk (y) ∧ gk (z)] 1−k 2

(since g is an (∈, ∈ ∨qk )-fuzzy left ideal of S)   1−k = fk (a)∧ g(a) ∧ = fk (a) ∧ gk (a) 2 = (fk ∩ gk )(a) = (f ∩k g)(a), which means that f ∩k g ⊆ f ◦k g. (2) ⇒ (3). Clearly. (3) ⇒ (1). Let B and L be a bi-ideal and a left ideal of S, respectively. Let a ∈ B ∩ L. Then, by Theorems 4.6, 4.8, fB is an (∈, ∈ ∨qk )-fuzzy bi-ideal of S and fL is an (∈, ∈ ∨qk )-fuzzy left ideal of S. Thus, by hypothesis and Lemma 4.2, we have ((fB )k ◦ (fL )k )(a) = (fB ◦k fL )(a) ≥ (fB ∩k fL )(a) = ((fB )k ∩ (fL )k )(a) = (fB )k (a) ∧ (fL )k (a) =

1−k 2

/ ∅. By Definition 4.1, for all a ∈ B ∩ L, and Aa = ((fB )k ◦ (fL )k )(a) ≤ 1−k for all a ∈ S. Then 2 

[(fB )k (y) ∧ (fL )k (z)]

(y,z)∈Aa

= ((fB )k ◦ (fL )k )(a) =

(h ◦k f ◦k g)(a) = (hk ◦ fk ◦ gk )(a)  = [hk (y), (fk ◦ gk )(z)] (y,z)∈Aa

(y,z)∈Aa

≥ fk (a) ∧ gk (xa) = fk (a) ∧ g(xa) ∧   1−k 1−k ≥ fk (a) ∧ g(a) ∧ ∧ 2 2

a ∈ S. Then, since S is regular, there exists x ∈ S such that a ≤ axa. Thus (ax, a), (a, xa) ∈ Aa , and

1−k > 0, 2

which implies that there exist b, c ∈ S such that a ≤ bc, 1−k (fB )k (b) = 1−k 2 and (fL )k (c) = 2 . Then a ≤ bc ∈ BL, and so B ∩ L ⊆ (BL]. By Lemma 5.7, S is regular. (1) ⇒ (4). Let f, g and h be an (∈, ∈ ∨qk )-fuzzy generalized bi-ideal, an (∈, ∈ ∨qk )-fuzzy left ideal and an (∈, ∈ ∨qk )-fuzzy right ideal of S, respectively. Let

≥ hk (ax) ∧ (fk ◦ gk )(a)   1−k = h(ax) ∧ 2 ⎧ ⎫ ⎨   ⎬ ∧ fk (p) ∧ gk (q) ⎩ ⎭ (p,q)∈Aa



 1−k ≥ h(a) ∧ ∧ [fk (a) ∧ gk (xa)] 2    1−k = hk (a) ∧ fk (a) ∧ g(xa) ∧ 2   1−k ≥ hk (a) ∧ fk (a) ∧ g(a) ∧ 2 = hk (a) ∧ fk (a) ∧ gk (a) = (hk ∩ fk ∩ gk )(a) = (h ∩k f ∩k g)(a). Therefore, h ∩k f ∩k g ⊆ h ◦k f ◦k g. (4) ⇒ (5). Clearly. (5) ⇒ (1). The proof is similar to that of (3) ⇒ (1) with suitable modification. An ordered semigroup (S, ·, ≤) is called intraregular if, for each element a of S, there exist x, y ∈ S such that a ≤ xa2 y. Equivalent definition: a ∈ (Sa2 S], ∀a ∈ S (cf. [19]). Lemma 5.9. [38]. Let S be an ordered semigroup. Then the following statements are equivalent: (1) S is intra-regular. (2) R ∩ L ⊆ (LR] for every left ideal L and every right ideal R of S. Now we give a characterization of an intra-regular ordered semigroup by (∈, ∈ ∨qk )-fuzzy left ideals and (∈, ∈ ∨qk )-fuzzy right ideals. Theorem 5.10. Let S be an ordered semigroup. Then the following statements are equivalent: (1) S is intra-regular. (2) f ∩k g ⊆ g ◦k f for every (∈, ∈ ∨qk )-fuzzy left ideal g and every (∈, ∈ ∨qk )-fuzzy right ideal f of S. Proof. (1) ⇒ (2). Suppose that S is an intra-regular ordered semigroup, f and g are an (∈, ∈ ∨qk )-fuzzy

J. Tang and X. Xie / Characterizations of regular ordered semigroups by generalized fuzzy ideals

right ideal and an (∈, ∈ ∨qk )-fuzzy left ideal of S, respectively. Let a ∈ S. Then there exist x, y ∈ S such that a ≤ xa2 y, i.e., (xa, ay) ∈ Aa . Thus (g ◦k f )(a) = (gk ◦ fk )(a)  = [gk (y) ∧ fk (z)] ≥ gk (xa)∧fk (ay) (y,z)∈Aa

    1−k 1−k = g(xa) ∧ ∧ f (ay) ∧ 2 2     1−k 1−k ≥ g(a) ∧ ∧ f (a) ∧ 2 2 = gk (a) ∧ fk (a) = (gk ∩ fk )(a) = (fk ∩ gk )(a) = (f ∩k g)(a), which means that f ∩k g ⊆ g ◦k f. (2) ⇒ (1). Assume that (2) holds. Let R and L be any right ideal and left ideal of S, respectively, and a ∈ R ∩ L. Then , by Theorem 4.6, fR and fL be an (∈, ∈ ∨qk )-fuzzy right ideal and an (∈, ∈ ∨qk )-fuzzy left ideal of S, respectively. Thus , by hypothesis and Lemma 4.4, we have (fR∩L )k = fR ∩k fL ⊆ fL ◦k fR = (f(LR] )k , which implies that R ∩ L ⊆ (LR]. It thus follows, by Lemma 5.9, that S is intra-regular. Theorem 5.11. An ordered semigroup S is intra-regular if and only if (∀a ∈ S) fk (a) = fk (a2 ) for every (∈, ∈ ∨qk )-fuzzy ideal f of S. Proof. (=⇒) Let f be an (∈, ∈ ∨qk )-fuzzy ideal of S and a ∈ S. Then, by hypothesis, there exist x, y ∈ S such that a ≤ xa2 y, and 1−k 1−k ≥ f (xa2 y) ∧ 2 2   1−k 1−k ≥ f (a2 y) ∧ ∧ 2 2 1−k = f (a2 y) ∧ 2  1 − k 1 − k ≥ f (a2 ) ∧ ∧ 2 2 1−k = f (a2 ) ∧ 2  1 − k 1 − k ≥ f (a) ∧ ∧ 2 2 1−k = f (a) ∧ = fk (a), 2

fk (a) = f (a) ∧

251

2 which implies that fk (a) = f (a2 ) ∧ 1−k 2 = fk (a ). (⇐=) By Theorem 4.6, fI(a2 ) is an (∈, ∈ ∨qk )fuzzy ideal of S. By hypothesis, (fI(a2 ) )k (a) = 1−k 1−k (fI(a2 ) )k (a2 ) = fI(a2 ) (a2 ) ∧ 1−k 2 = 1 ∧ 2 = 2 , so 2 2 2 2 2 a ∈ I(a ) = (a ∪ Sa ∪ a S ∪ Sa S]. Thus a ≤ t for some t ∈ a2 ∪ Sa2 ∪ a2 S ∪ Sa2 S. If t = a2 , then a ≤ a2 ≤ a4 ∈ Sa2 S, that is a ∈ (Sa2 S]. If t = xa2 for some x ∈ S, then a ≤ xa2 ≤ x(xa2 )a = x2 a2 a ∈ Sa2 S. If t = a2 y for some y ∈ S, then a ≤ a2 y ≤ a(a2 y)y = aa2 y2 ∈ Sa2 S. If t ∈ Sa2 S, then a ∈ (Sa2 S]. Thus S is intra-regular.

Lemma 5.12. [38]. An ordered semigroup S is regular and intra-regular if and only if B = (B2 ] for every biideal B of S. Now we shall give some characterizations of an ordered semigroup which is both regular and intra-regular by (∈, ∈ ∨qk )-fuzzy left ideals, (∈, ∈ ∨qk )-fuzzy right ideals and (∈, ∈ ∨qk )-fuzzy bi-ideals. Theorem 5.13. Let S be an ordered semigroup. Then the following conditions are equivalent: (1) S is regular and intra-regular. (2) f ◦k f = fk for every (∈, ∈ ∨qk )-fuzzy bi-ideal f of S. (3) f ∩k g ⊆ (f ◦k g) ∩ (g ◦k f ) for any (∈, ∈ ∨qk )fuzzy bi-ideals f and g of S. (4) f ∩k g ⊆ (f ◦k g) ∩ (g ◦k f ) for every (∈, ∈ ∨qk )-fuzzy bi-ideal f and every (∈, ∈ ∨qk )-fuzzy left ideal g of S. (5) f ∩k g ⊆ (f ◦k g) ∩ (g ◦k f ) for every (∈, ∈ ∨qk )-fuzzy right ideal f and every (∈, ∈ ∨qk )fuzzy bi-ideal g of S. (6) f ∩k g ⊆ (f ◦k g) ∩ (g ◦k f ) for every (∈, ∈ ∨qk )-fuzzy right ideal f and every (∈, ∈ ∨qk )fuzzy left ideal g of S. Proof. The proof is similar to that of Theorem 5.8, we omit it. 6. Conclusion As we know, fuzzy ideals of an ordered semigroup with special properties always play an important role in the study of ordered semigroups structure. In this paper we have introduced the concepts of (∈, ∈ ∨qk )-fuzzy ideals, (∈, ∈ ∨qk )-fuzzy bi-ideals and (∈, ∈ ∨qk )-fuzzy generalized bi-ideals of an ordered semigroup, and investigated their related properties. Furthermore, we have given some characterizations of regular ordered semigroups by (∈, ∈ ∨qk )-fuzzy left

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