Characterizations of Regular Ordered Semigroups by ( ∈ , ∈ ∨ ( k ∗ , q k ) )-Fuzzy Quasi-Ideals

In this paper, some properties of the (k∗, k)-lower part of (∈,∈ ∨(k∗, qk))-fuzzy quasi-ideals are obtained. Then, we characterize regular ordered semigroups in terms of its (∈,∈ ∨(k∗, qk))-fuzzy quasi-ideals, (∈,∈ ∨(k∗, qk))-fuzzy generalized bi-ideals, (∈,∈ ∨(k∗, qk))-fuzzy left ideals and (∈,∈ ∨(k∗, qk))-fuzzy right ideals, and an equivalent condition for (∈,∈ ∨(k∗, qk))-fuzzy left (resp. right) ideals is obtained. Finally, the existence theorems for an (∈,∈ ∨(k∗, qk))-fuzzy quasi-ideal as well as for the minimality of an (∈,∈ ∨(k∗, qk))-fuzzy quasi-ideal of an ordered semigroup are provided.


Introduction
Since the concept of fuzzy sets was introduced by Zadeh in 1965 [1], the theories of fuzzy sets and fuzzy systems developed rapidly.The study of fuzzy algebraic structures started in the pioneering paper of Rosenfeld [2] in 1971.Rosenfeld introduced the notion of fuzzy groups and showed that many results in groups can be extended in an elementary manner to develop the theory of fuzzy groups.Since then, the literature on various fuzzy algebraic concepts has been growing very rapidly.Kuroki [3] introduced fuzzy sets in semigroup theory.Fuzzy (left, right) ideals and fuzzy bi-ideals in semigroups were introduced and studied by Kuroki [3,4].As an extension of the concept of a fuzzy bi-ideal, Kuroki [5] introduced the concept of a fuzzy generalized bi-ideal of a semigroup and characterized different classes of ordered semigroups in terms of fuzzy generalized bi-ideals.Furthermore, interior ideals and a semiprime subset in a semigroup were extended [6] to introduce the concept of fuzzy interior ideals and the fuzzy semiprimality of a fuzzy set in a semigroup, and then characterize different classes of a semigroup in terms of fuzzy semiprime interior ideals.
Chang [7] was the first to initiate and extend the concept of general theory of fuzzy sets to topological spaces.After that, several authors [8][9][10][11][12][13] had investigated the different topological properties of a set in terms of its fuzzy subsets.
A generalized notion of a one-sided ideal is a notion of a quasi-ideal.In 1953, Steinfeld introduced quasi-ideals for rings [22] and, in 1956, for semigroups [23].After that, quasi-ideals were widely studied in different algebraic structures by several authors.In his study of fuzzy ideals in semigroups, Kuroki [4] investigated some properties of fuzzy quasi-ideals of semigroups [6] and characterized different classes of semigroups in terms of fuzzy semiprime quasi-ideals.The concept of fuzzy quasi-ideals in ordered semigroups had been considered by Kehayoupu [24].Shabir and Khan [25] characterized left (resp.right) simple and completely regular ordered semigroups in terms of fuzzy quasi-ideals.They also defined semiprime fuzzy quasi-ideals of ordered semigroups and characterized completely regular ordered semigroups in terms of fuzzy semiprime quasi-ideals.
Motivated by the above, in the present paper we introduce, as a generalization of the notion of the k-lower part of an (∈, ∈ ∨q k )-fuzzy quasi-ideal defined in an ordered semigroup, the notion of the (k * , k)-lower part of an (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal in an ordered semigroup.Our first aim is to provide some different characterizations of regular ordered semigroups in terms of (∈, ∈ ∨(k * , q k ))-fuzzy left ideals, (∈, ∈ ∨(k * , q k ))-fuzzy right ideals, (∈, ∈ ∨(k * , q k ))-fuzzy generalized bi-ideals, and (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideals.Secondly, we present the relationships between (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideals and similar types of fuzzy left/right ideals, while the last section offers concluding remarks and some ideas for future work on the topic.

Preliminaries
In this section, we give some basic definitions and results that are needed to develop our paper.For more details and unexplained notions, readers may consult [17,21].
Our study is based on the notion of the ordered semigroup, that is, a semigroup (S, •) endowed with a partial order ≤ which is compatible with the • operation, i.e., for each x, y ∈ S, the relation x ≤ y implies that zx ≤ zy and xz ≤ yz for all z ∈ S.
For a subset A of an ordered semigroup S, we denote (A] = {t ∈ S | t ≤ a for some a ∈ A}.If A = {a}, we write (a] instead of ({a}].For any non-empty subsets A and B of an ordered semigroup S, the following properties hold: (1) A ⊆ (A]; (2) All of these notions have also been transferred into the fuzzy sets theory.A non-empty subset A of an ordered semigroup S is called a subsemigroup of S if for all x, y ∈ A, xy ∈ A; while A is called a left (resp.right) ideal of S if it satisfies the conditions SA ⊆ A (AS ⊆ A) and (A] ⊆ A. In addition A is called an ideal of S if it is both a left and a right ideal of S. A subsemigroup Let (S, •, ≤) be an ordered semigroup.A mapping η from S to the real closed interval [0, 1] is called a fuzzy subset of S. We denote by η A the characteristic function of the subset A of S, which is defined by Moreover, the identical function 1(x) = 1, for all x ∈ S, is a fuzzy subset of S. Let η and ξ be two fuzzy subsets of S.Then, for each x ∈ S, the intersection, union, and composition between these two fuzzy subsets are defined as where A x = {(y, z) ∈ S × S | x ≤ yz}.Moreover, on the set of all fuzzy subsets of S, one may define an order relation by η ξ ⇔ η(x) ≤ ξ(x) for all x ∈ S.
Now we recall the basic properties of fuzzy ideals on ordered semigroups, in accordance with [15,25].A fuzzy subset η of an ordered semigroup S is called a fuzzy subsemigroup of S if η(xy) ≥ min{η(x), η(y)} for all x, y ∈ S, while it is called a fuzzy left (resp.right) ideal of S if it satisfies the following two relations: (1) x ≤ y ⇒ η(x) ≥ η(y) and (2) η(xy) ≥ η(y) (resp.η(xy) ≥ η(x)) for all x, y ∈ S. η is called a fuzzy ideal of S if it is both a fuzzy left and right ideal of S. A fuzzy subsemigroup η of S is called a fuzzy bi-ideal of S if there are (1) x ≤ y ⇒ η(x) ≥ η(y) and (2) η(xyz) ≥ min{η(x), η(z)} for all x, y, z ∈ S.
Let S be an ordered semigroup, a ∈ S and u ∈ (0, 1].An ordered fuzzy point a u of S is defined by For any fuzzy subset η of S, we denote a u ⊆ η by a u ∈ η.Thus, a u ∈ η if and only if η(a) ≥ u.

Theorem 1 ([21]
).Let η be a fuzzy subset of S. Then η is an (∈, ∈ ∨(k * , q k ))-fuzzy left ideal of S if and only if for all r, s ∈ S.

Theorem 2 ([21]
).Let η be a fuzzy subset of S. Then η is an (∈, ∈ ∨(k * , q k ))-fuzzy right ideal of S if and only if for all r, s ∈ S. Theorem 3 ([21]).Let η be a fuzzy subset of S. Then η is an (∈, ∈ ∨(k * , q k ))-fuzzy generalized bi-ideal of S if and only if for all r, s, w ∈ S.

Definition 2 ([21]
).For any fuzzy subset η of S, the (k * , k)-lower part η k * k of η is defined as follows: For any subset T( = ∅) of S and fuzzy subset η of S, (η T ) k * k , the (k * , k)-lower part of the characteristic function η T will be denoted by (η k * k ) T in the sequel.

Lemma 1 ([21]
).The (k * , k)-lower part (η k * k ) T of the characteristic function η T of any subset T( = ∅) of S is an (∈, ∈ ∨(k * , q k ))-fuzzy left (resp.right) ideal of S if and only if T is a left (resp.right) ideal of S.

Lemma 2 ([21]
).The (k * , k)-lower part (η k * k ) T of the characteristic function η T of any subset T( = ∅) of S is an (∈, ∈ ∨(k * , q k ))-fuzzy generalized bi-ideal of S if and only if T is a generalized bi-ideal of S.
Example 1.Let S = {w, x, y, z}.Define a binary operation " • " and order " ≤ " in the following way: • w x y z w w w w w x w w w w y w w w x z w w x y ≤:= {(w, w), (x, x), (y, y), (z, z), (w, x)}.
Then (S, •, ≤) is an ordered semigroup.Now define a fuzzy subset η on S as follows: Theorem 5 ([34]).A fuzzy subset η of an ordered semigroup S is an (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal of S if and only if for each a, b ∈ S.

Lemma 7 ([21]
).Let H and J be any non-empty subsets of S. Then In the following result, a correspondence between quasi-ideals and the (k * , k)-lower part of the (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideals of an ordered semigroup is discussed.
Let r, s, t, p, q ∈ S and u, v ∈ (0, 1] be such that r ≤ sp, r ≤ qt and Then by Theorem 5, Proof.Let r, s ∈ S, and r ≤ s.Since η is an (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal of S and r ≤ s, we have η(r) ≥ min{η(s), Next, we suppose that r, s, z, p, q ∈ S such that r ≤ sp and r ≤ qz, and we have Recall the fuzzy sets from [21] defined as follows: for all a ∈ S and 0 ≤ k < k * ≤ 1.These fuzzy sets help us to obtain different characterizations of (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideals of ordered semigroups.Their immediate properties are expressed in the following results.
Proof.(⇒) Take any a ∈ S.Then, there exists r ∈ S such that a ≤ ara.Thus, (ar, a) ∈ A a .Now, we have Therefore, there exists (r, s) ∈ A a such that ((( Theorem 7. The following assertions are equivalent in S: (1) S is regular; (2) Proof.
Theorem 8.The following assertions are equivalent in S: (1) S is regular; (2 Proof.(1) ⇒ (2).Suppose that η and ξ are an (∈, ∈ ∨(k * , q k ))-fuzzy right ideal and an (∈, ∈ ∨(k * , q k ))-fuzzy left ideal of S, respectively, and a ∈ S. Then there exists r ∈ S such that a ≤ ara.It follows that (ar, a) ∈ A a .Then we have (2) ⇒ (1).Suppose that A and B are a left ideal and a right ideal of S. Take any x ∈ A ∩ B. Then x ∈ A and x ∈ B. As A is a left ideal and B is a right ideal of S, by Lemma 1, (η k * k ) A is an (∈, ∈ ∨(k * , q k ))-fuzzy left ideal, and (η k * k ) B is an (∈, ∈ ∨(k * , q k ))-fuzzy right ideal of S. Thus, we have (η By Lemma 7(3), we have (1) S is regular; (2) ))-fuzzy right ideal η and for each (∈, ∈ ∨(k * , q k ))-fuzzy generalized bi-ideal ξ of S.
Proof.(1) ⇒ (2).Suppose that η and ξ are an (∈, ∈ ∨(k * , q k ))-fuzzy right ideal and an (∈, ∈ ∨(k * , q k ))-fuzzy generalized bi-ideal of S, respectively.Take any a ∈ S, then there exists r ∈ S such that a ≤ ara implies (a, ra) ∈ A a .Then we have (2) ⇒ (1).Suppose that C is a right ideal and D is a generalized bi-ideal of S. Take any x ∈ C ∩ D. As C is a right ideal and D is a generalized bi-ideal of S, by Lemmas 1 and 2, (η Proof.(⇒) The direct part follows from Lemma 5 because each quasi-ideal of S is a generalized bi-ideal of S.
(⇐) Let u ∈ S. As Q(u) is a quasi-ideal and S is a left ideal of S, we have Similarly, L(u) ⊆ (Su].As (uS] and (Su] are a quasi-ideal and left ideal of S, respectively, by hypothesis we have Hence S is regular.
Lemma 12.An ordered semigroup S is regular ⇔ H ∩ K ⊆ (HK] for each right ideal H and for each quasi ideal K of S.
Proof.Along similar lines to the proof of Lemma 11.
Theorem 10.The following assertions are equivalent in S: (1) S is regular; (2 Proof.(1) ⇒ (2).Suppose that η and ξ are an (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal and an (∈, ∈ ∨(k * , q k ))-fuzzy left ideal of S. Take any a ∈ S, and as S is regular, there exists r ∈ S such that a ≤ ara.It follows that (a, ra) ∈ A a .Then we have . Suppose that C is a left ideal and D is a quasi ideal of S. Take any x ∈ C ∩ D. Since D is a quasi-ideal and C is a left ideal of S, by Lemmas 9 and 7, ( (1) S is regular; (2) ))-fuzzy right ideal η and any (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal ξ of S.
Corollary 1.The following assertions are equivalent in S: (1) S is regular; (2) B ∩ L ⊆ (BL] for each generalized bi-ideal B and for each left ideal L of S; (3) Q ∩ L ⊆ (QL] for each quasi-ideal Q and for each left ideal L of S; (4) R ∩ B ⊆ (RB] for each right ideal R and for each generalized bi-ideal B of S; (5) R ∩ Q ⊆ (RQ] for each right ideal R and for each quasi-ideal Q of S; Corollary 2. The following assertions are equivalent in S: (1) S is regular; ))-fuzzy right-ideal η and for each (∈, ∈ ∨(k * , q k ))-fuzzy generalized bi-ideal ξ of S; (5) ))-fuzzy right ideal η and for each (∈, ∈ ∨(k * , q k ))-fuzzy generalized bi-ideal ξ of S.
The next two results provide the existence of the notion of an (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal and minimal (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal of an ordered semigroup.
To show that g is an (∈, ∈ ∨(k * , q k ))-fuzzy left ideal of S, first we show that for each a, b ∈ S such that a ≤ b, we have It is sufficient to show that for each (y, z) ∈ A b , it holds that Let (y, z) ∈ A b .As a ≤ b, and a ≤ b ≤ yz, then we have Next, we show that 1(•) k * k g ⊆ g.Since ab ≤ ab, it follows that (a, b) ∈ A ab .Then we have Similarly, From ( 1) and ( 2), it follows that, (η(∪) and so, η = g(∩) k * k h.(⇐) Suppose that η is a fuzzy subset of S such that η = g(∩) k * k h for an (∈, ∈ ∨(k * , q k ))-fuzzy left ideal g and an (∈, ∈ ∨(k * , q k ))-fuzzy right ideal h of S. Let r, s ∈ S such that r ≤ s.Then clearly η Now we have . Therefore by (1), Hence, by Theorem 5, η is an (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal of S. Definition 4.An (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal f of S is said to be minimal if there does not exist any (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal g of S such that g f .Theorem 13.Let η be an (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal of S such that η(a) ≥ η(b) for each a, b with a ≤ b.Then η is a minimal (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideal of S ⇔ there exists a minimal (∈, ∈ ∨(k * , q k ))-fuzzy left ideal g and a minimal (∈, ∈ ∨(k * , q k ))-fuzzy right ideal h of S such that η = g(∩) k * k h.

Conclusions and Future Work
The aim of the present paper is to enhance the understanding of ordered semigroups and regular ordered semigroups by considering the structural influence of (∈, ∈ ∨q k )-fuzzy quasi-ideals.In this view, we obtain several characterizations of regular ordered semigroups in terms of (∈, ∈ ∨(k * , q k ))-fuzzy right ideals, (∈, ∈ ∨(k * , q k ))-fuzzy left ideals, (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideals, and (∈, ∈ ∨(k * , q k ))-fuzzy generalized bi-ideals.In addition, we characterize the minimal of (∈, ∈ ∨(k * , q k ))-fuzzy quasi-ideals in terms of (∈, ∈ ∨(k * , q k ))-fuzzy left ideals and (∈, ∈ ∨(k * , q k ))-fuzzy right ideals.Following are the particular cases of the present paper: One may also conclude that: (1) If we put k * = 1, then most of the results of this paper reduce in the setting of (∈, ∈ ∨q k )-fuzzy quasi-ideals.
(2) If we put k * = 1 and k = 0, then most of the results of this paper reduce in the setting of (∈, ∈ ∨q)-fuzzy quasi-ideals.
and so x ∈ (DC].This implies that D ∩ C ⊆ (DC].Hence, by Lemma 11, S is regular.Similarly, we may prove the following theorem: Theorem 11.The following assertions are equivalent in S: