# Ordered Semigroups Based on∈,∈∨qkδ-Fuzzy Ideals

Authors: Faiz Muhammad Khan; Nie Yufeng; Hidayat Ullah Khan; Bakht Muhammad Khan
Journal: Advances in Fuzzy Systems
 (2018)

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## Abstract

A new trend of using fuzzy algebraic structures in various applied sciences is becoming a central focus due to the accuracy and nondecoding nature. The aim of the present paper is to develop a new type of fuzzy subsystem of an ordered semigroupS. This new type of fuzzy subsystem will overcome the difficulties faced in fuzzy ideal theory of an ordered semigroup up to some extent. More precisely, we introduce ∈,∈∨qkδ-fuzzy left (resp., right, quasi-) ideals of S. These concepts are elaborated through appropriate examples. Further, we are bridging ordinary ideals and ∈,∈∨qkδ-fuzzy ideals of an ordered semigroup S through level subset and characteristic function. Finally, we characterize regular ordered semigroups in terms of ∈,∈∨qkδ-fuzzy left (resp., right, quasi-) ideals.


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## Body

## 1. Introduction

The breakthrough of Zadeh’s pioneering idea [1] of fuzzy sets changed our traditional tools for formal modeling, reasoning, and computing. These traditional tools are crisp in nature. Crisp means dichotomous, i.e., yes or no type rather than more or less type. In set theory, an element can either belong to a set or not. Zadeh’s seminal paper on fuzzy set opens a new direction for researchers to use fuzzy sets in applied sciences like automata theory, computer science, artificial intelligence, coding theory, robotics, and much more. Keeping this inspiration in view, Rosenfeld [2] introduced fuzzy subgroups. The inception of fuzzy subgroups provides a platform for other researchers to use this pioneering idea in other algebraic structures along with several diverse applications. Among other algebraic structures, semigroups (especially ordered semigroups) are having a lot of applications in error correcting codes, control engineering, performance of super computer, and information sciences. Mordeson et al.’s idea in [3] gives birth to an up to date account of fuzzy subsemigroups and fuzzy ideals of semigroups while Kehayopulu and Tsingelis [4–6] used fuzzy sets in ordered semigroups to develop fuzzy ideal theory. Shabir and Khan [7] gave a characterization of ordered semigroups by the properties of fuzzy ideals and fuzzy generalized bi-ideals.In 1996, the idea of a quasi-coincidence of a fuzzy point with a fuzzy set [8, 9] is presented which played a vital role to generate different types of fuzzy subgroups. Bhakat and Das [8] gave the concept of α,β-fuzzy subgroups and introduced ∈,∈∨q-fuzzy subgroups by using the “belongs to” relation (∈) and “quasi-coincident with” relation (q) between a fuzzy point and a fuzzy subgroup. In fact this is an important and useful generalization of Rosenfeld’s idea of fuzzy subgroup [2]. These new generalizations provide a motivation to other researchers for further development in other algebraic structures. Since then a variety of research has been carried out using this icebreaking idea. More precisely, the aforementioned notion is used by Ma et al. [10, 11] in R0-algebras and Davvaz and Mozafar [12] in Lie algebra, and Davvaz and coauthors used the idea of generalized fuzzy sets in rings [13–15], Jun et al. [16], Khan and Shabir [17], and Khan et al. [18] in ordered semigroups, and Shabir et al. [19, 20] in semigroups. For further details, the readers are referred to [21–26].In 2009, Jun [27] initiated a more general form of quasi-coincident with relation (q) and provided (qk), where k∈[0,1). The notion is further strengthened by other researchers to use in various algebraic structures [28–31].Recently, Jun et al. [32] presented another comprehensive generalization of ∈,∈∨q-fuzzy subgroups and discussed fuzzy subgroups in light of generalized quasi-coincident with relation. Further, Khan et al. [33] elaborated ordered semigroups in terms of fuzzy generalized bi-ideals using this idea [32]. Also, Khan et al. [34] determined fuzzy filters of ordered semigroups for the said notion.In this paper, we apply Jun et al.’s idea [32] to ordered semigroups and discuss ordered semigroups through fuzzy left (resp., right) ideals of type ∈,∈∨qkδ, where k∈[0,1) and δ∈(0,1], and obtained several important results. More precisely, ∈,∈∨qkδ-fuzzy left (resp., right) ideals are introduced. Ordinary fuzzy ideals and ∈,∈∨qkδ-fuzzy ideals of an ordered semigroup S are linked. Finally, regular ordered semigroups by the properties of these newly developed fuzzy left (resp., right) ideals are characterized.


## 2. Preliminaries

Algebraic structures like semigroups, ordered semigroups, and monids are extensively used in applied fields such as automata theory, coding theory, and theoretical computer science. The present section discusses various fundamental definitions and results essential for the proceeding article.A system(S,·,≤), in which (S,·) is a semigroup, (S,≤) is a poset with both left and right compatibility, i.e., a≤b→ax≤bx, xa≤xb for all a,b,x∈S, is known as ordered semigroup. For A⊆S, we denote A≔{a∈S∣a≤x for some x∈A}. If A={a}, then we write a instead of a. For A,B⊆S, we denote AB≔ab∣a∈A,b∈B. If A⊆B, then (A]⊆(B], (A](B]⊆(AB] and ((A]] [18]. Later, S will denote an ordered semigroup, unless otherwise stated. For a nonempty subset A of S, a∈A and S∋b≤a, then b∈A is called a right (resp., left) ideal [18] of S if AS⊆A (resp., SA⊆A). If A is both a left and a right ideal of S then A is called a two-sided ideal or simply an ideal of S. An ordered semigroup S is regular [22] if for every a∈S there exists x∈S such that a≤axa. Let S be an ordered semigroup and ∅≠A⊆S. Then the characteristic function χA of A is defined by (1)χAx≔1ifx∈A,0ifx∉A.By a fuzzy subsetξ of an ordered semigroup S, we mean a function ξ:X→[0,1].A fuzzy subsetξ of S is called a fuzzy left (resp., right) ideal of S if, for all x,y∈S,(i)
x≤y→ξ(x)≥ξ(y),(ii)
ξ(xy)≥ξ(y) (resp. ξ(xy)≥ξ(x)).A fuzzy subsetξ of S is called a fuzzy ideal of S if it is both a fuzzy left and a fuzzy right ideal of S.For a nonempty subsetX of S, defineXa=y,z∈S×S∣a≤yz (see [18]).Transfer principle in fuzzy set theory [35] plays a remarkable role by providing the connection between classical and fuzzy subsets through level subset. A subset of the form (2)Uξ;t≔x∈S∣ξx≥t,where t∈(0,1], is known as level subset of S. For the said purpose, the following lemma is used.Lemma 1 (see [35]).
A fuzzy subsetξ defined on S has the property P if and only if all nonempty level subsets U(ξ;t) have the property P.
As a consequence of the above property, the following useful theorem is given.Theorem 2 (see [14]).
A fuzzy subsetξ of S is a fuzzy left (resp., right) ideal of S if and only if each nonempty level subset U(ξ;t), for all 0<t≤1, is a left (resp., right) ideal of S.
We denote byξS the set of all fuzzy subsets of S and define an order relation “⊆” on the fuzzy subsets of ξS as follows:(3)ξ⊆Giffξx≤Gx∀x∈S,ξ,G∈ξS.If ξ, G∈ξS, the intersection ξ∩G, union ξ∪G, and product ξ∘G can be defined as follows.
For allx∈S, (4)ξ∩Gx=ξx∧Gx,ξ∪Gx=ξx∨Gx,ξ∘Ga=⋁y,z∈Xaξy∧GzifXa≠∅,0otherwise.Theorem 3 (see [4]).
A nonempty subsetA of an ordered semigroup S is a left (resp., right) ideal of S if and only if the characteristic function χA of A is a fuzzy left (resp., right) ideal of S.


## 3. Fuzzy Ideals Based onδ-Quasi-Coincident with Relation

A verity of latest research, conducting multiple conferences and workshops on fuzzy mathematics, shows the importance of this particular field of study. It is worth mentioning that the researchers are more interested to use algebraic structures in upcoming completed problems not even in mathematics but in an engineering, computer science, robotics coding theory, and others. The idea of quasi-coincident with relation was based on the aforementioned concept, where Jun’s [27] generalization of quasi-coincident with relation played a vital role in generating several types of fuzzy subsystems which have already been used in various applied fields which leads to a verity of productive research [20, 28–31, 33, 34]. Jun et al. [32] further generalized his idea [27] and initiated an essential generalization of ∈,∈∨q-fuzzy subgroups. Keeping in view Jun et al.’s idea [32], we introduce a new generalization called δ-quasi-coincident with relation. In this section, new classification of ordered semigroups based on ∈,∈∨qkδ -fuzzy ideals, where k is an arbitrary element of [0,1) and δ∈(0,1], is determined. A fuzzy subset ξ of S of the form (5)ξ:S⟶0,1,y⟼aify=x,0ify≠x,is fuzzy point [36] with support x and value a∈(0,1] and is denoted by xa. A fuzzy point xa is said to belong to (resp., quasi-coincident with) a fuzzy set ξ, written as xa∈ξ (resp., xaqξ) if ξx≥a (resp., ξx+a>1). If xa∈ξ or xaqξ, then we write xa∈∨qξ. The symbol ∈∨q¯ means ∈∨q does not hold. Generalizing the concept of xaqξ, in ordered semigroups, we define xaqkδξ, as ξx+a+k>δ, where k∈[0,1) and δ∈(0,1]. Note that xaqkξ implies xaqkδξ, but the converse of the statement is not always true which will later be elaborated through examples. Particularly, if δ=1, then every δ-quasi-coincident with relation will lead to quasi-coincident with relation, symbolically xaqk1ξ=xaqkξ. Also, xa∈∨qkδξresp.xa∈∧qkδξ means that xa∈ξ or xaqkδξresp.xa∈ξandxaqkδξ.Definition 4.
A fuzzy subsetξ of S is called an ∈,∈∨qkδ-fuzzy left (resp., right) ideal of S if it satisfies the conditions:(1)
(∀x,y∈S)(∀a∈(0,1])(x≤y→ya∈ξ→xa∈∨qkδξ).(2)
(∀x,y∈S)(∀a∈(0,1])(ya∈ξ→(xy)a∈∨qkδξresp.(yx)a∈∨qkδξ).
Note that every∈,∈∨qkδ-fuzzy left (resp., right) ideal with δ=1 is an ∈,∈∨qk-fuzzy left (resp., right) ideal of S [30].Example 5.
Consider an ordered semigroupS={a1,a2,a3,a4,a5} with multiplication Table 1 and order relation: (6)≤≔a1,a1,a1,a3,a1,a4,a1,a5,a2,a2,a3,a3,a3,a5,a4,a4,a5,a5.
This order relation is represented by Figure1.Table 1
Multiplication table of ordered semigroupS.
⋅ a 1 a 2 a 3 a 4 a 5 a 1 a 1 a 4 a 1 a 4 a 4 a 2 a 1 a 2 a 1 a 4 a 4 a 3 a 1 a 4 a 3 a 4 a 5 a 4 a 1 a 4 a 1 a 4 a 4 a 5 a 1 a 4 a 3 a 4 a 5Figure 1
Order relation ofS.Letξ:S→[0,1] be a fuzzy subset of S, for x∈S, and define a fuzzy subset ξ as follows: (7)
S
a
1
a
2
a
3
a
4
a
5
ξ
x
0.60
0.20
0.50
0.40
0.45Thenξ is an (∈,∈∨qkδ)-fuzzy ideal of S with δ=0.8.Obviously, every∈,∈∨qk-fuzzy left (resp., right) ideal is an ∈,∈∨qkδ-fuzzy left (resp., right) ideal of S, but the converse in not true in general. Because in the aforementioned example, if a=0.42 and k∈0,0.18 then ξ is an (∈,∈∨qkδ)-fuzzy ideal of S with δ=0.8 but not ∈,∈∨qk-fuzzy ideal. a50.42∈ξ but a1a50.42∈∨qkδ¯ξ. If δ∈0.8,1, then ξ is not an ∈,∈∨qkδ-fuzzy ideal of S.Theorem 6.
LetI be a left (resp., right) ideal of S and ξ a fuzzy subset in S defined by (8)ξx=≥δ-k2ifx∈I,0otherwise.
Then(1)
ξ is a (q,∈∨qkδ)-fuzzy ideal of S.(2)
ξ is an (∈,∈∨qkδ)-fuzzy ideal of S.Proof.
(1) We discuss only the case of left ideals since the case for right ideals can be proved similarly. Letx,y∈S, x≤y, and a∈(0,1] be such that yaqξ. Then ξ(y)+a>1 and so y∈I. Since I is a left ideal of S and x≤y∈I, we have x∈I. Thus ξ(x)≥δ-k/2. If a≤δ-k/2, then ξ(x)≥a and so xa∈ξ. If a>δ-k/2, then ξ(x)+a+k>δ-k/2+δ-k/2+k=δ-k+δ-k+2k/2=δ and so xaqkδξ. Therefore xa∈∨qkδξ.
Letx,y∈S and a∈(0,1] be such that yaqξ. Then ξ(y)+a>1 so y∈I. Since I is a left ideal of S, we have xy∈I. Thus ξ(xy)≥δ-k/2. Now, if a>δ-k/2, then ξ(xy)+a+k>δ-k/2+δ-k/2+k=δ and so (xy)aqkδξ. If a≤1-k/2, then ξ(xy)≥a and so (xy)a∈ξ. Therefore, (xy)a∈∨qkδξ.
(2) Letx,y∈S, x≤y, and a∈(0,1] be such that ya∈ξ. Then ξ(y)≥a and y∈I. Since I is a left ideal of S and x≤y∈I, we have x∈I. Thus ξ(x)≥δ-k/2. If a≤δ-k/2, then ξ(x)≥a and so xa∈ξ. If a>δ-k/2, then ξ(x)+a+k>δ-k/2+δ-k/2+k=δ and so xaqkδξ. Therefore xa∈∨qkδξ.
Ifx,y∈S and a∈(0,1] be such that ya∈ξ. Then x,y∈I and so xy∈I. Thus ξ(xy)≥δ-k/2. If a>δ-k/2, then ξ(xy)+a+k>δ-k/2+δ-k/2+k=δ and so (xy)aqkδξ. If a≤δ-k/2, then ξ(xy)≥a and we have (xy)a∈ξ. Therefore (xy)a∈∨qkδξ. Consequently, ξ is an (∈,∈∨qkδ)-fuzzy left ideal of S.If we takeδ=1 in Theorem 6, then we get the following corollary.Corollary 7 (see [30]).
LetI be a left (resp., right) ideal of S and ξ, a fuzzy subset in S defined by (9)ξx=≥1-k2ifx∈I,0otherwise.
Then(1)
ξ is a (q,∈∨qk)-fuzzy ideal of S.(2)
ξ is an (∈,∈∨qk)-fuzzy ideal of S.Theorem 8.
A fuzzy subsetξ of S is an (∈,∈∨qkδ)-fuzzy left (resp., right) ideal of S if and only if(1)
(∀x,y∈S)(x≤y→ξ(x)≥ξ(y)∧δ-k/2),(2)
(∀x,y∈S)(ξ(xy)≥ξ(y)∧δ-k/2resp.ξxy≥ξx∧δ-k/2).Proof.
Suppose thatξ is an (∈,∈∨qkδ)-fuzzy ideal of S. On the contrary, assume that there exist x,y∈S, x≤y such that ξx<ξy∧δ-k/2. Choose a∈(0,1] such that ξ(x)<a≤ξ(y)∧δ-k/2. Then ya∈ξ, but ξ(x)<a and ξ(x)+a+k<δ-k/2+δ-k/2+k=δ, so xa∈∨qkδ¯ξ, a contradiction. Hence ξ(x)≥ξ(y)∧δ-k/2 for all x,y∈S with x≤y.
If there existx,y∈S such that ξ(xy)<ξ(y)∧δ-k/2, choose a∈(0,1] such that ξ(xy)<a≤ξ(y)∧δ-k/2. Then ya∈ξ but ξ(xy)<a and ξxy+a+k<δ-k/2+δ-k/2+k=δ, so (xy)aqkδ¯ξ. Thus, (xy)a∈∨qkδ¯ξ, a contradiction. Therefore, ξ(xy)≥ξ(y)∧δ-k/2 for all x,y∈S.
Conversely, letya∈ξ for some a∈(0,1]. Then ξ(y)≥a. Now, ξ(x)≥ξ(y)∧δ-k/2≥a∧δ-k/2. If a>δ-k/2, then ξ(x)≥δ-k/2 and ξ(x)+a+k>δ-k/2+δ-k/2+k=δ, and xaqkδξ follows. If a≤δ-k/2, then ξ(x)≥a and so xa∈ξ. Thus, xa∈∨qkδξ.
Letya∈ξ, then ξ(y)≥a and so ξ(xy)≥ξ(y)∧δ-k/2≥a∧δ-k/2. If a>δ-k/2, then ξ(xy)≥δ-k/2 and ξ(xy)+a+k>δ-k/2+δ-k/2+k=δ and so (xy)aqkδξ. If a≤δ-k/2, then ξ(xy)≥a implies that (xy)a∈ξ. Thus (xy)a∈∨qkδξ and consequently, ξ is an (∈,∈∨qkδ)-fuzzy left ideal of S. Similarly we can prove that ξ is an (∈,∈∨qkδ)-fuzzy right ideal of S.If we takeδ=1 in Theorem 8, then we have the following corollary [30].Corollary 9.
A fuzzy subsetξ of S is an (∈,∈∨qk)-fuzzy left (resp. right) ideal of S if and only if(1)
(∀x,y∈S)(x≤y→ξx≥ξy∧1-k/2),(2)
(∀x,y∈S)(ξ(xy)≥ξ(y)∧1-k/2respξ(xy)≥ξ(x)∧1-k/2).
Using Theorem8, we have the following characterization of fuzzy left (resp., right) ideals of ordered semigroups.Proposition 10.
Let(S,·,≤) be an ordered semigroup and ∅≠I⊆S. Then I is a left (resp., right) ideal of S if and only if the characteristic function χI of I is an (∈,∈∨qkδ)-fuzzy left (resp., right) ideal of S.Corollary 11.
A fuzzy subsetξ of an ordered semigroup S is an (∈,∈∨qkδ)-fuzzy left (resp., right) ideal of S if and only if it satisfies conditions (1) and (2) of Theorem 8.Remark 12.
Every fuzzy left (resp., right) ideal of an ordered semigroup is an(∈,∈∨qkδ)-fuzzy left (resp., right) ideal of S. But the converse is not true.Example 13.
Consider the ordered semigroup as given in Example5, and define a fuzzy subset ξ:S→[0,1] as follows: (10)
S
a
1
a
2
a
3
a
4
a
5
ξ
x
0.60
0.20
0.50
0.40
0.45
Thenξ is an (∈,∈∨qkδ)-fuzzy ideal with δ=0.8 but Uξ;a={a1,a3} for all a∈(0.45,0.50] is not an ideal of S. Hence by Theorem 2, ξ is not a fuzzy ideal of S for all a∈(0.45,0.50].Theorem 14.
Every(∈,∈)-fuzzy left (resp., right) ideal is an (∈,∈∨qkδ)-fuzzy left (resp., right) ideal of S.Proof.
The proof is straightforward and is omitted here.The converse of the above theorem is not true in general as shown in Example15.Example 15.
Consider the ordered semigroup as given in Example5, and define a fuzzy subset ξ:S→[0,1] as follows: (11)
S
a
1
a
2
a
3
a
4
a
5
ξ
x
0.80
0.20
0.70
0.50
0.60
Thenξ is an (∈,∈∨qkδ)-fuzzy ideal but not an (∈,∈)-fuzzy ideal of S. This is because (a1)0.50∈ξ but (a1a2)0.50=(a4)0.50∈¯ξ.Theorem 16.
Every(∈∨qkδ,∈∨qkδ)-fuzzy left (resp., right) ideal is an (∈,∈∨qkδ)-fuzzy left (resp., right) ideal of S.Proof.
Suppose thatξ is an (∈∨qkδ,∈∨qkδ)-fuzzy left ideal of S. Let x,y∈S, and a∈(0,1] be such that x≤y, ya∈ξ, then ya∈∨qkδξ. So by hypothesis, we have xa∈∨qkδξ. Now let x,y∈S, and a∈(0,1] be such that ya∈ξ. Then ya∈∨qkδξ. Hence xya∈∨qkδξ. Similarly we can prove that yxa∈∨qkδξ.Ifδ=1 in Theorem 16, then the following corollary [30] can be obtained.Corollary 17.
Every(∈∨qk,∈∨qk)-fuzzy left (resp., right) ideal is an (∈,∈∨qk)-fuzzy left (resp., right) ideal of S.
The importance of the following theorem is to provide the bridge between ordinary ideals and fuzzy ideals of type∈,∈∨qkδ. This linkage is very much essential for the characterization of ordered semigroups. Some time it is quite difficult to understand weather a particular ideal is fuzzy (resp., ordinary) ideal or not. In such a situation, the following result plays its role by providing the required information.Theorem 18.
A fuzzy subsetξ of S is an ∈,∈∨qkδ-fuzzy left (resp., right) ideal of S if and only if Uξ;a(≠∅) is left (resp., right) ideal of S for all a∈(0,δ-k/2].Proof.
Assume thatξ is an (∈,∈∨qkδ)-fuzzy left ideal of S. Let x,y∈S, x≤y be such that y∈U(ξ;a) for some a∈(0,δ-k/2]. Then ξ(y)≥a and by hypothesis (12)ξx≥ξy∧δ-k2≥a∧δ-k2=a.
Hencex∈U(ξ;a). Let x,y∈S be such that y∈U(ξ;a) for some a∈(0,δ-k/2]. Then ξ(y)≥a and we have (13)ξxy≥ξy∧δ-k2≥a∧δ-k2=a.Thus xy∈U(ξ;a).
Conversely, suppose thatU(ξ;a)(≠∅) is a left ideal of S for all a∈(0,δ-k/2]. If there exist x,y∈S, x≤y such that ξ(x)<ξ(y)∧δ-k/2, choose a∈(0,δ-k/2] such that ξ(x)<a≤ξ(y)∧δ-k/2, then, ξ(y)≥a implies that y∈U(ξ;a) but x∉U(ξ;a), a contradiction. Hence, ξ(x)≥ξ(y)∧δ-k/2. If x,y∈S such that ξ(xy)<ξ(y)∧δ-k/2, then choose a∈(0,δ-k/2], and ξ(xy)<a≤ξ(y)∧δ-k/2. Thus, y∈U(ξ;a) but xy∉U(ξ;a), a contradiction. Hence, ξ(xy)≥ξ(y)∧δ-k/2 for all x,y∈S,  k∈[0,1) and δ∈(0,1]. Therefore, ξ is an (∈,∈∨qkδ)-fuzzy left ideal of S. By a similar way, we can prove that ξ is an ∈,∈∨qkδ-fuzzy right ideal of S.Note thatqk1=qk. Therefore, if δ=1 in Theorem 18, then this new result can produce the following corollary of the literature [30].Corollary 19.
A fuzzy subsetξ of S is an ∈,∈∨qk-fuzzy left (resp., right) ideal of S if and only if Uξ;a(≠∅) is left (resp., right) ideal of S for all a∈(0,1-k/2].Proposition 20.
Letξ be a nonzero (∈,∈∨qkδ)-fuzzy left (resp., right) ideal of S. Then the set ξ0=x∈S∣ξx>0 is a left (resp., right) ideal of S.Proof.
Letξ be an (∈,∈∨qkδ)-fuzzy left ideal of S. If ,y∈S, x≤y, and y∈ξ0, then, ξ(y)>0. Since ξ is an (∈,∈∨qkδ)-fuzzy left ideal of S, we have (14)ξx≥ξy∧δ-k2>0,becauseξy>0.
Thusξ(x)>0 and so x∈ξ0. Let y∈ξ0, then, ξ(y)>0 and by hypothesis, (15)ξxy≥ξy∧δ-k2>0.
Thusxy∈ξ0. Consequently, ξ0 is a left ideal of S. Similarly, we can prove that ξ0 is a right ideal of S.Ifδ=1, then Proposition 20 leads to the following result [30].Result 1.
Letξ be a nonzero (∈,∈∨qk)-fuzzy left (resp., right) ideal of S. Then the set ξ0 is a left (resp., right) ideal of S.
In the following lemma, we establish a relationship between(∈,∈∨qkδ)-fuzzy left (resp., right) ideals and (∈,∈)-fuzzy left (resp., right) ideals of S.Lemma 21.
Suppose thatξ is an (∈,∈∨qkδ)-fuzzy left (resp., right) ideal of S such that ξ(x)<δ-k/2 for all x∈S. Then ξ is an (∈,∈)-fuzzy left (resp., right) ideal of S.Proof.
Letx,y∈S and a∈(0,1] be such that x≤y, ya∈ξ. Then ξy≥a. Since ξ is an (∈,∈∨qkδ)-fuzzy left ideal and ξ(x)<δ-k/2, we have (16)ξx≥ξy∧δ-k2≥a∧δ-k2=a.Hence, xa∈ξ. If x,y∈S and a∈(0,1] be such that ya∈ξ. Then ξy≥a and we have (17)ξxy≥ξy∧δ-k2≥a∧δ-k2=a.Thus, (xy)a∈ξ. By a similar way, we can prove that (yx)a∈ξ.Ifδ=1, then Lemma 21 leads to the following result [30].Result 2.
Ifξ is an (∈,∈∨qk)-fuzzy left (resp., right) ideal of S such that ξ(x)<1-k/2 for all x∈S, then ξ is an (∈,∈)-fuzzy left (resp., right) ideal of S.
For any fuzzy subsetξ of S and a∈(0,1], we denote Ξkδξ;a as follows: (18)Ξkδξ;a≔x∈S∣xaqkδξ,Ωkδξ;a≔x∈S∣xa∈∨qkδξ.Clearly Ωkδξ;a=Uξ;a∪Ξkδξ;a.
We callΩkδξ;a an ∈∨qkδ-fuzzy level left (resp., right) ideal I and Ξkδξ;a a qkδ-level left (resp., right) ideal of S.
We give another characterization of∈,∈∨qkδ-fuzzy left (resp., right) ideals by using Ωkδξ;a.Theorem 22.
A fuzzy subsetξ of S is an (∈,∈∨qkδ)-fuzzy left ideal of S if and only if Ωkδξ;a is a left (resp., right) ideal of S for all a∈(0,1].Proof.
Assumeξ is an (∈,∈∨qkδ)-fuzzy left (resp., right) ideal of S. Let x,y∈S, x≤y, and a∈(0,1] be such that y∈Ωkδξ;a. Then ya∈∨qkδξ, that is, ξy≥a or ξy+a+k>δ. Since ξ is an (∈,∈∨qkδ)-fuzzy left ideal of S and x≤y, we have ξx≥ξy∧δ-k/2. We discuss the following cases.
Case  1 ( ξ y ≥ a ). If a>δ-k/2, then ξx≥ξy∧δ-k/2=δ-k/2 and so (19)ξx+a+k>δ-k2+δ-k2+k=δ,so xaqkδξ. If a≤δ-k/2, then ξx≥ξy∧δ-k/2≥a. Hence xa∈ξ.
Case  2 ( ξ y + a + k > δ ). If a>δ-k/2, then (20)ξx≥ξy∧δ-k2>δ-a-k∧δ-k2=δ-a-k,that is, ξx+a+k>δ. Thus xaqkδξ. If a≤δ-k/2, then (21)ξx≥ξy∧δ-k2>δ-a-k∧δ-k2=δ-k2≥a.Therefore xa∈ξ. Hence xa∈∨qkδξ. Let x∈S and y∈Ωkδξ;a for a∈(0,1]. Then ya∈∨qkδξ, that is, ξy≥a or ξy+a+k>δ. Since ξ is an (∈,∈∨qkδ)-fuzzy left ideal of S, we have ξxy≥ξy∧δ-k/2.
Case  1 ( ξ y ≥ a ). If a>δ-k/2, then ξxy≥ξy∧δ-k/2=δ-k/2 and so (22)ξxy+a+k>δ-k2+δ-k2+k=δ,so xyaqkδξ. If a≤δ-k/2, then ξxy≥ξy∧δ-k/2≥a. Hence xya∈ξ.
Case  2 ( ξ y + a + k > δ ). If a>δ-k/2, then (23)ξxy≥ξy∧δ-k2>δ-a-k∧δ-k2=δ-a-k,that is, ξxy+a+k>δ. Thus xyaqkδξ. If a≤δ-k/2, then (24)ξxy≥ξy∧δ-k2>δ-a-k∧δ-k2=δ-k2≥a.Therefore, xya∈ξ. Hence xya∈∨qkδξ. Thus Ωkδξ;a is left ideal. Similarly, we can prove that yxa∈∨qkδξ. Consequently, Ωkδξ;a is left (resp., right) ideal of S.
Conversely, assume thatξ is a fuzzy subset of S and let a∈(0,1] be such that Ωkδξ;a is left ideal of S. If there exist x,y∈S, x≤y, and a∈(0,δ-k/2] such that (25)ξx<a≤ξy∧δ-k2,then y∈Uξ;a⊆Ωkδξ;a. Hence y∈Ωkδξ;a, since Ωkδξ;a is left ideal and x≤y. Thus x∈Ωkδξ;a and we have ξx≥a or ξx+a+k>δ, a contradiction. Hence ξx≥ξy∧δ-k/2 for all x,y∈S with x≤y. Let x,y∈S be such that (26)ξxy<a≤ξy∧δ-k2for some a∈(0,δ-k/2]. Then y∈Uξ;a⊆Ωkδξ;a. Hence y∈Ωkδξ;a, since Ωkδξ;a is left ideal. Thus xy∈Ωkδξ;a and we have ξxy≥a or ξxy+a+k>δ, a contradiction. Hence ξxy≥ξy∧δ-k/2 for all x,y∈S. Thus ξ is an (∈,∈∨qkδ)-fuzzy left ideal of S. Similarly we can prove that ξ is an (∈,∈∨qkδ)-fuzzy right ideal of S.Ifδ=1, then, a fuzzy subset ξ of S is an (∈,∈∨qk)-fuzzy left ideal of S if and only if Ωkξ;a is a left (resp., right) ideal of S for all a∈(0,1] [30].From Theorems2 and 22, Uξ;a and Ωkδξ;a are left ideals of S for all a∈(0,δ-k/2], but Ξkδξ;a is not left ideal of S for a∈(0,δ-k/2] in general, as shown in the next example.Example 23.
Consider an ordered semigroup as given in Example5, and define a fuzzy subset ξ:S→[0,1] as follows: (27)
S
a
1
a
2
a
3
a
4
a
5
ξ
x
0.60
0.40
0.50
0.30
0.10Then Ξkδξ;a={a1,a2,a3,a4} is left ideal for a∈(0.1,1-k/2], δ=0.8, and k=0.3 but neither Uξ;a or Ωkδξ;a is left ideal because of (a2)0.20qkδξ but a1a20.20=(a4)0.20qkδ¯ξ.


## 4. Characterization of Regular Ordered Semigroups byδ-Quasi-Coincident with Relation

Regular ordered semigroups are known to be the most important classes of semigroup theory. In this section, fuzzy quasi-ideals of type(∈,∈∨qkδ) are introduced. Further, the connection between (∈,∈∨qkδ)-fuzzy left (resp., right) ideals and (∈,∈∨qkδ)-fuzzy quasi-ideals is provided. The classification of regular ordered semigroups by the properties of (∈,∈∨qkδ)-fuzzy left (resp., right) ideals is also the milestone of the present section.Definition 24.
A fuzzy subsetξ of S is called an (∈,∈∨qkδ)-fuzzy quasi-ideal of S if it satisfies the conditions:(1)
(∀x,y∈S)(x≤y→ξ(x)≥ξ(y)∧δ-k/2),(2)
(∀x∈S)(ξ(x)≥((ξ∘1)∩(1∘ξ))(x)∧δ-k/2).Theorem 25.
Letξ be a nonzero (∈,∈∨qkδ)-fuzzy quasi-ideal of S, then the set ξ0=x∈S∣ξx>0 is a quasi-ideal of S.Proof.
Letx,y∈S be such that x≤y. If y∈ξ0, then ξ(y)>0. Since ξ is an (∈,∈∨qkδ)-fuzzy quasi-ideal of S, therefore (28)ξx≥ξy∧δ-k2>0,becauseξy>0.Thus ξ(x)>0 and so x∈ξ0. Let a∈(ξ0S]∩(Sξ0], then a∈(ξ0S] and a∈(Sξ0]. Hence there exist r,s∈S and x,y∈ξ0 such that a≤xs, and a≤ry. Then x,s,r,y∈Xa. Since Xa≠ϕ, we have (29)ξx≥ξ∘1∩1∘ξx∧δ-k2=⋁p,q∈Xaξp∧1q∧⋁q,p∈Xa1q∧ξp∧δ-k2≥ξx∧1r∧1r∧ξy∧δ-k2=ξx∧ξy∧δ-k2>0,becausex,y∈ξ0.Hence a∈ξ0 implies that (ξ0S]∩(Sξ0]⊆ξ0.The proof of the following proposition is straightforward and is omitted.Proposition 26.
Every(∈,∈∨qkδ)-fuzzy left (resp., right) or two-sided ideal of S is an (∈,∈∨qkδ)-fuzzy quasi-ideal of S.
In the following example, it is shown that the converse of Proposition26 is not true in general.Example 27.
Consider an ordered semigroupS={0,a,b,c} with multiplication Table 2 and order relation: (30)≤≔0,0,a,a,b,b,c,c,0,a,0,b,0,c.
The order relation is given by Figure2.
Then{0,a} is a quasi-ideal of S, but not a left (resp., right) ideal of S.
Define a fuzzy subsetξ:S→0,1 as follows: (31)
S
0
a
b
c
ξ
x
0.25
0.25
0
0Then ξ is fuzzy quasi-ideal of S but not an (∈,∈∨qkδ)-fuzzy left ideal of S for δ=0.8 and k=0.4 since ξca=ξc=0<ξa∧δ-k/2.Table 2
Multiplication table of ordered semigroupS.
⋅ 0 a b c 0 0 0 0 0 a 0 a b 0 b 0 0 0 0 c 0 c 0 0Figure 2
Order relation ofS.Definition 28.
LetS be an ordered semigroup and ξ,G fuzzy subsets of S. Then the δ-k/2-product of ξ and G is defined by (32)ξ∘δ-k/2Ga=⋁y,z∈Xaξy∧Gz∧δ-k2ifXa≠∅,0ifXa=∅.Proposition 29.
LetS,·,≤ be an ordered semigroup and ξ1,ξ2,G1,G2 fuzzy subsets of S such that ξ1⊆ξ2 and G1⊆G2. Then ξ1∘δ-k/2G1⊆ξ2∘δ-k/2G2.Lemma 30.
Ifξ and G are (∈,∈∨qkδ)-fuzzy left (resp., right) ideals of an ordered semigroup S, then ξ∩δ-k/2G is an (∈,∈∨qkδ)-fuzzy left (resp., right) ideal of S, where ξ∩δ-k/2Gx≔ξx∧Gx∧δ-k/2 for all x∈S.Proof.
Letx,y∈S such that x≤y. Since ξ and G are (∈,∈∨qkδ)-fuzzy left ideals of S, hence ξx≥ξy∧δ-k/2 and Gx≥Gy∧δ-k/2. Therefore (33)ξ∩1-k/2Gx=ξx∧Gx∧δ-k2≥ξy∧δ-k2∧Gy∧δ-k2∧δ-k2=ξy∧Gy∧δ-k2∧δ-k2=ξ∩δ-k/2Gy∧δ-k2.Hence (ξ∩δ-k/2G)x≥(ξ∩δ-k/2G)y∧δ-k/2.
Letx,y∈S. Since ξ and G are (∈,∈∨qkδ)-fuzzy left ideals of S. Then, ξxy≥ξy∧δ-k/2 and Gxy≥Gy∧δ-k/2. Thus (34)ξ∩δ-k/2Gxy=ξxy∧Gxy∧δ-k2≥ξy∧δ-k2∧Gy∧δ-k2∧δ-k2=ξy∧Gy∧δ-k2∧δ-k2=ξ∩δ-k/2Gy∧δ-k2.Hence ξ∩δ-k/2G is an (∈,∈∨qkδ)-fuzzy left ideal of S. Similarly we can prove that ξ∩δ-k/2G is an (∈,∈∨qkδ)-fuzzy right ideal of S.Letξ and G be fuzzy subsets of S; we define ξ∪G as follows: (35)ξ∪δ-k/2Gx=ξx∨Gx∧δ-k2for all x∈S.Puttingδ=1 in Lemma 30 leads to a result [30]: “If ξ and G are (∈,∈∨qk)-fuzzy left (resp., right) ideals of an ordered semigroup S, then ξ∩1-k/2G is an (∈,∈∨qk)-fuzzy left (resp., right) ideal of S”.Lemma 31.
Ifξ is an (∈,∈∨qkδ)-fuzzy right of S and G is an (∈,∈∨qkδ)-fuzzy left ideal of S, respectively, then ξ∘δ-k/2G⊆ξ∩δ-k/2G.Proof.
Leta∈S. If Xa=ϕ, then ξ∘δ-k/2Ga=0≤ξ∩δ-k/2Ga. Assume that Xa≠ϕ, then (36)ξ∘δ-k/2Ga≔⋁y,z∈Xaξy∧Gz∧δ-k2.Since a≤yz, and ξ is an (∈,∈∨qkδ)-fuzzy right and G is an (∈,∈∨qkδ)-fuzzy left ideal of S, thus (37)ξa≥ξyz∧δ-k2≥ξy∧δ-k2∧δ-k2=ξy∧δ-k2,Ga≥Gyz∧δ-k2≥ξy∧δ-k2∧δ-k2=Gy∧δ-k2.Hence, ξy∧Gy∧δ-k/2≤ξa∧Ga∧δ-k/2=(ξ∩δ-k/2G)a.Regular ordered semigroups are characterized by right and left ideals in [5]. In the following we give a characterization of regular ordered semigroups in terms of (∈,∈∨qkδ)-fuzzy right and (∈,∈∨qkδ)-fuzzy left ideals.Lemma 32 (see [5]).
LetS be an ordered semigroup. Then the following statements are equivalent:(1)
S is regular.(2)
R∩L=(RL] for every right ideal R and every left ideal L of S.Lemma 33.
LetS be an ordered semigroup and ϕ≠A, B⊆S. Then, we have(i)
(χA∘δ-k/2χB)a=χ(AB]a∧δ-k/2 for all a∈S.(ii)
χA∩δ-k/2χB=χA∩B∧δ-k/2.(iii)
χA∪δ-k/2χB=χA∪B∧δ-k/2.Theorem 34.
An ordered semigroupS is regular if and only if, for every (∈,∈∨qkδ)-fuzzy right ideal ξ and every (∈,∈∨qkδ)-fuzzy left ideal G of S, one has ξ∘δ-k/2G=ξ∩δ-k/2G.Proof.
LetS be regular ordered semigroup and a∈S. Then there exists x∈S such that a≤axa. Hence ax,a∈Xa and we have (38)ξ∘δ-k/2Ga=⋁y,z∈Xaξy∧Gz∧δ-k2≥ξax∧Ga∧δ-k2.Since ξ is an (∈,∈∨qkδ)-fuzzy right ideal of S, therefore ξax≥ξa∧δ-k/2. Thus (39)ξax∧Ga∧δ-k2≥ξa∧δ-k2∧Ga∧δ-k2=ξa∧Ga∧δ-k2=ξ∩δ-k/2Ga.Hence ξ∩δ-k/2Ga≤(ξ∘δ-k/2G)a. On the other hand, by Lemma 31, we have (ξ∘δ-k/2G)a≤(ξ∩δ-k/2G)a. Therefore (ξ∘δ-k/2G)a=(ξ∩δ-k/2G)a.
Conversely, suppose for every(∈,∈∨qkδ)-fuzzy right ideal ξ and every (∈,∈∨qkδ)-fuzzy left ideal G of S we have ξ∘δ-k/2G=ξ∩δ-k/2G. By Lemma 32, we need to show that R∩L=(RL] for every right ideal R and left ideal L of S. Let y∈R∩L, then y∈R and y∈L. Since R is right ideal and L is left ideal of S, thus by Lemma 33, χR is a fuzzy right and χL is a fuzzy left ideal S. Therefore by Proposition 10, χR is an (∈,∈∨qkδ)-fuzzy right ideal and χL is an (∈,∈∨qkδ)-fuzzy left ideal of S. By hypothesis, (χR∘δ-k/2χL)y=(χR∩δ-k/2χL)y. Since y∈R and y∈L so χRy=1 and χLy=1, then (χR∩δ-k/2χL)y=χRy∧χLy∧δ-k/2=1∧1∧δ-k/2=δ-k/2. It follows that (χR∘δ-k/2χL)y=δ-k/2. By Lemma 33, (χR∘δ-k/2χL)y=χ(RL]y∧δ-k/2. Thus χ(RL]y=δ-k/2. It follows that y∈(RL]. Hence R∩L⊆(RL] but (RL]⊆R∩L, always holds. Therefore R∩L=(RL]. Consequently S is regular.


## 5. Conclusion

Since the traditional mathematical tools are dichotomous in nature, i.e., yes or no type rather than more or less type, therefore, these tools can not be used to handle complex problems involving uncertainties of various applied fields like decision-making problems, control engineering, coding theory, theoretical physics, computer sciences, and fuzzy automata theory. For the said problems, fuzzy sets and their various generalizations are used. A quasi-coincident with relation is one of the most important relations used in algebraic structures along with a verity of applications in the aforementioned areas. The aim of the present paper was to introduce a comprehensive type of relation known asδ-quasi-coincident with relation which can lead to the existing systems and also some new types of subsystems involving quasi-coincident with relation. More precisely, ∈,∈∨qkδ-fuzzy left (resp., right, quasi-) ideals of S are developedwhich were elaborated through examples. The bridging of ordinary ideals and ∈,∈∨qkδ-fuzzy ideals of an ordered semigroup S is constructed while regular ordered semigroups were characterized through ∈,∈∨qkδ-fuzzy left (resp., right, quasi-) ideals. As δ∈(0,1], therefore for particular value δ=1, our algebraic system reduced to [30].



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