Completely weakly quasi-regular semigroups characterized by soft union Quasi ideals generalized bi-i

Abstract

Keywords: Soft Set; Soft Union Quasi-ideals; Soft Union (Generalized) Bi-ideals; Soft Union Semiprime Ideals; Completely Regular Semigroups; Soft Union Semiprime Ideals; Completely Regular Semigroups

Introduction

Molodtsov [1] introduced the principle concept of soft set to find solutions for uncertainty and vagueness problems in 1999. Since then, set theoretical aspects of soft sets especially operations of soft sets are studied in [2], [3], [4]. The theory of soft set has also many applications in different kinds of algebraic structures such as groups,

semigroups, rings, semirings, near-rings, BCK/BCI algebras and BL-algebras [5–14]. In [15], Dar and Ali described the structure of generalized Jordan *-derivations of prime rings with involution. Further as a consequence, it was shown that any generalized Jordan *-derivation on a prime ring with involution is generalized X-inner, provided R is not a PI-ring. In cite [16],

*Corresponding author. *E-mail address: aslihan.sezgin@amasya.edu.tr This paper was recommended for publication in revised form by Regional Editor Sania Qureshi Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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Dar and Khan studied the generalized derivations in rings with involution which behave like strong commutativity preserving mappings. In [17], certain identities involving multiplicative (generalized)-derivations on some appropriate subsets of the ring R was investigated. The results obtained in the paper [17] extends, unifies and complements several known results, since a great deal of calculation with commutators and anti-commutators are done. In [18], Abu Arqub and Al-Smadi discussed a new definition of fuzzy fractional derivative, so-called fuzzy conformable and they proposed fuzzy conformable fractional integral softly. Also, uniqueness, existence, and other properties of solutions of certain fuzzy conformable fractional differential equations under strongly generalized differentiability were utilized. Furthermore, all needed requirements for characterizing solutions by equivalent systems of crisp conformable fractional differential equations were discussed. In [19], Abu Arqub et al. proposed a new method for solving fuzzy differential equations based on the reproducing kernel theory under strongly generalized differen- tiability. In [20], the analytic and approximate solutions of second-order, two-point fuzzy boundary value problems based on the reproducing kernel theory under the assumption of strongly generalized differentiability were discussed. In [21], Abu Arqub proposed the reproducing kernel Hilbert space method to obtain the exact and the numerical solutions of fuzzy Fredholm-Volterra integro-differential equations. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions in which the constraint initial condition is satisfied. In the early 20th century, the researchers started to study the formal properties of semigroups since semigroups have a vital applications in language theory, coding theory, combinatorics, automata theory and mathematical analysis. A semigroup is an algebraic structure consisting of a nonempty set S together with an associative binary operation [24]. The theory of finite semigroups has been of particular importance in theoretical computer science since the 1950s because of the natural link between finite semigroups and finite automata via the syntactic monoid. In other areas of applied mathematics, semigroups are fundamental models for linear time-invariant systems. While in partial differential equations, a semigroup is associated to any equation whose spatial evolution is independent of time; in probability theory, semigroups are associated with well-konwn Markov processes. The researchers not only studied the general structure of the semigroup, but also their ideals and different kinds of semigroups as regards regularities such as regular semigroups, intra- regular semigroups, completely regular semigroups, quasi-regular semigroups and weakly regular semigroups [24–29]. In [30], Sezgin defined soft union semigroups, soft union left (right, two- sided) ideals, bi-ideals and soft semiprime ideals of a semigroup and obtained their basic properties. Also, regular and intra-regular semigroups were characterized by soft union semigroups and soft union left

(right, two-sided) ideals and soft union bi-ideals. In [31], soft union interior ideals, quasi-ideals, generalized bi-ideals are defined, their basic properties with respect to soft set operations and soft union product are obtained and the interrelations of them are investigated. Also, regular and intra-regular semigroups are characterized by the properties of soft union interior ideals, soft union quasi-ideals and soft union generalized bi-ideals. In this paper, soft right (left) ideals, soft quasi-ideals, soft (generalized) bi-ideals of a semigroup and soft union semiprime ideals are characterized by completely regular, weakly regular and quasi-regular semigroups. In Section 2, some basic definitions about semigroups, soft sets, different kinds of soft union ideals of a semigroup and their interrelations are recalled. In Section 3, completely regular semigroups are characterized as regards soft union quasi-ideals, soft union (generalized) bi-ideals and soft union semiprime ideals. It is shown that if every soft union quasi-ideal of S is soft union semiprime, then every quasi-ideal of S is semiprime and so if every quasi-ideal of S is semiprime, then S is completely regular. In Section 4, quasi-regular semigroups are studied in terms of soft union left (right) ideals, soft union quasi ideals and soft union (generalized) bi-ideals. In Section 5, weakly regular semigroups are characterized by soft union quasi- ideals and soft union bi-ideals of a semigroup. Our intent in this paper is to bring a new perspective to the regularities of semigroup theory via soft set theory.

Preliminaries

In this part, some basic conceptions related to semigroups and certain kinds of semigroups, soft sets, soft union left (right) ideals, soft union quasi-ideals and soft union (generalized) bi-ideals of a semigroup are recalled. S denotes a semigroup throughout this paper. A semigroup is an algebraic structure consisting of a set together with an associative binary operation. A nonempty subset C of S is called a subsemigroup of S if CC ⊆ C; generalized bi-ideal of S if CSC ⊆ C; a quasi-ideal of S if CS ∩ SC ⊆ C; semiprime if ∀a ∈ S, a2 ∈ C implies that a ∈ C. A subsemigroup D of S is called a bi-ideal of S if DSD ⊆ D. Quasi-ideal of S generated by a ∈ S is defined as following: Q[a] = {a} ∪ (aS ∩ Sa) and denoted by Q[a]. An element b of S is completely regular, if there exists an element y in S such that b = byb and by = yb. If all element of S is completely regular, then S is called a completely regular semigroup. S is left regular if for each element b of S, there exists an element y in S such that b = yb2 if for each element b of S, there exists an element y in S satisfying b = b2y, then S is called right regular semigroup. S is left (right) quasi-regular if all left (right) ideal of S is idempotent and is called quasi-regular if all both left ideal and right ideal of S is idempotent ([28]). For the other definitions of quasi-regular semigroups, we refer to [22, 23]. S is weakly-regular if for all y in S, y (yS)2. S is called intra-regular if for all element b of S there exist elements y and z in S such that b =

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yb2z. For more about semigroups, ideals of semigroups and regularity of semigroups, we refer to [24–29]. From now on, U refers to an initial universe, E is a set of parameters, P (U) is the power set of U and A, B, C ⊆ E. Definition 1 A soft set fA over U is a set given by fA : E → P (U ) such that fA(x) = ∅ if x is not in A. We can represent a soft set over U by the set of ordered pairs as following: fA = {(x, fA(x)) : x ∈ E, fA(x) ∈ P (U)} [1, 32]. It is clear to see that a soft set is a parametrized family of subsets of the set U. It is worth noting that the sets fA(x) may be arbitrary. Some of them may be empty, some may have nonempty intersection. If we define more then one soft set in a subset A of the set of parameters E, then the soft sets will be denoted by fA, gA, hA etc. If we define more then one soft set in some subsets A, B, C etc. of parameters E, then the soft sets will be denoted by fA, fB, fC etc., respectively. From now on, all the soft sets over U is denoted by S(U) and all the soft sets defined in this paper is the element of S(U). Definition 2 A soft set by fA is called a soft subset of fB and denoted by fA fB if fA(x) ⊆ fB(x) for all x ∈ E and is called a soft supset of fB and denoted by fA fB if fA(x) ⊇ fB(x) and is called soft equal to fB and denoted by fA = fB if fA(a) = fB(x) for all x ∈ E [32] . Definition 3 Let fA and fB be soft sets over U. Then, union of fA and fB, denoted by fA fB, is defined as by (fA fB)(a)=fA(a) ∪ fB(a) for all a ∈ E [32]. Definition 4 Let fS and gS be soft sets over U. Then, soft union product fS ∗ gS is defined by

Definition 11 Let X be a subset of S. The soft characteristic function of the complement X, denoted by SXc, is defined as

for every a ∈ S [30]. In [30], it is proved that soft union product is associative. Definition 5 A soft set fS is called a soft union semigroup of S, if fS(ab) ⊆ fS(a) ∪ fS(b) for all x, y ∈ S [30]. Definition 6 A soft set fS is called a soft union left ideal of S over U if if fS(ab) ⊆ fS(b) and is called a soft union right ideal of S over U if fS(ab) ⊆ fS(a) for all a, b ∈ S. A soft set fS is a soft union two-sided ideal (soft union ideal) of S if it is both soft union left and soft union right ideal of S [30]. If fS(x) = ∅ for all x ∈ S, then fS is a soft union left (right) ideal of S over U. We denote such a kind of soft union left and ∗ fS is obvious. And fS is (right) ideal by . fS ∗ an soft union left ideal of S if and only if ∗ fS fS and soft fS [30]. union right ideal of S over U if and only if fS ∗ Definition 7 A soft union semigroup fS is called a soft union bi-ideal of S over U if fS(abc) ⊆ fS(a) ∪ fS(c) for all a, b, c ∈ S [30]. Definition 8 A soft set fS is called a soft union quasi-ideal of S over U if fS (fS∗ ) ( ∗ fS) [31]. Definition 9 A soft set fS is called a soft union generalized bi-ideal of S over U if fS(abc) ⊆ fS(a) ∪ fS(c) for all a, b, c ∈ S [31]. Definition 10 A soft set fS is called soft union semiprime if fS(a) ⊆ fS(a2) for all a ∈ S [30].

[30]. In [31], it is proved that when X is a nonempty subset of S, X is a quasi-ideal of S if and only if SXc is a soft union quasi-ideal of S over U. In [31], it is also proved that all soft union bi-ideals of S are also soft union generalized bi-ideals of S, all soft union quasi-ideals of S are also soft union bi-ideals of S, and all soft union left (right) ideals of S are also soft union quasi-ideals of S.

Completely Regular Semigroups

In this part, a completely regular semigroup is characterized as regards soft quasi-ideals, soft (generalized) bi-ideals of a semigroup and soft union semiprime ideals. In [33], for a semigroup S, it is proved that the condition where S is completely regular is equivalent to the condition where S is both left and right regular, namely, x ∈ Sx2 and x ∈ x2 S for all x∈S. Hence, we have the following:

Theorem 1 If S is a completely regular semigroup, then for every soft union generalized bi-ideal kS of S, kS(x) = kS(x2) for oll all x ∈ S.

Proof Let S be a completely regular semigroup. Assume that kS is a soft union generalized bi-ideal of S. Then, there is an element y ∈ S such that x = x2yx2 since S is completely regular by hypothesis. Hence, kS(x) =kS(x2yx2) ⊆kS(x2)∪kS(x2)

Hence, kS(x) ⊆kS(x2). Now, kS(x2) = kS(xx) = kS(x(x2yx2) = kS(x(x2yx)x) ⊆ kS(x) ∪ kS(x) = kS(x).

Thus, kS(x2) ⊆ kS(x). By double-sided inclusion, kS(x) = kS(x2). Theorem 2 Let S be a semigroup. If every soft union quasi-ideal of S is soft union semiprime, then every quasi-ideal of S is semiprime. Proof Suppose that every soft union quasi-ideal of S is soft union semiprime and assume that Q is a quasi-ideal of S. Let x2 ∈ Q and x ∉ Q. As the soft characteristic function SQc is a soft union quasi-ideal of S, by hypothesis it is soft union semiprime. Thus, by Definition 4, SXc (x) = U and

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SQc (x2)= ∅. Now, by the definition of soft union semiprime, SQc (x) = U⊆SQc (x2) = ∅. But this is a contradiction. Thus, x ∈ Q and hence Q is semiprime, implying that every quasi-ideal of S is semiprime. Theorem 3 Let S be a semigroup. If every quasi-ideal of S is semiprime, then S is completely regular. Proof Suppose that every quasi-ideal of S is semiprime and let x ∈ S. Since the principal ideal Q[x2] generated by x2 is a quasi-ideal and by the hypothesis a semiprime, and since x2 ∈ Q[x2], then x ∈ Q[x2] by the definition of semiprime ideal. It is known that the soft characteristic function S(Q[x2])c is a soft union quasi-ideal of S when Q[x2] is quasi-ideal [31]. Then, S(Q[x2])c(x2)= S(Q[x2]) c(x)= ∅ implying that x ∈ Q[x2]= { x2} ∪ (x2S ∩ S x2). Since x ≠ x2 and so x ∉ {x2}, then x ∈(x2S ∩ S x2) implying that S is completely regular according to [33]. Theorem 4 Let S be a semigroup S. Then, the followings are equivalent: 1) S is completely regular. 2) kS(x) = kS(x2) for every soft union generalized bi-ideal kS of S and for all x ∈ S. 3) kS(x) = kS(x2) for every soft union bi-ideal kS of S and for all x ∈ S. 4) kS(x) = kS(x2) for every soft union quasi-ideal kS of S and for all x ∈ S. 5) Every soft union generalized bi-ideal of S is soft union semiprime. 6) Every soft union bi-ideal of S is soft union semiprime. 7) Every soft union quasi-ideal of S is soft union semiprime. 8) Every generalized bi-ideal of S is semiprime. 9) Every bi-ideal of S is semiprime. 10) Every quasi-ideal of S is semiprime. Proof By Theorem 1, Theorem 2 and Theorem 3, (1) implies (2), (7) implies (10) and (10) implies (1), respectively. Since every soft union bi-ideal S of is a soft union generalized bi-ideal of S, (2) implies (3), (5) implies (6) and (8) implies (9). And since every soft union quasi-ideal of S is a soft union bi-ideal of S, (3) implies (4), (6) implies (7) and (9) implies (10). And by the definition of soft union semiprime, (4) implies (7), (3) implies (6), (2) implies (5). This completes the proof.

Theorem 5. First, assume that kS is a soft union right ideal of S. Since kS is a soft union right ideal of S, kS ∗ kS holds [30]. Thus, ( ∗ kS) (kS ∗ ) kS ∗ kS. Hence, kS is a soft union quasi-ideal of S. Now, let kS = (kS ∗ )2 ( ∗ kS)2 holds for the soft union quasi ideal kS of S. Since kS is a soft union right ideal of S and kS ∗ kS holds, we have kS = (kS ∗ )2 ∪˜( ∗kS)2 (kS ∗ )2 = (kS ∗ ) ∗(kS ∗ ) kS ∗ kS .

Hence, kS ⊇ kS ∗ kS. Now, we need to show that kS ∗ kS kS. Since kS θ and kS is a soft right ideal of S, kS ∗ kS kS ∗ θ kS and so by double inclusion kS = (kS)2. Thus, we obtain that every soft right ideal of S is idempotent, so S is right quasi-regular by Theorem 5. One can similarly show the left quasi-regularity. Thus, S is quasi-regular. Theorem 7 If S is both left quasi-regular and intra-regular, then lS mS kS lS∗ mS ∗ kS for every soft union generalized bi-ideal kS, for every soft union left ideal lS and every soft union right ideal mS of S. Proof Suppose that S is both left quasi-regular and intra-regular. Let kS be any soft union generalized bi-ideal, lS be any soft union left ideal and mS be any soft union right ideal of S and x be any element of S. Since S is intra- regular, there exist elements a, b ∈ S such that x = ax2b. Also, as S is left quasi-regular, there exist k, l ∈ S such that x = kxlx. Therefore, (4)

Quasi-Regular Semigroups

In this part, quasi-regular semigroups are studied in terms of soft union left (right) ideals, soft union quasi-ideals and soft union (generalized) bi-ideals of a semigroup S. In ([28]), it is proved that S is left (right) quasi-regular if and only if x ∈ S x S x (x ∈ x S x S), namely, there exist elements a, b ∈ S such that x = axbx (x = xaxb). Theorem 5 [30] A semigroup S is left (right) quasi-regular if and only if every soft union left (right) ideal is idempotent. Proof In order to show that S is quasi-regular, we need to show that every soft union ideal of S is idempotent by

Theorem 8 If lS mS kS lS∗ mS ∗ kS for every for every soft union quasi-ideal kS, for every soft union left ideal lS and every soft union right ideal mS of S, then S is both intra-regular and left quasi-regular. Proof Suppose that lS mS kS lS∗ mS ∗ kS holds for every soft union-quasi- ideal kS, for every soft union left ideal lS and every soft union ideal mS of S. Since lS is a soft union left ideal of S, ∗ lS lS holds. Thus, ( ∗ lS) (lS ∗

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) ( ∗ lS) lS. Hence, lS is a soft union quasi-ideal of S. Moreover, since it self is a soft union right ideal of S, lS = lS mS kS = lS ∗ ∗ l = lS ∗ ( ∗ lS) lS ∗ lS

Hence, we have lS lS ∗ lS for the soft union left ideal lS of S. Now, lS ∗ lS θ ∗ lS lS. Hence, we obtained that lS ∗ lS lS. By double-sided inclusion, lS = lS∗lS for the soft union left ideal lS of S. Hence S is left quasi-regular by Theorem 5. Now, since itself is a soft union left ideal of S, hence soft union quasi-ideal of S, lS mS = lS mS = lS∗ mS ∗ = lS ∗ (mS ∗ ) lS ∗ mS.

Theorem 9 Let S be a semigroup S. Then, the followings are equivalent: 1) S is both left quasi-regular and intra-regular. 2) lS mS kS lS∗ mS ∗ kS for every union quasi-ideal kS, for every soft union left ideal lS and every soft union right ideal mS of S. 3) lS mS kS lS∗ mS ∗ kS for every soft union bi-ideal kS, for every soft union left ideal lS and every soft union right ideal mS of S. 4) lS mS kS lS∗ mS ∗ kS for every soft union generalized bi-ideal kS, for every soft union left ideal lS and every soft union right ideal mS of S. Proof (1) implies (4) is from Theorem 7, and (2) implies (1) is from Theorem 8. Since, every soft union bi-ideal S of is a soft union generalized bi-ideal of S, (4) implies (3), and since every soft union quasi-ideal of S is a soft union bi-ideal of S, (3) implies (2).

Weakly Regular Semigroups

In this part, weakly regular semigroup is characterized as regards soft union quasi-ideals and soft union (generalized) bi-ideals of a semigroup. Theorem 10 [30] The following conditions are equivalent for a monoid S: 1) S is weakly regular. 2) kS lS kS ∗ lS for every soft union right ideal kS of S and for every soft union ideal lS of S. Theorem 11 Let S be a monoid. If S weakly regular, then kS lS kS∗ lS for every soft union generalized bi-ideal kS of S and for every soft union ideal lS of S. Proof Assume that S is a weakly regular monoid and kS lS kS∗ lS holds for every soft union generalized bi-ideal kS of S, for every soft union ideal lS of S. Let x S. Then, a ∈ (aS)2. Hence, a = axay for some x, y∈S. Also, since

Theorem 12 Let S be a monoid. If kS lS kS ∗ lS for every soft union quasi- ideal kS of S and for every soft union ideal lS of S, then S is weakly regular. Proof Let S be a monoid and kS lS kS ∗ lS hold for every soft union quasi- ideal kS of S and for every soft union ideal lS of S. Since every soft union right ideal of S is a soft union quasi-ideal of S, then kS lS kS ∗ lS holds for every soft union right kS of S and for every soft union ideal lS of S. Hence, S is a weakly regular semigroup by Theorem 10. Theorem 13 The following conditions are equivalent for a monoid S: 1)S is weakly regular. 2)kS lS kS ∗ lS for all soft union generalized bi-ideal kS of S and for all soft union ideal lS of S. 3) kS lS kS ∗ lS for all soft union bi-ideal kS of S and for all soft union ideal lS of S. 4) kS lS kS ∗ lS for all soft union quasi-ideal kS of S and for all soft union ideal lS of S. Proof (1) implies (2) is by Theorem 11. Since every soft union bi-ideal of S is a soft union generalized bi-ideal of S, (2) implies (3) and since every soft union quasi ideal of S is a soft union bi-ideal of S, (3) implies (4) and by Theorem 12, (4) implies (1). Theorem 14 Let S be a monoid. If S is weakly regular, then kS lS mS kS ∗ lS ∗ mS for all soft union generalized bi-ideal kS, for all soft union ideal lS and for all soft union right ideal mS of S. Proof Assume that S is a weakly regular monoid and kS lS mS kS ∗ lS ∗ mS holds for all soft union generalized bi-ideal kS, for all soft union ideal lS and for all soft union right ideal mS of S. Let a ∈ S. Then, a ∈ (aS)2. Hence, a = axay for some x, y ∈ S. Also, since xay = x(axay)y = (xax) (ay2).

(9) Theorem 15 Let S be a monoid. If kS lS mS kS ∗ lS ∗ mS for all soft union quasi-ideal kS, for all soft union ideal lS and for all soft union right ideal mS of S, then S is weakly regular.

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Proof Let kS lS mS kS ∗ lS ∗ mS hold for all soft union quasi ideal of kS, soft union ideal lS and soft union right ideal mS of S. Since soft union right ideal mS of S is also a soft union quasi-ideal of S, the soft union ideal lS of S is also a soft union right ideal of S and θ itself is a soft union ideal of S, lS =mS θ lS = mS ∗ ∗ lS = (mS ∗ ) ∗ lS mS ∗ lS.

Thus, S is weakly regular by Theorem 10. Theorem 16 The following conditions are equivalent for a monoid S: 1) S is weakly regular. 2) kS lS mS kS ∗ lS ∗ mS for all soft union generalized bi-ideal kS of S, for all soft union ideal lS of S and for all soft union right ideal mS of S. 3) kS lS mS kS ∗ lS ∗ mS for all soft union bi-ideal kS of S, for all soft union ideal lS of S and for all soft union right ideal mS of S. 4) kS lS mS kS ∗ lS ∗ mS for all soft union quasi-ideal kS of S, for all soft union ideal lS of S and for all soft union right ideal mS of S. Proof (1) implies (2) is by Theorem 14. Since every soft union bi-ideal of S is a soft union generalized bi-ideal of S, (2) implies (3) and since every soft union quasi-ideal of S is a soft union bi-ideal of S (3) implies (4). (4) implies (1) is by Theorem 15.

Conclusion

In this paper, we have characterized completely regular, weakly regular, quasi- regular semigroups by soft union quasi-ideals, soft union (generalized) bi-ideals of a semigroup and soft union semiprime ideals. A different point of view has been brought to semigroup theory as regards regularity via soft set theory. In the future works, completely regular, weakly regular, quasi-regular semigroups can be characterized by soft union tri quasi ideals, soft union m-quasi ideals, soft union m-bi-ideals, soft union left bi-quasi ideals, soft union lateral bi-quasi ideals, soft union right bi-quasi ideals, soft union left tri-quasi ideals, soft union lateral tri-quasi ideals, soft union right tri-quasi ideals of a semigroup.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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