Group and semigroup theory

In this paper, we introduce the concept of tri-quasi hyperideal in Γ-semihyperring
generalizing the classical ideal, left ideal, right ideal, bi-ideal, quasi ideal, interior ideal,
bi-interior ideal, weak interior ideal, bi-quasi ideal, tri-ideal, quasi-interior ideal and biquasi-
interior ideal of Γ-semihyperring and semiring. Furthermore, charecterizations of
Γ-semihyperring, regular Γ-semihyperring and simple Γ-semihyperring with relative triquasi
hyperideals are provided discussing the characteristics of Γ-semihyperring of relative
tri-quasi hyperideals.

The notion of a hesitant fuzzy semigroup with a frontier is introduced, and several properties are investigated. Characterizations of a hesitant fuzzy semigroup with a frontier are considered, and a condition for a special set to be a subsemigroup is provided. The hesitant union and hesitant intersection of two hesitant fuzzy semigroups with a frontier are dealt with, and the hesitant fuzzy pre-image and hesitant fuzzy image of a hesitant fuzzy semigroup with a frontier under the homomorphism are discussed.

The concept of 'weak finitely regular relations' was introduced by Shuzhen Luo and Xiaoquan Xu in 2019 and applied to partially ordered sets. In this article, this idea is extended to weak finality of dually quasi-conjugative relations was introduced in 2013 by this author.

In this paper, concepts of quasi-normal and dually quasi-normal relations are introduced. Characterizations of these relations are obtained. In addition, particulary we show that the anti-order relation (= C) is a (dually) quasi-normal relation if and only if the partially ordered set (X,) is an anti-chain.

Following Jiang Guanghao and Xu Luoshan?s concept of conjugative, dually conjugative, normal and dually normal relations on sets, the concept of quasi-regular relations is introduced. Characterizations of quasi-regular relations are obtained and it is shown when an anti-order relation is quasiregular. Some nontrivial examples of quasi-regular relations are given. At the end we introduce dually quasi-regular relations and give a connection between these two types of relations.

In this paper, we investigate such a regular elements α and idempotents of the complete semigroup of binary relations B X (D) defined by semilattices of the class lower incomplete nets, for which V (D, α) = Q. Also we investigate right units of the semigroup B X (Q). For the case where X is a finite set we derive formulas by means of which we can calculate the numbers of regular elements, idempotents and right units of the respective semigroup.

In this paper, we have applied rough set theory to ordered Γ-ideals mainly quasi(bi)-Γideals in ordered LA Γ-semigroups by using pseudoorder of relations. Based on this relation we have studied properties of rough ideals, rough interior ideals, rough quasi-ideals, rough bi-ideals and rough prime ideals in ordered LA-Γ-semigroups and have shown that lower and upper approximations of an LA-Γ-subsemigroups (resp., left ideals, right ideals, interior ideals, two-sided ideals, quasi-ideals, bi-ideals and prime ideals) in an ordered LA-Γ-semigroup is an LA-Γ subsemigroups (resp., left ideals, right ideals, interior ideals, two-sided ideals, quasi-ideals, bi-ideals and prime ideals).

Using the characteristic family of sets, the characteristic mapping and base sources of , we characterize the class whose elements are each isomorphic to. We generate some advanced formulas in order to calculate the number of regular elements of () satisfying (,) = , in an e cient way.

Difficulties encountered in studying generators of semigroup () X B D of binary relations defined by a complete X-semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup () X B D , which are defined by semilattice D, can be generated by the set

then we characterize the class each element of which is isomorphic to Q by means of the characteristic family of sets, the characteristic mapping and the generate set of Q. Moreover, we calculate the number of regular elements of B X (D) for a finite set X.

In this paper, we investigate such a regular elements α and idempotents of the complete semigroup of binary relations B X (D) defined by semilattices of the class lower incomplete nets, for which V (D, α) = Q. Also we investigate right units of the semigroup B X (Q). For the case where X is a finite set we derive formulas by means of which we can calculate the numbers of regular elements, idempotents and right units of the respective semigroup.

In this paper, we take Q16 subsemilattice of D and we will calculate the number of right unit, idempotent and regular elements α of BX (Q16) satisfied that V (D, α) = Q16 for a finite set X. Also we will give a formula for calculate idempotent and regular elements of BX (Q) defined by an X-semilattice of unions D.

In this paper we investigate idempotents of complete semigroups of binary relations defined by semilattices of the class Z−elementary X−semilattice of unions. For the case where X is a finite set we derive formulas by calculating the numbers of idempotents of the respective semigroup.

This paper is a contribution to the study of the effective content of LAhyperstructure. In this paper, we introduce the notion of soft interior-hyperideals. Further, we give several basic properties of these notions and provide different important characterizations in terms of soft interior hyperideals.

Issues in daily life, where making the best decisions is crucial, are frequently encountered. But, in the majority of these situations, the best course of action is uncertain. We must take into account a number of parameters in order to find the best possible solution to these difficulties. The best mathematical instrument for this is fuzzy soft set FSS theory in decision making. Nutrition is the process of supplying cells and organisms with the nutrients they need to grow and thrive and to sustain life. A healthy diet has the potential to prevent or mitigate numerous prevalent health issues. The purpose of this paper is to select a burning problem for the nutrition of students and successfully apply the FSS theory in decision making. We aim to prove that the approach to decision-making problems with imprecise data via FSSs is more accurate than other types of approaches, and we present a new approach to the FSS model and its applications in decision-making problems.

Issues in daily life, where making the best decisions is crucial, are frequently encountered.
But, in the majority of these situations, the best course of action is uncertain. We must take into
account a number of parameters in order to find the best possible solution to these difficulties. The
best mathematical instrument for this is fuzzy soft set FSS theory in decision making. Nutrition is
the process of supplying cells and organisms with the nutrients they need to grow and thrive and to
sustain life. A healthy diet has the potential to prevent or mitigate numerous prevalent health issues.
The purpose of this paper is to select a burning problem for the nutrition of students and successfully
apply the FSS theory in decision making. We aim to prove that the approach to decision-making
problems with imprecise data via FSSs is more accurate than other types of approaches, and we
present a new approach to the FSS model and its applications in decision-making problems.

Enumeration and classification of mathematical entities is an important part of mathematical research in particular in finite algebra. For algebraic structures that are more general than groups this task is often only feasible by use of computers due to the sheer number of structures that have to be considered. In this paper we present the enumeration and partial classification of AG-groupoids-groupoids in which the identity (ab)c = (cb)a holds-of up to order 6. The results are obtained with the help of the computer algebra system GAP and the constraint solver Minion by making use of both algebraic techniques as well as search pruning via symmetry breaking.

The concept of prime and weakly prime ideal in semigroups has been introduced by G. Szasz [4]. In this paper, we define the involution in po-Γ-semigroups, then we extend some results on prime, semiprime and weakly prime ideals to the involution po-Γ-semigroup S. Also, we characterize intra-regular involution po-Γ-semigroups. We establish that in the involution po-Γ-semigroup S such that the involution preserves the order, an ideal of S is prime if and only if it is both weakly prime and semiprime and if S is commutative, then the prime and weakly prime ideals of S coincide. Finally, we prove that if S is a po-Γ-semigroup with order preserving involution, then the ideals of S are prime if and only if S is intra-regular.

Enumeration and classification of mathematical entities is an important part of mathematical research in particular in finite algebra. For algebraic structures that are more general than groups this task is often only feasible by use of computers due to the sheer number of structures that have to be considered. In this paper we present the enumeration and partial classification of AG-groupoids-groupoids in which the identity (ab)c = (cb)a holds-of up to order 6. The results are obtained with the help of the computer algebra system GAP and the constraint solver Minion by making use of both algebraic techniques as well as search pruning via symmetry breaking.

In this paper we study the conditions under which an additively inverse semiring is a band and commutative. Also in the case of totally ordered additive inverse semiring in which (,) S  is positively totally ordered, we prove the additive structure is max(,) a b b a a b     for all , abin. S We have also framed an example for this result by considering 4 elements.

We introduce the concept of normal Γ-ideal and bi-Γ-ideal in normal Γsemigroups. We characterize the (normal) Γ-semigroup and normal regular Γ-semigroup in terms of elementary properties of bi-Γ-ideal proving the various equivalent conditions. In particular, we establish, among the other things, that if I1, I2 are any two normal Γideals of a Γ-semigroup S, then their product I1ΓI2 and I2ΓI1 are also normal Γ-ideals of S and I1ΓI2 = I2ΓI1. Finally, we show that the minimal normal Γ-ideal of a Γ-semigroup S is a Γ-group.

Difficulties encountered in studying generators of semigroup () X B D of binary relations defined by a complete X-semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup () X B D , which are defined by semilattice D, can be generated by the set

Introduction. Suppose that 5 is a topological semigroup which contains the bicyclic semigroup B as a subsemigroup. Let T denote the closure of B in 5. We investigate the structure of the semigroup T and the extent to which B determines this structure. In §1, two properties of F are established which hold for arbitrary 5; namely, that B is a discrete open subspace of F and T\B is an ideal of T if it is nonvoid. In §11, we introduce the notion of a topological inverse semigroup and establish several properties of such objects. Some questions are posed. In §111, it is shown that if 5 is a topological inverse semigroup, then T\B is a group with a dense cyclic subgroup. §IV contains a description of three examples of a topological semigroup which contains B as a dense proper subsemigroup. Finally, in §V, we assume that 5 is a locally compact topological inverse semigroup and show that either B is closed in 5 or F is isomorphic with the last of the examples described in §IV. A corollary about homomorphisms from B into a locally compact topological inverse semigroup is obtained which generalizes a result due to A. Weil [1, p. 96] concerning homomorphisms from the integers into a locally compact group. All spaces are topological Hausdorff in this paper. We state the definitions of Green's equivalence relations in a semigroup and the definition of an inverse semigroup. Green's relations £f, &, Jf, and 3i on a semigroup 5 are defined by : a@b if and only if a u aS=b\J bS, a£Cb if and only if a u Sa=b u Sb, JP=<e n 0t and &=£ ° 0t. The notations Ra, La, Ha, and Da stand for the appropriate equivalence class of a in 5. A semigroup 5 is an inverse semigroup provided each element x of 5 has a unique inverse ; that is, an element x~l of 5 such that xx " 1x=x and x " 1xx~1 = x~1. For details about inverse semigroups and Green's relations, see [2]. We assume a certain familiarity with these notions. I. The bicyclic semigroup. B is the semigroup with identity 1 generated by two elements p and q subject only to the condition that pq-l. The distinct elements

It is known that if we know all XI-subsemilattices of a given X-semilattice of unions, then we can determine all idempotent elements of the semigroup, and the structure of idempotent elements is characterized. In this work, we find idempotent elements of the semigroup corresponding to Xsemilattices of unions of the class Σ16(X, 6). Moreover, we give formulas for the number of idempotent elements, where X is finite.

In this paper, we prove, among others, that in right chains, the proper ideals are completely prime if and only if they are completely semiprime and that the proper right ideals are prime if and only if they are semiprime. Moreover, the proper non-zero idempotent ideals are completely prime.

In this article we present a brief history and some applications of semirings,  the structure of compact monothetic semirings. The classification of these semirings be based on known description of discrete cyclic semirings and compact monothetic semigroups.

The aim of this paper is to introduce a new concept of an ordered (m; (p; q); n)-bi ideal of an ordered tenary semigroup. Some classical results in ordered ternary semigroups are given. We also consider the minimal ordered (m; (p; q); n)-bi ideals in ordered ternary semigroups. In particular, the (m; (p; q); n)-bi simple ordered ternary semigroups are dened and some of their properties are explored. As a conse- quence, we will show that the regular ordered ternary semigroups can be characterized by using various generalised ideals.

In this paper, we take Q = { T1, T 2, . . . , Tm−3, Tm−2, Tm−1, Tm } subsemilattice of X−semilattice of unions D where the elements Ti’ s are satisfying the following properties, T1⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−2⊂ Tm, T1⊂ T 3⊂ · · · ⊂ Tm−3 ⊂ Tm−1⊂ Tm, T2⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−2⊂ Tm, T2⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−1 ⊂ Tm, T1\T 2 6= ∅, T2\T 1 6= ∅, Tm−2\Tm−1 6= ∅, Tm−1\Tm−2 6= ∅, T1∪T 2= T 3, Tm−2∪Tm−1= Tm. We will investigate the properties of regular element α ∈ BX(D) satisfying V (D,α) = Q. Moreover, we will calculate the number of regular elements of BX(D) for a finite set X.

Difficulties encountered in studying generators of semigroup () X B D of binary relations defined by a complete X-semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup () X B D , which are defined by semilattice D, can be generated by the set

In this paper, we take Q16 subsemilattice of D and we will calculate the number of right unit, idempotent and regular elements α of BX (Q16) satisfied that V (D, α) = Q16 for a finite set X. Also we will give a formula for calculate idempotent and regular elements of BX (Q) defined by an X-semilattice of unions D.

We show in an additive inverse regular semiring (S,+,·) with E • (S) as the set of all multiplicative idempotents and E + (S) as the set of all additive idempotents, the following conditions are equivalent: (i) For all e,f∈E • (S), ef∈E + (S) implies fe∈E + (S). (ii) (S,·) is orthodox. (iii) (S,·) is a semilattice of groups. This result generalizes the corresponding result for regular rings.

This paper bases on the well-studied semigroup T(X; Y ) of all transfor- mations on X with restricted range Y X. We introduce the semi- group TP (X; Y ) of all non-empty subsets of T(X; Y ) under the operation AB := fab : a 2 A;B 2 Bg. We determine the idempotent and regular ele- ments in TP (X; Y ) for the case that jY j = 2. In particular, we characterize the (maximal) regular subsemigroups of TP (X; Y ), the largest semiband, and the (maximal) idempotent subsemigroups of TP (X; Y ).

In this paper, we study some useful results of ordered (p; q)-lateral ideals in ordered ternary semigroups. Also, some properties of (p; q)-lateral simple ordered ternary semigroup have been examined. Further, we characterize the relationship between minimal (resp., maximal) ordered (p; q)- lateral ideals and (p; q)-lateral simple ordered ternary semigroups

We introduce the notions of ,-hesitant fuzzy ideal, ,-hesitant fuzzy bi-ideal and ,-hesitant fuzzy interior ideal on semigroup and characterizations of a hesitant fuzzy ideal (bi-ideal, interior ideal) with two frontiers are considered. We provide an example to show that) , ( -hesitant fuzzy ideal on S need not be a -hesitant fuzzy ideal on S. We provide conditions for a) , ( -hesitant fuzzy ideal on S to be a -hesitant fuzzy ideal on S. Moreover, we show that the hesitant intersection of two hesitant fuzzy left (right, bi-, interior) ideal with two frontiers is a hesitant fuzzy left (right, bi-, interior) ideal respectively with two frontiers.

The aim of this paper is to introduce a new concept of an ordered (m; (p; q); n)-bi ideal of an ordered tenary semigroup. Some classical results in ordered ternary semigroups are given. We also consider the minimal ordered (m; (p; q); n)-bi ideals in ordered ternary semigroups. In particular, the (m; (p; q); n)-bi simple ordered ternary semigroups are dened and some of their properties are explored. As a conse- quence, we will show that the regular ordered ternary semigroups can be characterized by using various generalised ideals.

In this paper, we introduce generalised quasi-ideals in ordered ternary semigroups. Also, we define ordered m-right ideals, ordered (p, q)-lateral ideals and ordered n-left ideals in ordered ternary semigroups and studied the relation between them. Some intersection properties of ordered (m, (p, q), n)-quasi ideals are examined. We also characterize these notions in terms of minimal ordered (m, (p, q), n)-quasi-ideals in ordered ternary semigroups. Moreover, m-right simple, (p, q)-lateral simple, n-left simple, and (m, (p, q), n)-quasi simple ordered ternary semigroups are defined and some properties of them are studied.

This paper deals with a class of algebraic hyperstructures called ordered ternary semihypergroups which are studied in terms of int-soft hyperideals. We introduce the notion of int-soft hyperideals in ordered ternary semihypergroups and investigate some properties of them. We also introduce the concepts of convex soft sets and int-soft points and discuss their properties. Moreover, the classes of regular and intra-regular ordered ternary semihpergroups are characterized in terms of int-soft hyperideals.

In this paper, we introduce relative hyperideals, relative quasi-hyperideals, relative bi-hyperideals, relative prime hyperideal, relative semiprime hyperideal, relative quasi-prime hyperideal, relative quasi-semiprime hyperideal, relative quasi-irreducible hyperideal, and relative regularity in ordered LA-semihypergroups. The results in this paper cleanly generalize those in LA-semigroups, LA-Γ-semigroups, ordered LA-semigroups, ordered LA-Γ-semigroups, LA-semihypergroups, LA-Γ-semihypergroups, ordered LA-Γsemihypergroups as well as in ordered LA-semihypergroups by these novel relative hyperideals.

In this paper, we introduce some mathematical objects, viz., relative ordered prime, relative ordered completely prime, relative ordered semiprime, and relative ordered completely semiprime right Γ-hyperideals in ordered Γ-semihypergroups. We also introduce the concepts of relative ordered prime right associated Γ-hyperideals. Moreover, we characterize ordered Γ-semihypergroups by relative ordered prime, relative ordered completely prime, relative ordered semiprime, and relative ordered completely semiprime right Γ-hyperideals.

In this paper, we take Q = { T1, T 2, . . . , Tm−3, Tm−2, Tm−1, Tm } subsemilattice of X−semilattice of unions D where the elements Ti’ s are satisfying the following properties, T1⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−2⊂ Tm, T1⊂ T 3⊂ · · · ⊂ Tm−3 ⊂ Tm−1⊂ Tm, T2⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−2⊂ Tm, T2⊂ T 3⊂ · · · ⊂ Tm−3⊂ Tm−1 ⊂ Tm, T1\T 2 6= ∅, T2\T 1 6= ∅, Tm−2\Tm−1 6= ∅, Tm−1\Tm−2 6= ∅, T1∪T 2= T 3, Tm−2∪Tm−1= Tm. We will investigate the properties of regular element α ∈ BX(D) satisfying V (D,α) = Q. Moreover, we will calculate the number of regular elements of BX(D) for a finite set X.

Difficulties encountered in studying generators of semigroup () X B D of binary relations defined by a complete X-semilattice of unions D arise because of the fact that they are not regular as a rule, which makes their investigation problematic. In this work, for special D, it has been seen that the semigroup () X B D , which are defined by semilattice D, can be generated by the set

In this paper, we take Q16 subsemilattice of D and we will calculate the number of right unit, idempotent and regular elements α of BX (Q16) satisfied that V (D, α) = Q16 for a finite set X. Also we will give a formula for calculate idempotent and regular elements of BX (Q) defined by an X-semilattice of unions D.

In this paper we investigate idempotents of complete semigroups of binary relations defined by semilattices of the class Z−elementary X−semilattice of unions. For the case where X is a finite set we derive formulas by calculating the numbers of idempotents of the respective semigroup.