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We prove the existence and uniqueness (up to isomorphism) of a regular semigroup F (X) weakly generated by a set X of elements such that all other regular semigroups weakly generated by X are homomorphic images of F (X) . The semigroup F (X) is introduced by a presentation and the word problem for that presentation is solved. The structure of the semigroup F (X) is also studied. [ABSTRACT FROM AUTHOR]

Copyright of Palestine Journal of Mathematics is the property of Palestine Polytechnic University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper we study multiplicative semigroups of $2\times 2$ matrices over a linearly ordered abelian group with an externally added bottom element. The multiplication of such a semigroup is defined by replacing addition and multiplication by join and addition in the usual formula defining matrix multiplication. We show that there are four types of idempotents in this semigroup and we determine which of them are $0$-primitive. We also prove that the poset of idempotents with respect to the natural order is a lattice. It turns out that this matrix semigroup is inverse or orthodox if and only if the abelian group is trivial.

In this paper, we show that semigroups containing middle units can be constructed from semigroups containing one-sided identity elements. More-over, we show that regular semigroups containing middle units can be obtained from regular monoids. [ABSTRACT FROM AUTHOR]

In this paper we study the multiplicative semigroup of 2 × 2 matrices over a linearly ordered abelian group with an externally added bottom element. The multiplication of such a semigroup is defined by replacing addition and multiplication by join and addition in the usual formula defining matrix multiplication. We show that there are four types of idempotents in such a matrix semigroup and we determine which of them are 0-primitive. We also prove that the poset of idempotents of such a matrix semigroup with respect to the natural order is a lattice. It turns out that such a matrix semigroup is inverse or orthodox if and only if the abelian group is trivial. [ABSTRACT FROM AUTHOR]

Copyright of Journal of South China Normal University (Natural Science Edition) / Huanan Shifan Daxue Xuebao (Ziran Kexue Ban) is the property of Journal of South China Normal University (Natural Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we introduce the normal category ℒ () [ ℛ () ] of principal left [right] ideals of the normed algebra of all finite rank bounded operators on a Hilbert space H and show that ℒ () and ℛ () are isomorphic using Hilbert space duality. We also prove that the semigroup of all normal cones in ℒ () is isomorphic to the semigroup of all finite rank operators on H. Further, we construct bounded normal cones in ℒ () such that the set of all bounded normal cones in ℒ () forms a normed algebra isomorphic to the normed algebra . [ABSTRACT FROM AUTHOR]

In this paper we study the regular semigroups weakly generated by a single element x, that is, with no proper regular subsemigroup containing x. We show there exists a regular semigroup F 1 weakly generated by x such that all other regular semigroups weakly generated by x are homomorphic images of F 1 . We define F 1 using a presentation where both sets of generators and relations are infinite. Nevertheless, the word problem for this presentation is decidable. We describe a canonical form for the congruence classes given by this presentation, and explain how to obtain it. We end the paper studying the structure of F 1 . In particular, we show that the 'free regular semigroup FI 2 weakly generated by two idempotents' is isomorphic to a regular subsemigroup of F 1 weakly generated by { x x ′ , x ′ x } . [ABSTRACT FROM AUTHOR]

The concept of normal category was introduced by Nambooripad in his theory of cross connections for describing the structure of regular semigroups. These are essentially the categories of principal left ideals and the categories of principal right ideals of regular semigroups with appropriate morphisms. These categories also have an associated partial order on the set of objects. Discrete normal categories are normal categories in which the partial order on the object set is the identity relation. A cross connection relates two normal categories and and produces a regular semigroup S such that one is the category of principal left ideals and the other the category of principal right ideals of the semigroup S. This paper considers the question of compatibility of two normal categories in order that they can be connected by a cross connection. It is known that given any normal category there is a normal category = ∗ called the normal dual of such that is compatible with . The question of finding all normal categories that are compatible with a given normal category remains unsettled. Such categories are called compatible duals of . For the class of discrete normal categories the characterization of all compatible duals has been done here. [ABSTRACT FROM AUTHOR]

A regular semigroup is weakly generated by a set X if it has no proper regular subsemigroup containing X. In this paper, we study the regular semigroups weakly generated by idempotents. We show there exists a regular semigroup FI (X) weakly generated by | X | idempotents such that all other regular semigroups weakly generated by | X | idempotents are homomorphic images of FI (X). The semigroup FI (X) is defined by a presentation 〈 G (X) , ρ e ∪ ρ s 〉 and its structure is studied. Although each of the sets G (X) , ρ e , and ρ s is infinite for | X | ≥ 2 , we show that the word problem is decidable as each congruence class has a "canonical form". If FI n denotes FI (X) for | X | = n , we prove also that FI 2 contains copies of all FI n as subsemigroups. As a consequence, we conclude that (i) all regular semigroups weakly generated by a finite set of idempotents, which include all finitely idempotent generated regular semigroups, strongly divide FI 2 ; and (ii) all finite semigroups divide FI 2 . [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to introduce the \((\alpha,\beta)\)-hesitant fuzzy set and apply it to semigroups. In this paper, as a generalization of the concept of hesitant fuzzy sets to semigroup theory, the concept of \((\alpha,\beta)\)-hesitant fuzzy subsemigroups of semigroups is introduced, and related properties are discussed. Furthermore, we define and study \((\alpha,\beta)\)-hesitant fuzzy ideals on semigroups. In particular, we investigate the structure of \((\alpha,\beta)\)-hesitant fuzzy ideal generated by a hesitant fuzzy ideal in a semigroup. In addition, we also introduce the concepts of \((\alpha,\beta)\)-hesitant fuzzy semiprime sets of semigroups, and characterize regular semigroups in terms of \((\alpha,\beta)\)-hesitant fuzzy left ideals and \((\alpha,\beta)\)-hesitant fuzzy right ideals. Finally, several characterizations of regular and intra-regular semigroups by the properties of \((\alpha,\beta)\)-hesitant ideals are given.

The aim of this manuscript is to introduce the (α, β)-hesitant fuzzy set and apply it to semigroups. In this paper, as a generalization of the concept of hesitant fuzzy sets to semigroup theory, the concept of (α, β)-hesitant fuzzy subsemigroups of semigroups is introduced, and related properties are discussed. Furthermore, we define and study (α, β)-hesitant fuzzy ideals on semigroups. In particular, we investigate the structure of (α, β)-hesitant fuzzy ideal generated by a hesitant fuzzy ideal in a semigroup. In addition, we also introduce the concepts of (α, β)-hesitant fuzzy semiprime sets of semigroups, and characterize regular semigroups in terms of (α, β)-hesitant fuzzy left ideals and (α, β)-hesitant fuzzy right ideals. Finally, several characterizations of regular and intra-regular semigroups by the properties of (α, β)-hesitant ideals are given. [ABSTRACT FROM AUTHOR]

It is proved that for an IC abundant semigroup (a primitive abundant semigroup; a primitively semisimple semigroup) S and a field K, if K 0[S] is right (left) self-injective, then S is a finite regular semigroup. This extends and enriches the related results of Okniński on self-injective algebras of regular semigroups, and affirmatively answers Okniński’s problem: does that a semigroup algebra K[S] is a right (respectively, left) self-injective imply that S is finite? (Semigroup Algebras, Marcel Dekker, 1990), for IC abundant semigroups (primitively semisimple semigroups; primitive abundant semigroups). Moreover, we determine the structure of K 0[S] being right (left) self-injective when K 0[S] has a unity. As their applications, we determine some sufficient and necessary conditions for the algebra of an IC abundant semigroup (a primitively semisimple semigroup; a primitive abundant semigroup) over a field to be semisimple.

The structure theorems for (Completely) simple semigroups and (Completely) 0-simple semigroups have proved a powerful tool in the investigation of such semigroups. In this paper, first of all, we define weak simple semigroups and weak 0-simple semigroups and compare them with simple semigroups and 0-simple semigroups respectively. Then we give examples of these semigroups and describe the structure of them. Also, we define completely weak simple semigroup and completely weak 0-simple semigroup. Finally, by using Green's equivalences, we prove some results and give equivalences, for these semigroups. [ABSTRACT FROM AUTHOR]

Abstract In this paper the concepts of intuitionistic anti fuzzy interior ideal in semigroups is introduced and some important characterizations have been obtained. [ABSTRACT FROM AUTHOR]

We generalize the regularity concept for semigroups in two ways simultaneously: to higher regularity and to higher arity. We show that the one-relational and multi-relational formulations of higher regularity do not coincide, and each element has several inverses. The higher idempotents are introduced, and their commutation leads to unique inverses in the multi-relational formulation, and then further to the higher inverse semigroups. For polyadic semigroups we introduce several types of higher regularity which satisfy the arity invariance principle as introduced: the expressions should not depend of the numerical arity values, which allows us to provide natural and correct binary limits. In the first definition no idempotents can be defined, analogously to the binary semigroups, and therefore the uniqueness of inverses can be governed by shifts. In the second definition called sandwich higher regularity, we are able to introduce the higher polyadic idempotents, but their commutation does not provide uniqueness of inverses, because of the middle terms in the higher polyadic regularity conditions. Finally, we introduce the sandwich higher polyadic regularity with generalized idempotents. [ABSTRACT FROM AUTHOR]

We investigate the class BA of ordered regular semigroups in which each element has a biggest associate x† = max {y | xyx = x}. This class properly contains the class PO of principally ordered regular semigroups (in which there exists x⋆ = max {y | xyx ⩽x}) and is properly contained in the class BI of ordered regular semigroups in which each element has a biggest inverse x◦. We show that several basic properties of the unary operation x ↦ x⋆ in PO extend to corresponding properties of the unary operation x ↦ x† in BA. We consider naturally ordered semigroups in BA and prove that those that are orthodox contain a biggest idempotent. We determine the structure of some such semigroups in terms of a principal left ideal and a principal right ideal. We also characterise the completely simple members of BA. Finally, we consider the naturally ordered semigroups in BA that do not have a biggest idempotent.

In this paper, we introduce a generalization of a bipolar fuzzy subsemigroup, namely a (𝛼1, 𝛼2; 𝛽1, 𝛽2)-bipolar fuzzy subsemigroup. The notions of (𝛼1, 𝛼2; 𝛽1, 𝛽2)-bipolar fuzzy left (right, bi-) ideals are discussed. Some necessary and sufficient conditions of (𝛼1, 𝛼2; 𝛽1, 𝛽2)-bipolar fuzzy left (right, bi-) ideals are obtained. Furthermore, any regular semigroup is characterized in terms of generalized bipolar fuzzy semigroups

In this paper, we introduce the notion a cubic bi-quasi ideal of semigroup and we characterize the regular semigroup in terms of a cubic bi-quasi ideal of a semigroup. [ABSTRACT FROM AUTHOR]

In this article, we shall give the concepts of bipolar fuzzy weakly interior ideals of semigroups, and we provide some interesting properties of bipolar fuzzy weakly interior ideals of semigroups. The relationship between bipolar fuzzy weakly interior ideals and bipolar fuzzy left (right) ideals, and the relationship between bipolar fuzzy weakly interior ideals and bipolar fuzzy interior ideals are also discussed. In the results, we proceed to characterize some semigroups by using bipolar fuzzy weakly interior ideals. Finally, we discuss the image and pre-image of bipolar fuzzy weakly interior ideals of semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we first show the well-posedness of SDEs driven by Lévy noises under mild conditions. Then, we consider the existence and uniqueness of periodic solutions of the SDEs. To establish the ergodicity and uniqueness of periodic solutions, we investigate the strong Feller property and the irreducibility of the corresponding time-inhomogeneous semigroups when both small and large jumps are allowed in the equations. Doob’s celebrated theorem on the uniqueness of invariant measures for time-homogeneous Markov processes has been generalized to obtain the uniqueness of periodic measures for time-inhomogeneous Markov processes. Some examples are presented to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, as a further generalization of ideals, we introduce the notion of bi-interior ideal as a generalization of quasi ideal, bi-ideal and interior ideal of semigroup and study the properties of bi-interior ideals of semigroup, simple semigroup and regular semigroup.

In this paper, we introduce the notion of left (right) bi-quasi ideal of a semigroup which is a generalization of bi-ideal and quasi -ideal of a semigroup. We characterize the left bi-quasi simple semigroups, regular semigroups using left bi-quasi ideals and study the properties of left bi-quasi ideals of semigroups. [ABSTRACT FROM AUTHOR]

In this paper, as a further generalization of ideals, we introduce the notion of bi- quasi-interior ideals as a generalization of bi ideal, quasi ideal, interior ideal, bi-interior ideal and bi-quasi ideal of semigroup and study the properties of bi-quasi-interior ideals of semigroup.

In this paper, we study a kind of soft rough semigroups according to Shabir’s idea. We define the upper and lower approximations of a subset of a semigroup. According to Zhan’s idea over hemirings, we also define a kind of new C-soft sets and CC-soft sets over semigroups. In view of this theory, we investigate the soft rough ideals (prime ideals, bi-ideals, interior ideals, quasi-ideals, regular semigroups). Finally, we give two decision making methods: one is for looking a best a parameter which is to the nearest semigroup, the other is to choose a parameter which keeps the maximum regularity of regular semigroups.

This survey aims to give an overview of several substantial developments of the last 50 years in the structure theory of regular semigroups and to shed light on their impact on other parts of semigroup theory. [ABSTRACT FROM AUTHOR]

We introduce a new class of semigroups called strict abundant semigroups, which are concordant semigroups and subdirect products of completely J ∗ -simple abundant semigroups and completely 0- J ∗ -simple primitive abundant semigroups. A general construction and a tree structure of such semigroups are established. Consequently, the corresponding structure theorems for strict regular semigroups given by Auinger in 1992 and by Grillet in 1995 are generalized and extended. Finally, an example of strict abundant semigroups is also given. [ABSTRACT FROM AUTHOR]

Copyright of Sultan Qaboos University Journal for Science is the property of Sultan Qaboos University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case.

In this paper the notions of intuitionistic anti fuzzy bi-ideal, intuitionistic anti fuzzy interior ideal, intuitionistic anti fuzzy (1,2)-ideal in semigroups are introduced and some important characterizations have been obtained.

There are two major structure theorems for an arbitrary regular semigroup using categories, both due to Nambooripad. The first construction using inductive groupoids departs from the biordered set structure of a given regular semigroup. This approach belongs to the realm of the celebrated Ehresmann-Schein-Nambooripad Theorem and its subsequent generalisations. The second construction is a generalisation of Grillet's work on cross-connected partially ordered sets, arising from the principal ideals of the given semigroup. In this article, we establish a direct equivalence between these two seemingly different constructions. We show how the cross-connection representation of a regular semigroup may be constructed directly from the inductive groupoid of the semigroup, and vice versa. [ABSTRACT FROM AUTHOR]

Weakly simplistic orthodox transversals are introduced in this paper and some characterizations associated with them are obtained. An example is given to demonstrate that weakly simplistic orthodox transversals are proper generalisations of left simplistic orthodox transversals. The related results of Blyth and Almeida Santos on left simplistic inverse transversals obtained in 1996 and Kong and Luo on left simplistic orthodox transversals obtained in 2011 are generalised and amplified. A structure theorem of regular semigroups with weakly simplistic orthodox transversals is also established. [ABSTRACT FROM AUTHOR]

Cross-connection theory developed by Nambooripad is the construction of a regular semigroup from its principal left (right) ideals using categories. We use the cross-connection theory to study the structure of the semigroup Sing(V) of singular linear transformations on an arbitrary vector space V over a field K. There is an inbuilt notion of duality in the cross-connection theory, and we observe that it coincides with the conventional algebraic duality of vector spaces. We describe various cross-connections between these categories and show that although there are many cross-connections, upto isomorphism, we have only one semigroup arising from these categories. But if we restrict the categories suitably, we can construct some interesting subsemigroups of the variants of the linear transformation semigroup. [ABSTRACT FROM AUTHOR]

Cross-connection theory propounded by Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant T θX of the full transformation semigroup (TX, ·) for an arbitrary θ ϵ TX is the semigroup T θ X = (TX, *) with the binary operation α* β = α· θ · β where α, β ϵ TX. In this article, we describe the ideal structure of the regular part Reg(T θ X) of the variant of the full transformation semigroup using cross-connections. We characterize the constituent categories of Reg(T θ X) and describe how they are cross-connected by a functor induced by the sandwich transformation θ. This leads us to a structure theorem for the semigroup and gives the representation of Reg(T θ X) as a cross-connection semigroup. Using this, we give a description of the biordered set and the sandwich sets of the semigroup. [ABSTRACT FROM AUTHOR]

A generalised D-semigroup is here defined to be a left E-semiabundant semigroup S in which the R¯E-class of every x∈S contains a unique element D(x) of E, made into a unary semigroup. Two-sided versions are defined in the obvious way in terms of R¯E and L¯E. The resulting class of unary (bi-unary) semigroups is shown to be a finitely based variety, properly containing the variety of D-semigroups (defined in an order-theoretic way in Communications in Algebra, 3979-4007, 2014). Important subclasses associated with the regularity and abundance properties are considered. The full transformation semigroup TX can be made into a generalised D-semigroup in many natural ways, and an embedding theorem is given. A generalisation of inverse semigroups in which inverses are defined relative to a set of idempotents arises as a special case, and a finite equational axiomatisation of the resulting unary semigroups is given. [ABSTRACT FROM AUTHOR]

In this paper, as a further generalization of ideals, we introduce the notion of bi-quasi-interior ideals as a generalization of biideal, quasi ideal, interior ideal, bi-interior ideal and bi-quasi ideal of semigroup and study the properties of bi-quasi-interior ideals of semigroup. [ABSTRACT FROM AUTHOR]

We introduce the relations and with respect to a subset U of idempotents. Based on and , we define a new class of semigroups which we name U-concordant semigroups. Our purpose is to describe U-concordant semigroups by generalized categories over a regular biordered set. We show that the category of U-concordant semigroups and admissible morphisms is isomorphic to the category of RBS generalized categories and pseudo functors. Our approach is inspired from Armstrong's work on the connection between regular biordered sets and concordant semigroups. The significant difference in strategy is by using RBS generalized categories equipped with pre-orders, we have no need to discuss the quotient of a category factored by a congruence. [ABSTRACT FROM AUTHOR]

The purpose of this note is to give a very elementary proof of a theorem of Graham that provides a structural description of finite 0-simple semigroups and its idempotent-generated subsemigroups.

For a given integer n=p1α1p2α2⋯pkαk$n = p_1^{\alpha _1 } p_2^{\alpha _2 } \cdots p_k^{\alpha _k }$ (k ≥ 2), we give here a class of finitely presented finite monoids of order n. Indeed the monoids Mon(π), where π=〈a1,a2,…,ak|aipiαi=ai, (i=1,2,…,k),aiai+1=ai, (i=1,2,…,k−1)〉.$$\pi = {\langle {a_1 ,a_2 , \ldots ,a_k |a_i^{p_i^{\alpha _i } } = {a_i}, {\left({i = 1,2, \ldots ,k} \right)}, a_i a_{i + 1} = {a_i}, \left({i = 1,2, \ldots ,k - 1} \right)} \rangle} .$$ As a result of this study we are able to classify a wide family of the k-generated p-monoids (finite monoids of order a power of a prime p). An interesting di erence between the center of finite p-groups and the center of finite p-monoids has been achieved as well. All of these monoids are regular and non-commutative.