Study Of Semigroups Research Articles

Idempotents and Ideals of Regular Rings

Multiplicative semigroups of rings form an important class of semigroups and one theme in the study of semigroups is how the structure of this semigroup affects the structure of the ring. An important tool in analyzing the structure of a semigroup is the Green’s relations. We study some properties of these relations on the multiplicative semigroup of a regular ring with unity. This also gives easier proofs of some known results on ring theory.

D-squares and E-squares

The concept of E-squares introduced by Prof. K.S.S. Nambooripad plays an important role in the study of structure of Semigroups. Multiplicative semigroups of rings form an important class of semigroups and one theme in the study of semigroups is how the structure of this semigroup affects the structure of the ring. An important tool in analyzing the structure of a semigroup are the Green's relations. In this paper, we study some properties of these relations on the multiplicative semigroup of a regular ring and hence the properties of \(E\)-squares and \(\mathcal{D}\)-squares.

One Sided Homomorphism and One Sided Congruence

The congruence on semigroup and the homomorphism of semigroup play an important role in the study of semigroup. As a generalization of congruence and homomorphism, in this paper we give the definition of one sided congruence and one sided homomorphism, and obtain the largest congruence contained in some one sided congruence.

Complete Asymptotic Expansions for Altomare Operators

In the present paper, we study an approximation process by a sequence of operators with a prescribed asymptotic behavior. The method of construction of these operators is due to Altomare and Amiar. The operators have several applications in semigroup theory. Our investigation provides an insight in their asymptotic behaviour. Finally, we apply our results to concrete examples of approximation operators recently defined and used by Altomare and Milella for the study of semigroups.

Anton Kazimirovich Suschkewitsch (1889–1961)

Anton Kazimirovich Suschkewitsch was a Russian mathematician who spent most of his working life at Kharkov State University in the Ukraine. In the 1920s, he embarked upon the first systematic study of semigroups, placing him at the very beginning of algebraic semigroup theory and, arguably, earning him the title of the world's first semigroup theorist. Owing to the political circumstances under which he lived, however, his work failed to find a wide audience during his lifetime. We give a brief account of his life and his researches into semigroup theory.

The Early Development of the Algebraic Theory of Semigroups

In the history of mathematics, the algebraic theory of semigroups is a relative new-comer, with the theory proper developing only in the second half of the twentieth century. Before this, however, much groundwork was laid by researchers arriving at the study of semigroups from the directions of both group and ring theory. In this paper, we will trace some major strands in the early development of the algebraic theory of semigroups. We will begin with the aspects of the theory which were directly inspired by, and were analogous to, existing results for both groups and rings, before moving on to consider the first independent theorems on semigroups: theorems with no group or ring analogues.

Studying semigroups of mappings using quasi-identities

We define pq-varieties as classes of finite semigroups closed under the formation of subsemigroups and finitary direct products. Different examples of pq-varieties arise naturally in the study of semigroups of mappings. We consider for these pq-varieties two typical problems of the theory of pseudovarieties.

Semigroups presented by one relation and satisfying the Church-Rosser property

This paper is devoted to the study of semigroups presented by a single defining relationA=B and satisfying the Church-Rosser property.

Some results on compact convergence semigroups defined by filters

In this paper the concept of convergence defined by filters is used and applied in the study of semigroups. Special emphasis is placed on compact convergence semigroups and their properties.

Groups of complexes of a representable lattice-ordered group

In 1954 N. Kimura proved that each idempotent in a semigroup is contained in a unique maximal subgroup of the semigroup and that distinct maximal subgroups are disjoint [13] (or see [6, pp. 21–23]). This generalized earlier results of Schwarz [14] and Wallace [15]. These maximal subgroups are important in the study of semigroups. If G is a group, then the collectionS(G) of nonempty complexes ofGis a semigroup and it is natural to inquire what properties ofGare inherited by the maximal subgroups ofS(G). There seems to be very little literature devoted to this subject. In [5, Theorem 2], with certain hypotheses placed on an idempotent, it was shown that ifGis a lattice-ordered group (“1-group”) then a maximal subgroup ofS(G) containing an idempotent satisfying these conditions admits a natural lattice-order. The main result of this note (Theorem 1) is that ifGis a representable1-group andEis a normal idempotent ofS(G) and a dual ideal of the latticeG, then the maximal subgroup ofS(G) containingEadmits a representable lattice-order.

Totally ordered 𝐷-class decompositions

Introduction. An irreducible semigroup is a compact connected semigroup with identity which contains no proper compact connected subsemigroup containing the identity and meeting the minimal ideal. Since every compact connected semigroup with identity has an irreducible subsemigroup joining its identity to its minimal ideal, the study of the category of irreducible semigroups is basic in any quest for information about the structure of compact semigroups with identity. In a previous announcement [4], the authors described the notion of a hormos and conjectured that an irreducible semigroup is an irreducible hormos. (The structure of the latter was completely described—without proofs—in that announcement.) We also announced a number of categories of semigroups for which we are able to obtain this result. One distinguishing feature of an irreducible hormos is that the Dclass (and in fact even the u-class) decomposition is a totally ordered semigroup—in fact an /-semigroup [6]. I t may be conjectured that a study of semigroups whose P-class decomposition is totally ordered might lead to a description of all irreducible semigroups, and it is this motive which brings us to the results of this announcement. We shall give a brief outline of the techniques we use to prove our results. The details will appear in our forthcoming book. For terminology, the reader is referred to [ l ] and [4],

Markov Processes and Semigroups of Operators

In this paper the relations between various semigroups of operators and between various infinitesimal operators connected with a homogeneous in t Markov process are investigated. General conditions are established under which the Markov process is determined by its corresponding infinitesimal operator.Let $U_t $ be a semigroup of linear operators in the Banach space L such that $\|U_t \| \leqq 1$. Let $T_t = U_t $ be an adjoint semigroup in the conjugate space $B = L^ * $. More abstractly the main object of this paper can be characterized as the study of semigroups $T_t $ and its infinitesimal operators in strong and weak topologies of space B.