Semigroups

The following sections are included:

• Introduction

• Contents

With the release of Version 4.1, GAP becomes an integrated environment for performing calculations with semigroups and developing algorithms for them. In the Introduction, we give a brief account of the state of computational semigroup theory and outline what we see as the main challenges for the future. The rest of the paper takes the form of a tutorial and is intended to be a semigroup theorist's introduction to the use of GAP, showcasing the features directly connected with semigroups.

A finite semigroup S is said to be efficient if it can be defined by a semigroup presentation 〈 A | R 〉 with |R| - |A| = rank(H₂(S¹)) where H₂(S¹)is the second integral homology of the monoid S¹ obtained from S by adjoining an identity. In this paper we show that certain classes of direct powers of finite groups are efficient as semigroups.

We offer an up-to-date survey of results concerning regular semigroups with inverse transversals, several of which are new. In Section 1 we derive the basic properties and in Section 2 we give a classification of the most important types arising. In Section 3 we list some interesting illustrative examples. Section 4 is devoted to various structure theorems, and Section 5 to congruences.

Putcha and Weissglass in [13] and [14] used variable identities to characterize periodic semigroups which are nilpotent extensions of unions of groups and semilattices of groups. In this paper they are used to describe periodic semigroups which are nil-extensions and retractive nil-extensions of unions of groups in the general and various special cases.' The obtained results generalize those of Putcha and Weissglass, as well as the results of Bell [2] concerning rings satisfying variable semigroup identities and the results of Ćirić and Bogdanović [7] concerning semigroups satisfying some ordinary identities.

The aim of this paper is to complete the characterization of the languages that are Boolean combinations (of a subset) of languages of the form wA*, A*w or L(w,r,t,n), where A is an alphabet, w ∈ A⁺, r,t ≥ 0, n ≥ 1 and L(w,r,t,n) denotes the set of all words u in A⁺ such that the number of occurrences of the factor w in u is congruent to r threshold t mod n. For each class C of languages such that A+C is a Boolean algebra generated by some of the following types of languages: wA*, A*w, A*wA* = L(w,1,1,1) or L(w,r,t,1), and such that C does not constitute a variety of languages, we compute the smallest variety of languages containing C and the largest variety of languages contained in C.

In [5] the concept of an ID-semiring (S;+,.) as an algebra of type (2,2) where (S;+) and (S;.) are idempotent semigroups (bands) and where four distributive laws: xy + z ≈ (x + z)(y + z), x + yz ≈ (x + y)(x + z), x(y + z) ≈ xy + xz, (x + y)z ≈ xz + xz are satisfied, was introduced. An ID-semiring (S; +,.) is normal if x + y + u + v ≈ x + u + y + v and xyuv ≈ xuyv are identities. In this paper we want to determine the least and the greatest solid variety of normal ID-semirings, i.e. varieties in which every identity is satisfied as a hyperidentity. The result is that the variety of all normal ID-semirings is solid. The variety RA_(2,2) of rectangular algebras of type (2,2), i.e. the variety generated by all projection algebras of type (2,2) is the least non-trivial solid variety of normal ID-semirings.

We give a simple way of showing that the problem of embeddability into finite inverse semigroups and finite semigroups of order-preserving mappings is decidable

In this paper we characterize some remarkable elements in semigroups containing very good left magnifiers. We generalize a result of N. Jacobson concerning a sufficient condition under which a semigroup does not contain very good magnifiers. A more general construction than that used for semigroups under a sandwich operation enables us to give a new meaning of semigroups with very good magnifiers. Furthermore we present several general methods for obtaining semigroups containing such elements. We also characterize the minimal right ideals associated with very good left magnifiers. Finally we give conditions under which two semigroups with very good left magnifiers are isomorphic.

Uniformly π-regular semigroups, defined as π-regular semigroups whose any regular element is completely regular, form an important kind of semigroups that have been first investigated in 1977 by Shevrin, who announced that they are characterized as completely π-regular semigroups whose any regular -class is a subsemigroup, and also as semilattices of completely Archimedean semigroups. The proofs of these results were first published in [10], together with quite a number of other results concerning the same semigroups as well as semigroups of a more general type, whereas some similar results were obtained independently by Veronesi in [11]. Various other characterizations,ns of these semigroups have been given in a series of papers by Bogdanović and Ćirić (see Theorem 1, the book [1], and the survey papers [2] and [5]). The main purpose of this paper is to characterize a more general kind of semigroups – π-regular semigroups whose any local submonoid is uniformly π-regular.

• Definitions and examples

• Characterizations

• Cancellativity

• Cancellativity conditions

• Restrictions on idempotents

• Congruences

• Covers

• Introduction

• Definitions and Examples

• Problem to be studied: results on inverse semigroups rings

• Semiprimeness and semiprimitivity

• Primeness and right primitivity

• Converses?

• Acknowledgment

• References

The description of the lattice of varieties of bands is probably the best single achievement in the theory of varieties (with possible modification as pseudo, existence, …) of semigroups (and their variants). It was determined by Biryukov, Gerhard and Fennemore (within a short period of time). Only the last of these authors provided a system of identities for bases of these varieties. Their proofs are long and involved. This subject was "revisited" by Gerhard and myself with a more transparent proof, a new system of identities and an explicit solution of the word problem for free objects in proper varieties of bands. Polák derived the lattice of band varieties from his theorem on varieties of completely regular semigroups. Various aspects of the varieties of bands and their lattice were studied by many researches. They included word problems for free objects in all band varieties, relationship with Malcev products, construction of bands belonging to specified varieties, quasivarieties of (special) bands, groupoids of varieties and some particular quasivarieties, monoid and *-band varieties, relatively free bands, and relationship of bands with formal languages. Many, but not all, of these results are complete with considerable esthetic apeal.

The semidirect product G = H λ ((G₁ × … × G_n) λ K) of groups, where < G_i, K >= G_i λ K (i = 1, 2,…, n), is characterized by the properties of the semigroup End G (Theorem 2.1). This characterization makes it possible to give conditions for the summability of orthogonal idempotents of the semigroup End G for an arbitrary group G (Theorem 3.3). This leads to the fo!owing result. Let G and be groups such that the semigroups End G and End are isomorphic: , φ - isomorphism. If x₁, …, x_n are orthogonal and summable idempotents of End G, then the idempotents x₁φ ,…, x_nφ of End are also summable and (x_l + … + x_n)φ = x₁φ + … + x_nφ (Theorem 3.4).

This paper introduces the concept of generalized -semigroup, shows a method to obtain this class of semigroups and gives theorems of structure for them.

This paper gives a proof, using geometric techniques, of the theorem of Henckell and Rhodes that J * G = J ⓜ G, a key ingredient of the proof that PG = BG. The purpose of this paper is twofold: firstly, it gives a proof of the result using less machinery and providing, we hope, more insight than the original proof; secondly, it serves as an introduction to the techniques used by the author to show that, in fact, J * H = J ⓜ H for extension-closed and, even, arborescent pseudovarieties of groups H. At the end of the paper, we sketch what changes are needed to prove this stronger result.

From the 1940s to the early 1960s, the Russian school of semigroups produced some major results on transformation semigroups which concerned generators, morphisms, ideals and congruences. Then in 1966 Howie published his seminal paper on idempotent transformations and in 1975 Sullivan initiated the study of G(X)-normal semigroups of transformation.

Throughout the last two decades, many papers have been written on these and related ideas, as well as on links with linear algebra, ring theory and general topology. As a consequence, the theory of transformation semigroups now constitues a substantial body of knowledge in its own right. In this paper, I briefly discuss some achievements of the past and activities of the present, and suggest avenues for further work in the future.

We provide an overview of recent research on the natural question what makes a finite semigroup have finite or infinite identity basis. An emphasis is placed on results published since 1985 when the previous survey of the area had appeared. We also present a few unpublished results and formulate several open problems.

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