On inverse semigroups the closure of whose set of idempotents is a clifford semigroup (original) (raw)

arXiv (Cornell University), 2022

This paper concerns a class of semigroups that arise as products U S, associated to what we call 'action pairs'. Here U and S are subsemigroups of a common monoid and, roughly speaking, S has an action on the monoid completion U 1 that is suitably compatible with the product in the over-monoid. The semigroups encapsulated by the action pair construction include many natural classes such as inverse semigroups and (left) restriction semigroups, as well as many important concrete examples such as transformational wreath products, linear monoids, (partial) endomorphism monoids of independence algebras, and the singular ideals of many of these. Action pairs provide a unified framework for systematically studying such semigroups, within which we build a suite of tools to ensure a comprehensive understanding of them. We then apply our abstract results to many special cases of interest. The first part of the paper constitutes a detailed structural analysis of semigroups arising from action pairs. We show that any such semigroup U S is a quotient of a semidirect product U ⋊S, and we classify all congruences on semidirect products that correspond to action pairs. We also prove several covering and embedding theorems, each of which naturally extends celebrated results of McAlister on proper (a.k.a. E-unitary) inverse semigroups. The second part of the paper concerns presentations by generators and relations for semigroups arising from action pairs. We develop a substantial body of general results and techniques that allow us to build presentations for U S out of presentations for the constituents U and S in many cases, and then apply these to several examples, including those listed above. Due to the broad applicability of the action pair construction, many results in the literature are special cases of our more general ones.

O ct 2 01 8 CONJUGACY IN INVERSE SEMIGROUPS

In a group G, elements a and b are conjugate if there exists g ∈ G such that g −1 ag = b. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements a and b in an inverse semigroup S, a is conjugate to b, which we will write as a ∼ i b, if there exists g ∈ S 1 such that g −1 ag = b and gbg −1 = a. The purpose of this paper is to study the conjugacy ∼ i in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAllister P-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid and stable inverse semigroups.

Inverse semigroups with idempotent-fixing automorphisms

Semigroup Forum, 2014

A celebrated result of J. Thompson says that if a finite group G has a fixed-point-free automorphism of prime order, then G is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixedpoint-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.

Products of quasi-idempotents in finite symmetric inverse semigroups

Semigroup Forum, 2015

Let X n = {1, 2,. .. , n} and let S n and I n be the symmetric group and symmetric inverse semigroup on X n respectively. In this paper, we show that the semigroup SI n = I n \S n , of all strictly partial one-to-one maps on X n , is generated by quasi-idempotent elements (non-idempotent elements α satisfying α 4 = α 2). Also, we give the least upper bound for the minimum length factorisation of each α ∈ SI n into a product of quasi-idempotents.

Direct product and wreath product of transformation semigroups

GANIT: Journal of Bangladesh Mathematical Society, 2012

In this paper direct product and wreath product of transformation semigroups have been defined, and associativity of both the products and distributivity of wreath product over direct product have been established.DOI: http://dx.doi.org/10.3329/ganit.v31i0.10303GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 31 (2011) 1-7

Free products of inverse semigroups II

Glasgow Mathematical Journal, 1991

Let S and T be inverse semigroups. Their free product S inv T is their coproduct in the category of inverse semigroups, defined by the usual commutative diagram. Previous descriptions of free products have been based, like that for the free product of groups, on quotients of the free semigroup product S sgp T. In that framework, a set of canonical forms for S inv T consists of a transversal of the classes of the congruence associated with the quotient. The general result [4] of Jones and previous partial results [3], [5], [6] take this approach.

A note on amalgams of inverse semigroups

Journal of the Australian Mathematical Society, 2001

This note gives a necessary condition, in terms of graded actions, for an inverse semigroup to be a full amalgam. Under a mild additional hypothesis, the condition becomes sufficient.