Semigroups have layers: a generalisation of stratified semigroups

Abstract

This thesis is a work of two parts. In the first part, we attempt to study the structure of E−inversive (also known as E−dense) semigroups. As these semigroups can be viewed as a generalisation of regular semigroups, our approach aims to adapt methods used to understand regular semigroups to the E−dense setting. Our first method is a geometric approach inspired by work due to K. S. S. Nambooripad, while the second is based on T. S. Blyth’s work on inverse transversals. We also briefly examine these results in the context of some simple wreath products, motivated by the Krohn-Rhodes decomposition of finite semigroups.

Due to a number of factors, including the Covid-19 pandemic, our focus then shifts to a generalisation of work by Pierre Grillet on stratified semigroups. The main body of this second part of the thesis consists largely of joint work with James Renshaw, namely the following papers, elements of which also appear in the introductory chapter:
[1] James Renshaw & William Warhurst, Semilattices of Stratified Semigroups, preprint, available at arXiv:2305.11535 [math.GR], 2023.
[2] James Renshaw & William Warhurst, The multiplicative semigroup of a Dedekind domain, preprint, available at arXiv:2309.02831 [math.GR], 2023.

In [1], we introduce stratified extensions as a generalisation of Grillet’s stratified semigroups, which we describe these in terms of ideal extensions of semigroups. While many commonly studied semigroups (such as monoids or regular semigroups) are stratified extensions only in a fairly trivial sense, we provide a number of interesting examples of semigroups which can be decomposed as semilattices of stratified semigroups. In [2], we continue this work by first showing that the multiplicative semigroup of any commutative ring can be viewed as a semilattice of some semigroups. We then show that if the ring is a Dedekind domain then the semilattice consists of the group of units and a stratified extension of the trivial group and hence the multiplicative semigroup can be viewed as a semilattice of stratified extensions. Further, if the ring is any quotient of a Dedekind domain, the multiplicative semigroup is again a semilattice of stratified extensions, with potentially a much more complex structure than in the non-quotient case.

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Identifiers

ORCID for William Lee Warhurst: orcid.org/0009-0001-4431-4234

ORCID for Ian Leary: orcid.org/0000-0001-8300-4979

ORCID for Jim Renshaw: orcid.org/0000-0002-5571-8007

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