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Product decompositions of semigroups induced by action pairs

Volume 587 / 2023

Scott Carson, Igor Dolinka, James East, Victoria Gould, Rida-e Zenab Dissertationes Mathematicae 587 (2023), 1-180 MSC: Primary 20M10; Secondary 20M05, 20M20, 20M30, 08A05, 08A30, 08A35. DOI: 10.4064/dm871-8-2023 Published online: 18 October 2023


This paper concerns a class of semigroups that arise as products $US$, 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 an 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 $US$ is a quotient of a semidirect product $U\rtimes S$, 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 (also called $E$-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 $US$ 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.


  • Scott CarsonMathematical Sciences Department
    George Mason University
    Fairfax, VA 22030, USA
  • Igor DolinkaDepartment of Mathematics and Informatics
    University of Novi Sad
    21101 Novi Sad, Serbia
  • James EastCentre for Research in Mathematics and Data Science
    Western Sydney University
    Penrith NSW 2751, Australia
  • Victoria GouldDepartment of Mathematics
    University of York
    York YO10 5DD, UK
  • Rida-e ZenabDepartment of Mathematics and Related Studies
    Sukkur IBA University
    65200, Sindh, Pakistan

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