Finite coverings of semigroups and related structures

Document Type : Ischia Group Theory 2020/2021


1 Department of Mathematics, Montana State University, Havre, MT, 59501, USA

2 Department of Mathematical Sciences, Binghamton University, Binghamton, NY, 13902-6000, USA


For a semigroup $S$, the covering number of $S$ with respect to semigroups, $\sigma_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all $n\geq 2$, there exists an inverse semigroup with covering number $n$, similar to the case of loops. Finally, a monoid that is neither a group nor a semigroup with an identity adjoined has covering number two as well.


Main Subjects

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Volume 12, Issue 3 - Serial Number 3
Proceedings of the Ischia Group Theory (2020/2021) - Part 3
September 2023
Pages 205-222
  • Receive Date: 19 November 2021
  • Revise Date: 02 July 2022
  • Accept Date: 04 July 2022
  • Published Online: 01 September 2023