Finite coverings of semigroups and related structures

Document Type : Ischia Group Theory 2020/2021

Authors

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

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

Abstract

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.

Keywords

Main Subjects


[1] J. H. E. Cohn, On n-sum groups, Math. Scand., 75 (1994) 44–58.
[2] C. Donoven, Groups that are the union of two semigroups have left-orderable quotients, In preparation, 2021.
[3] C. Donoven and Ch. Eppolito, Notes on semigroup covering numbers and limits of monogenic semigroups, in
preparation, 2021.
[4] J. East, Idempotents and one-sided units in infinite partial Brauer monoids, J. Algebra, 534 427-482.
[5] S. M. Gagola III and L.-Ch. Kappe, On the covering number of loops, Expo. Math., 34 (2016) 436–447.
[6] M. Garonzi, L.-Ch. Kappe and E. Swartz, On Integers that are Covering Numbers of Groups, Exp. Math., 31 (2022)
425–443.
[7] G. M. S. Gomes and J. M. Howie, On the ranks of certain finite semigroups of transformations, Math. Proc.
Cambridge Philos. Soc., 101 (1987) 395–403.
[8] N. Graham, R. Graham and J. Rhodes, Maximal subsemigroups of finite semigroups, J. Combinatorial Theory, 4
(1968) 203–209.
[9] J. M. Howie, Fundamentals of Semigroup Theory, London Mathematical Society Monographs, New Series, 12,
Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
[10] A. Lucchini and A. MaroĢti, Rings as the Unions of Proper Subrings, Algebr. Represent. Theory, 15 (2012) 1035–1047.
[11] D. Rees, On semi-groups, Proc. Cambridge Philos. Soc., 36 (1940) 387–400.
[12] S. Satoh, K. Yama and M. Tokizawa,Semigroups of order 8, Semigroup Forum, 49 (1994) 7–29.
[13] G. Scorza, I gruppi che possone pensarsi come somma di tre lori sottogruppi, Boll. Un. Mat. Ital., 5 (1926) 216–218.
[14] M. J. Tomkinson, Groups as the union of proper subgroups, Math. Scand., 81 (1997) 191–198.
[15] R. J. Warne, On the structure of semigroups which are unions of groups, Trans. Amer. Math. Soc., 186 (1973)
385–401.
[16] N. J. Werner, Covering numbers of finite rings, Amer. Math. Monthly, 122 (2015) 552–566.