Chains of subsemigroups

Cameron, Peter J.; Gadouleau, Maximilien; Mitchell, James D.; Peresse, Yann

doi:10.1007/s11856-017-1523-x

Chains of subsemigroups

Cameron, Peter J.; Gadouleau, Maximilien; Mitchell, James D.; Peresse, Yann

Authors

Peter J. Cameron

Dr Maximilien Gadouleau m.r.gadouleau@durham.ac.uk
Associate Professor

James D. Mitchell

Yann Peresse

Abstract

We investigate the maximum length of a chain of subsemigroups in various classes of semigroups, such as the full transformation semigroups, the general linear semigroups, and the semigroups of order-preserving transformations of finite chains. In some cases, we give lower bounds for the total number of subsemigroups of these semigroups. We give general results for finite completely regular and finite inverse semigroups. Wherever possible, we state our results in the greatest generality; in particular, we include infinite semigroups where the result is true for these. The length of a subgroup chain in a group is bounded by the logarithm of the group order. This fails for semigroups, but it is perhaps surprising that there is a lower bound for the length of a subsemigroup chain in the full transformation semigroup which is a constant multiple of the semigroup order.

Citation

Cameron, P. J., Gadouleau, M., Mitchell, J. D., & Peresse, Y. (2017). Chains of subsemigroups. Israel Journal of Mathematics, 220(1), 479-508. https://doi.org/10.1007/s11856-017-1523-x

Journal Article Type	Article
Acceptance Date	Aug 7, 2016
Online Publication Date	May 8, 2017
Publication Date	Jun 1, 2017
Deposit Date	Aug 26, 2016
Publicly Available Date	May 8, 2018
Journal	Israel Journal of Mathematics
Print ISSN	0021-2172
Electronic ISSN	1565-8511
Publisher	Springer
Peer Reviewed	Peer Reviewed
Volume	220
Issue	1
Pages	479-508
DOI	https://doi.org/10.1007/s11856-017-1523-x