Chains of subsemigroups

TL;DR

This paper studies the maximum length of chains of subsemigroups across various classes of semigroups, providing bounds and general results, including for infinite semigroups, revealing differences from group subgroup chains.

Contribution

It introduces bounds on the length of subsemigroup chains in different semigroup classes and extends results to infinite semigroups, highlighting novel structural insights.

Findings

Lower bounds for the number of subsemigroups in certain classes

A constant multiple bound for chain length in full transformation semigroups

General results applicable to finite and infinite semigroups

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.