Sequences with idempotent products from finite regular semigroups

Abstract

We find the smallest integer α( n) such that for every regular semigroup S of order n, every sequence of length α( n) of elements of S contains a consecutive subsequence whose product is an α-element, where α= ‘ idempotent’, ‘ core’ and ‘ subgroup and core’. For arbitrary semigroups of order n, we also find α( n) where α= ‘ regular’, ‘ group’, ‘ core’, ‘ regular and core’ and ‘ subgroup and core’.