On Finite Semigroups and Idempotents

Structure of the largest idempotent-product free sequences in semigroups

“…By Lemma 2.1, we may assume that A y is an ideal extension of a group G y by a nilsemigroup N y (note that G y or N y may be trivial which shall be reduced to the case that A y is a nilsemigroup or a group). Now we show that |G y ∩ supp(T )| ≤ 1 (8) and…”

A stronger connection between the Erdős-Burgess and Davenport constants

“…In 1969, Burgess [4] answered this question in the affirmative when S is commutative or contains only a single idempotent element. In 1972, Gillam, Hall, and Williams [11] proved the stronger result that I(S) ≤ |S| − |E| + 1 for all finite semigroups S, where E is the set of idempotent elements of S. They also showed that this bound is sharp in the sense that for any positive integers m < n, there exists a semigroup S with |S| = n, |E| = m, and I(S) = |S| − |E| + 1.…”

Ramsey's Theorem in algebraic semigroups

“…Another recent contribution is that of Bergelson and Hindman [9] in which Van der Waerden's Theorem is generalized by placing the question in the context of products in semigroups. Another result in similar vein to Lemma 1.5 is to be found in [20]: if 5 is a finite semigroup then any product A = a\a2 • • • ak where k = |S| -\E\ + 1 contains a sub word, (by which we mean a product formed from A by deletion of some of the a, ), which is idempotent. idempotents in the generating set B is bounded above by k, then mk is an upper bound for their lengths as products of the set A).…”