A Finiteness Condition for Semigroups Generalizing a Theorem of Coudrain and Schützenberger

Abstract

Let S be a semigroup. For s, t ∈ S we set s ≤Bt if s ∈ {t} ∪ tS1t; we say that S satisfies the condition minB, if and only if any strictly descending chain w.r.t. ≤B of elements of S has a finite length. The main result of the paper is the following theorem: Let T be a semigroup satisfying minB. Let T′ be a subsemigroup of T such that all subgroups of T are locally finite in T′. Then T′ is locally finite. This result is a noteworthy generalization of a theorem of Coudrain and Schützenberger. Moreover, as a corollary we obtain the theorem of McNaughton and Zalcstein which gives a positive answer to the Burnside problem for semigroups of matrices on a field.