A semigroup union of disjoint locally finite subsemigroups which is not locally finite

Thomas Craig Brown

Vol. 22 (1967), No. 1, 11–14

DOI: 10.2140/pjm.1967.22.11

Abstract

The semigroup S of the title is the free semigroup F on four generators factored by the congruence generated by the set of relations {w² = w³|w ∈ F}. The following lemma is proved by examining the elements of a given congruence class of F:

Lemma. If x,y ∈ S and x² = y², then either xy = x² or yx = x².

From the Lemma it then easily follows that the (disjoint) subsemigroups {y ∈ S|y² = x²} of S are locally finite.

Mathematical Subject Classification

Primary: 20.18

Milestones

Received: 19 December 1966

Published: 1 July 1967