An interesting combinatorial method in the theory of locally finite semigroups

Thomas Craig Brown

Vol. 36 (1971), No. 2, 285–289

DOI: 10.2140/pjm.1971.36.285

Abstract

Let X be a finite set, x∗ the free semigroup (without identity) on X, let M be a finite semigroup, and let φ be an epimorphism of x∗ upon M. We give a simple proof of a combinatorial property of the triple (X,φ,M), and exploit this property to ge t very simple proofs for these two theorems: 1. If φ is an epimorphism of the semigroup S upon the locally finite semigroup T such that φ⁻¹(e) is a locally finite subsemigroup of S for each idempotent element e of T, then S is locally finite. 2. Throughout 1, replace “locally finite” by “locally nilpotent”.

The method is simple enough, and yet powerful enough, to suggest its applicability in other contexts.

Mathematical Subject Classification

Primary: 20.93

Milestones

Received: 3 November 1969

Published: 1 February 1971