Vol. 36, No. 2, 1971

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An interesting combinatorial method in the theory of locally finite semigroups

Thomas Craig Brown

Vol. 36 (1971), No. 2, 285–289

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 xupon 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 φ1(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
Received: 3 November 1969
Published: 1 February 1971
Thomas Craig Brown