An interesting combinatorial method in the theory of locally finite semigroups

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 get 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