A note on the Burnside problem for semigroups (Q759839)

The author proves that a periodic semigroup S is finite if and only if there exist a finite alphabet A and a strongly repetitive epimorphism \(\phi\) from the free semigroup \(A^+\) on A onto S, where ''strongly repetive'' means that: for any map \(F: {\mathbb{N}}\to {\mathbb{N}}\) there exists an integer r such that any word in \(A^+\) of length greater than r may be factorized as \(w=hv_ 1...v_{F(p)}h'\), for some integer p and some words \(h,h',v_ i\) satisfying: \(1\leq | v_ i| \leq p\) and \(\nu_ 1\phi =\nu_ 2\phi =...=\nu_ p\phi\). The ''only if'' part is a theorem of \textit{J. Justin} [J. Comb. Theory, Ser. A 12, 357-367 (1972; Zbl 0248.05003)].