Some combinatorial properties of infinite words and applications to semigroup theory Original Research Article

This paper is concerned with finiteness conditions for finitely generated semigroups. First, we present a combinatorial result on infinite sequences from which an alternative proof of a theorem of Restivo and Reutenauer follows: a finitely generated semigroup is finite if and only if it is periodic and permutable. Then, generalizing notions studied in some papers of de Luca, Restivo, Hashiguchi and Varricchio, we introduce the notion of ω-iteration property and we prove that a finitely generated semigroup has the ω-iteration property if and only if it is finite.