Finiteness and iteration conditions for semigroups

AbstractLet S be a semigroup and m and n two integers such that m > 0 and n ⩾ 0. We say that S verifies the iteration property C(n, m) if the following condition is verified: For any sequence s1, s2, …, sm of m elements of S there exist i, j such that 1⩽i⩽j⩽m and s1…sm=s1…si−1 si…sj)nsj+1…sm. The main result of the paper is that if a finitely generated semigroup S satisfies C(2, m) or C(3, m) for a suitable m > 0 then S is finite. An application to the theory of regular languages is given. There exists a positive, uniform, “block-pumping property” which assures the regularity of a language. This result gives a partial answer to a question raised in Ehrenfeucht (1981)