FINITENESS AND ITERATION CONDITIONS FOR SEMIGROUPS

Let S be a semigroup and m and n two integers such that m > 0 and n greater-than-or-equal-to 0. We say that S verifies the iteration property C(n, m) if the following condition is verified: For any sequence s1, s2,...,s(m) of m elements of S there exist i, j such that 1 less-than-or-equal-to i less-than-or-equal-to j less-than-or-equal-to m and S1...S(m) = S1...S(i-1) (S(i)...S(j))(n)S(j+1)...S(m). 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 et al. (1981).