Semigroups of I-type

Assume that S is a semigroup generated by {x(1),..., x(n)}, and let U be the multiplicative free commutative semigroup generated by {u(1),..., u(n)}. We say that S is of I-type if there is a bijective upsilon : U --> S such that for all a is an element of U, {upsilon(u(1)a),..., upsilon(u(n)a)} = {x(1)upsilon(a),..., x(n)upsilon(a)}. This condition appeared naturally in the work on Sklyanin algebras by John Tate and the second author. In this paper we show that the condition for a semigroup to be of I-type is related to various other mathematical notions found in the literature. In particular we show that semigroups of I-type appear in the study of the set-theoretic solutions of the Yang-Baxter equation, in the theory of Bieberbach groups, and in the study of certain skew binomial polynomial rings which were introduced by the first author. (C) 1998 Academic Press.