Semigroups ofI-Type

Assume thatSis a semigroup generated by {x1,…,xn}, and let be the multiplicative free commutative semigroup generated by {u1,…,un}. We say thatSis ofI-typeif there is a bijectivev : → Ssuch that for alla , {v(u1a),…,v(una)} = {x1v(a),…,xnv(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 ofI-type is related to various other mathematical notions found in the literature. In particular we show that semigroups ofI-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.