Semigroups of I-type

Assume that $S$ is a semigroup generated by $\{x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n\}$, and let $\text{Uscr}$ be the multiplicative free commutative semigroup generated by $\{u_1,\mathrm{.}\mathrm{.}\mathrm{.},u_n\}$. We say that $S$ is of \emph{$I$-typ}e if there is a bijection $v:\text{Uscr}\mathring{S}$ such that for all $a\in \text{Uscr}$, ${v(u_1a),… v(u_na)}={x_1v(a),…,x_nv(a)}$. 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 settheoretic 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.