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