Semigroups of I-type

TL;DR

This paper explores semigroups of I-type, revealing their connections to solutions of the Yang-Baxter equation, Bieberbach groups, and skew binomial polynomial rings, thus unifying various mathematical concepts.

Contribution

It establishes the relationship between I-type semigroups and multiple areas such as algebraic solutions, geometric groups, and polynomial rings, highlighting their interconnectedness.

Findings

Semigroups of I-type are linked to set-theoretic solutions of the Yang-Baxter equation.

They are related to Bieberbach groups, which are fundamental in geometric group theory.

Connections are made with skew binomial polynomial rings introduced by the first author.

Abstract

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.