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Semigroups of I-type

Tatiana Gateva-Ivanova·Michel Van den Bergh
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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.

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Tatiana Gateva-Ivanova, Michel Van den Bergh. 2003-08-08. Semigroups of I-type. https://arxiv.org/abs/math/0308071

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