Semigroups of I-type

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 $a\in\Uscr$, $\{v(u_{1}a),...v(u_{n}a)\}=\{x_{1}v(a),...,x_{n}v(a)\}$. 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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• Finite quotients of groups of I-type
• Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation
• Enumeration of set-theoretic solutions to the Yang-Baxter equation
• From Braces to Hecke algebras & Quantum Groups
• Involutive latin solutions of the Yang-Baxter equation
• Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses
• Yang-Baxter operators in symmetric categories
• Combinatorial solutions to the reflection equation
• Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces
• Skew braces of size $p^{2}q$ I: abelian type
• "Set-theoretical" solutions of the quantum Yang-Baxter equation and a class of Garside groups
• New Solutions to the Reflection Equation with Braces
• Set-theoretic solutions of the Yang--Baxter equation, associated quadratic algebras and the minimality condition
• Finite dimensional Hopf actions on central division algebras
• Veronese and Segre morphisms between non-commutative projective spaces
• The matched product of set-theoretical solutions associated with shelves
• Cocommutative q-cycle coalgebra structures on the dual of the truncated polynomial algebra
• Simplicity and finite primitive level of indecomposable set-theoretic solutions of the Yang-Baxter equation
• Folding of set-theoretical solutions of the Yang-Baxter equation