Proceedings of the Edinburgh Mathematical Society (Series 2)

Table of Contents - June 2012 - Volume 55, Issue 02  

Research Article

Gröbner bases for quadratic algebras of skew type

Ferran Cedóa1 and Jan Oknińskia2

a1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

a2 Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warsaw, Poland (okninski@mimuw.edu.pl)

Abstract

Non-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and $\binom{n}{2}$ quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

(Received March 08 2011)

Keywords

  • finitely presented semigroup;
  • semigroup ring;
  • semigroup;
  • normal form;
  • regular language

2010 Mathematics subject classification

  • Primary 16S15;
  • 16S36;
  • Secondary 20M05;
  • 20M20;
  • 20M25;
  • 20M35;
  • 68Q70