# ALGB Seminar: Arkadiusz Mecel (University of Warsaw)

Title: Grobner basis and the automaton property of Hecke-Kiselman algebras.

Abstract:

In the work with an algebra A defined by generators and defining relations, we intuitively tend to work in the language of words, i.e. the elements of a free algebra. It is possible to do with the notions of normal words and the decomposition of a free algebra into an ideal and its normal complement. One method to obtain this decomposition utilizes the famous diamond lemma of Bergman and leads to the so-called minimal Grobner basis of A. If this Grobner basis is finite, the properties of a finitely generated algebra A become very natural. The notion, however, requires both the choice of generators and an ordering on monomials. It is highly nontrivial to determine if the algebra has finite Grobner basis, if the natural ordering fails to produce one. Another class of algebras is therefore considered to overcome this difficulty - the automaton algebras, whose language of normal words is regular. This allows the Grobner basis to be infinite but the properties of the algebras stay ,,nice".

In the work with J. Okninski and M. Wiertel (both from University of Warsaw) we consider Hecke-Kiselman algebras introduced as a generalization of 0-Hecke algebras of Coxeter groups. The combinatorics of these algebras is fairly simple to work with, yet it can be shown that these algebras provide a class with infinite Grobner basis which is automaton. Not many ,,small" examples of this kind were known. I will talk about recent results obtained in this area.