Gr\"obner basis and the automaton property of Hecke--Kiselman algebras

TL;DR

This paper proves that Hecke-Kiselman algebras from finite directed graphs are automaton algebras, leading to results on their Gelfand-Kirillov dimension and the existence of finite Gr"obner bases for certain cases.

Contribution

It establishes the automaton algebra property for Hecke-Kiselman algebras and derives implications for their Gelfand-Kirillov dimension and Gr"obner bases.

Findings

Hecke-Kiselman algebra is an automaton algebra for finite graphs.

Gelfand-Kirillov dimension is an integer if finite.

Finite Gr"obner basis exists for algebras from oriented cycles.

Abstract

It is shown that the Hecke-Kiselman algebra associated to a finite directed graph is an automaton algebra in the sense of Ufnarovskii. Consequently, its Gelfand-Kirillov dimension is an integer if it is finite. As a consequence, it is proved that the Hecke-Kiselman algebra associated to an oriented cycle admits a finite Gr\"obner basis.