Hecke-Kiselman Monoids of Small Cardinality

TL;DR

This paper characterizes small digraphs with at most four vertices for which the associated Hecke-Kiselman monoid is finite, revealing that acyclicity and Dynkin diagram components are necessary but not sufficient conditions.

Contribution

It provides a complete characterization of finite Hecke-Kiselman monoids for small digraphs and demonstrates that necessary conditions are not always sufficient.

Findings

Finite monoids require acyclic digraphs with Dynkin diagram components

Necessary conditions are not sufficient for finiteness in small cases

Constructed examples show exceptions to the conditions

Abstract

In this paper, we give a characterization of digraphs $Q, |Q|\leq 4$ such that the associated Hecke-Kiselman monoid $H_Q$ is finite. In general, a necessary condition for $H_Q$ to be a finite monoid is that $Q$ is acyclic and its Coxeter components are Dynkin diagram. We show, by constructing examples, that such conditions are not sufficient.