The Finite Basis Problem for Kiselman Monoids

TL;DR

This paper characterizes the identities of Kiselman monoids, showing they are nonfinitely based for all n ≥ 4, and provides finite bases for smaller cases, advancing understanding of their algebraic structure.

Contribution

It extends the description of identities to Hecke--Kiselman monoids, including Kiselman monoids, and determines their finite or infinite basis status.

Findings

Identities of Kiselman monoids are nonfinitely based for n ≥ 4.

Finite identity bases are found for Kiselman monoids with n=2 and n=3.

The paper resolves an open question from the initial submission.

Abstract

In an earlier paper, the second-named author has described the identities holding in the so-called Catalan monoids. Here we extend this description to a certain family of Hecke--Kiselman monoids including the Kiselman monoids $\mathcal{K}_n$. As a consequence, we conclude that the identities of $\mathcal{K}_n$ are nonfinitely based for every $n\ge 4$ and exhibit a finite identity basis for the identities of each of the monoids $\mathcal{K}_2$ and $\mathcal{K}_3$. In the third version a question left open in the initial submission has beed answered.