The Finite Basis Problem for Kiselman Monoids

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.