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.