On Kiselman's semigroup

We study the algebraic properties of the series ${\mathrm{K}}_n$ of semigroups, which is inspired by \cite{Ki} and has origins in convexity theory. In particular, we describe Green’s relations on ${\mathrm{K}}_n$, prove that there exists a faithful representation of ${\mathrm{K}}_n$ by $n\times n$ matrices with non-negative integer coefficients (and even explicitly construct such a representation), and prove that ${\mathrm{K}}_n$ does not admit a faithful representation by matrices of smaller size. We also describe the maximal nilpotent subsemigroups in ${\mathrm{K}}_n$, all isolated and completely isolated subsemigroups, all automorphisms and anti-automorphisms of ${\mathrm{K}}_n$. Finally, we explicitly construct all irreducible representations of ${\mathrm{K}}_n$ over any field and describe primitive idempotents in the semigroup algebra (which we prove is basic).