Combinatorics and structure of Hecke-Kiselman algebras

To every finite graph with $n$ vertices one can associate a finitely presented monoid, called the Hecke-Kiselman monoid. It is defined as a monoid generated by $n$ idempotents with relations of the form $xy=yx$, $xyx=yxy$ or $xyx=yxy=xy$, depending on the edges between vertices $x$ and $y$ in the graph. In my talk I will focus on structure of Hecke-Kiselman monoids and the associated semigroup algebras in the case of directed graphs.

We investigate the interplay between combinatorial and ring-theoretic properties. In the case of an oriented cycle certain structures of matrix type are discovered within the associated Hecke-Kiselman monoid. This allows us to describe the ring theoretical properties of the monoid algebra associated to the cycle and as a consequence characterize all Noetherian Hecke-Kiselman algebras $K[HK_{\theta}]$ in terms of graph $\theta$ for directed graphs. The talk is based on a joint work with J.Okniński.