On Kiselman's semigroup

Abstract

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).

26 pages

References in corpus (1)

• Isolated subsemigroups in the variants of T_n

Cited by in corpus (7)

• On Kiselman quotients of 0-Hecke monoids
• Combinatorial categorical equivalences of Dold-Kan type
• Effective representations of Hecke-Kiselman monoids of type A
• Excursions into Algebra and Combinatorics at $q=0$
• Algebraic and Affine Pattern Avoidance
• Categorification of the Catalan monoid
• The word problem for Hecke--Kiselman monoids of type $A_n$ and ${\tilde{A}}_n$