A complete classification of endomorphisms of Kiselman's semigroup

TL;DR

This paper provides a complete classification of all endomorphisms of Kiselman's semigroup $K_n$, including an isomorphic Boolean matrix monoid, advancing understanding of its algebraic structure.

Contribution

The paper offers the first full classification of endomorphisms of $K_n$ and constructs an isomorphic Boolean matrix monoid, filling a key gap in the algebraic theory of $K_n$.

Findings

Complete classification of endomorphisms of $K_n$

Construction of an isomorphic Boolean matrix monoid

Resolution of the classification question posed by Kudryavtseva and Mazorchuk

Abstract

Kiselman's semigroup $K_n$ was studied by Kudryavtseva and Mazorchuk, who posed the question of whether it is possible to classify all endomorphisms of $K_n$. In this paper, we provide a complete classification of endomorphisms of $K_n$ and present a Boolean matrix monoid that is isomorphic to $\text{End}(K_n)$.