A complete classification of endomorphisms of Kiselman's semigroup

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 $\text{End}(K_n)$.