Speaker
Description
Kiselman’s semigroup $K_n$ was introduced in the context of convexity theory and later studied in algebraic combinatorics. Kudryavtseva and Mazorchuk posed the open problem of classifying all endomorphisms of $K_n$, which remained unresolved. We present a complete description of the endomorphism monoid $\text{End}(K_n)$. We construct a Boolean matrix monoid $(Dn, ·)$, consisting of $n × n$ matrices over ${0, 1}$ avoiding the $2 × 2$ permutation matrix $P = \begin{pmatrix}0&1\\1&0\end{pmatrix}$ as a submatrix, and show that $\text{End}(K_n) \cong (D_n, ·)$. The classification proceeds via an intermediate monoid $(M_n, ∗)$ of so-called monotone sequences of subsets of $\{1, \ldots , n \}$, which captures the combinatorics of idempotents in $K_n$. These sequences are in bijection with Boolean matrices in $D_n$ via characteristic vectors, preserving monoid structure. As corollaries, we recover the known result that $\text{Aut}(K_n)$ is trivial, and compute the cardinality of $\text{End}(K_n)$ for small $n$ via existing results on pattern-avoiding matrices. This resolves an open problem and reveals a rich combinatorial structure underlying $\text{End}(K_n)$.
