  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).
