Słupecki Digraphs

Abstract

Call a finite relational structure $k$-Slupecki if its only surjective $k$-ary polymorphisms are essentially unary, and Slupecki if it is $k$-Slupecki for all $k\ge 2$. We present conditions, some necessary and some sufficient, for a reflexive digraph to be Slupecki. We prove that all digraphs that triangulate a 1-sphere are Slupecki, as are all the ordinal sums $m\oplus n$ ($m,n\ge 2$). We prove that the posets $P=m\oplus n\oplus k$ are not 3-Slupecki for $m,n,k\ge 2$, and prove there is a bound $B(m,k)$ such that $P$ is 2-Slupecki if and only if $n>B(m,k)+1$; in particular there exist posets that are 2-Slupecki but not 3-Slupecki.