Słupecki digraphs

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