S{\l}upecki Digraphs

TL;DR

This paper investigates the properties of Slupecki digraphs, providing conditions for reflexive digraphs to be Slupecki and analyzing specific classes like triangulating 1-spheres and ordinal sums, revealing nuanced polymorphism behaviors.

Contribution

It establishes new necessary and sufficient conditions for a reflexive digraph to be Slupecki and analyzes the polymorphism properties of various classes of digraphs and posets.

Findings

All digraphs triangulating a 1-sphere are Slupecki.

All ordinal sums m ⊕ n are Slupecki for m,n ≥ 2.

Certain posets m ⊕ n ⊕ k are not 3-Slupecki, but can be 2-Slupecki depending on parameters.

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 $k \geq 2$. 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 $m \oplus n$ ($m,n \geq 2$). We prove that the posets $P = m \oplus n \oplus k$ are not 3-Slupecki for $m,n,k \geq 2$, 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.