About posets for which no lower cover or no upper cover has the fixed point property

TL;DR

This paper studies specific posets within the set of all posets on a finite set, identifying conditions under which no immediate extension (cover) has the fixed point property, and confirms a related conjecture.

Contribution

It provides new conditions characterizing when no lower or upper cover of a poset has the fixed point property, and verifies a conjecture by Schröder.

Findings

Derived two conditions for fixed point property absence in covers

Conditions are equivalent to the absence of fixed point property in covers when the poset itself has it

Confirmed Schröder's conjecture regarding fixed point properties in posets

Abstract

For a finite non-empty set $X$, let $\mathfrak{P}(X)$ denote the set of all posets with carrier $X$, ordered by inclusion of their partial order relations. We investigate properties of posets $P \in \mathfrak{P}(X)$ for which no lower cover or no upper cover in $\mathfrak{P}(X)$ has the fixed point property. We derive two conditions, one of them sufficient for that no lower cover of $P$ has the fixed point property, the other one sufficient for that no upper cover of $P$ has the fixed point property. If $P$ itself has the fixed point property, the conditions are even equivalent to the respective total lack of lower or upper covers with the fixed point property. We use the results to confirm a conjecture of Schr\"{o}der.