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

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 Schröder.

Figures (3)

• Figure 1: An example of an $\mathfrak{ L }$- and $\mathfrak{ U }$-shielded poset with an FPP-graph.
• Figure 2: Illustration of Definition \ref{['def_dreiChProp']}. There are two possible locations for the points $a$ and $b$ in the 3-chain $C$, resulting in different posets $C \setminus (a,b)$. In the first case, we have $(x,y) \in \{ (a,w), (b,w) \}$, in the second one $(x,y) \in \{ (u,a), (u,b) \}$. Condition \ref{['xy_LU']} deals with $(x,y) \in \{ (a,w), (u,b) \}$, whereas the conditions \ref{['x_M']} and \ref{['y_M']} deal with $(x,y) = (b,w)$ and $(x,y) = (u,a)$, respectively.
• Figure 3: Schröder's poset $X_{16}$. For better readability, the indices of point labels are not lowered and the edges are partly dotted.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2: Schroeder_2021
• Definition 1
• Definition 2
• Proposition 1
• proof
• Lemma 3