About posets of height one as retracts

TL;DR

This paper explores the conditions under which certain height-one connected posets are retracts of finite posets, using graph homomorphisms between associated multigraphs to characterize these relationships.

Contribution

It introduces a novel graph-theoretic framework involving multigraphs to analyze retracts of finite posets of height one, linking poset structure to graph homomorphisms.

Findings

Retracts correspond to specific graph homomorphisms between multigraphs.

Characterization of retracts when the poset is an ordinal sum of two antichains.

Sparse improper 4-crowns relate to the structure of retracts.

Abstract

We investigate connected posets $C$ of height one as retracts of finite posets $P$. We define two multigraphs: a multigraph $\mathfrak{F}(P)$ reflecting the network of so-called improper 4-crown bundles contained in the extremal points of $P$, and a multigraph $\mathfrak{C}(C)$ depending on $C$ but not on $P$. There exists a close interdependence between $C$ being a retract of $P$ and the existence of a graph homomorphism of a certain type from $\mathfrak{F}(P)$ to $\mathfrak{C}(C)$. In particular, if $C$ is an ordinal sum of two antichains, then $C$ is a retract of $P$ iff such a graph homomorphism exists. Returning to general connected posets $C$ of height one, we show that the image of such a graph homomorphism can be a clique in $\mathfrak{C}(C)$ iff the improper 4-crowns in $P$ contain only a sparse subset of the edges of $C$.