Crowns as retracts

TL;DR

This paper characterizes crowns as retracts in finite posets using a graph-theoretic approach, providing criteria for their systematic identification, applicable also to certain infinite posets.

Contribution

It introduces a multigraph framework and homomorphism criteria to detect crowns as retracts in finite and some infinite posets, advancing understanding of their structural properties.

Findings

A multigraph $rak{F}(P)$ reflects improper 4-crowns in $P$

Existence of a certain graph homomorphism indicates a crown retract

Criteria for systematic investigation of crowns as retracts

Abstract

We investigate crowns as retracts of finite posets. We define a multigraph $\mathfrak{F}(P)$ reflecting the network of so-called improper 4-crowns contained in the extremal points of $P$, and we show that $P$ contains a 4-crown as retract iff there exists a graph homomorphism of a certain type from $\mathfrak{F}(P)$ to a multigraph $\mathfrak{C}$ not depending on $P$. Additionally we show that $P$ contains a retract-crown with more than four points iff the poset induced by the extremal points of $P$ contains such a retract-crown. As practical result we develop and apply criteria for the systematic investigation of crowns as retracts. Most of our results are valid for infinite posets without infinite chains, too.