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.

arXiv comment17 pages, 6 figures

References in corpus (1)

• The fixed point property of a poset and the fixed point property of the poset induced by its extremal points