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arXiv · 2111.09588

Crowns as retracts

Frank a Campo
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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.

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Frank a Campo. 2021-11-18. Crowns as retracts. https://doi.org/10.1007/s11083-021-09585-0

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