About posets of height one as retracts

2024/10/29 by Frank a Campo, Campo, Frank a
Engineering · Social Sciences · #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Historical Geography and Cartography

paper · pdf · doi:10.48550/arxiv.2410.22379

Abstract

We investigate connected posets C of height one as retracts of finite posets P. We define two multigraphs: a multigraph \mathfrakF(P) reflecting the network of so-called improper 4-crown bundles contained in the extremal points of P, and a multigraph \mathfrakC(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 \mathfrakF(P) to \mathfrakC(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 \mathfrakC(C) iff the improper 4-crowns in P contain only a sparse subset of the edges of C.

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