About posets of height one as retracts
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916459959025664 |
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| author | Campo, Frank a |
| author_facet | Campo, Frank a |
| contents | 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 $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22379 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | About posets of height one as retracts Campo, Frank a Combinatorics 06A07 (Primary) 06A06 (Secondary) 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 $C$. |
| title | About posets of height one as retracts |
| topic | Combinatorics 06A07 (Primary) 06A06 (Secondary) |
| url | https://arxiv.org/abs/2410.22379 |