The fixed point property of a poset and the fixed point property of the poset induced by its extremal points
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910675363692544 |
|---|---|
| author | Campo, Frank a |
| author_facet | Campo, Frank a |
| contents | For a connected finite poset $P$, let $E(P)$ be the poset induced by the extremal points of $P$. We show that the fixed point property of $E(P)$ implies the fixed point property of $P$. On the other hand, we show that a homomorphism $f : E(P) \rightarrow Q$ can be extended to $P$ if $Q$ is a flat poset not containing a 4-crown. We conclude that every retract-crown of $E(P)$ with more than four points is a retract-crown of $P$, too. We see that for $P$ having the fixed point property but $E(P)$ not, every edge of every crown in $E(P)$ must belong to a so-called improper 4-crown, with additional specifications if $P$ has height two. The results provide several sufficient and necessary conditions for $P$ having the fixed point property, and these conditions refer to objects simpler than $P$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_03122 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The fixed point property of a poset and the fixed point property of the poset induced by its extremal points Campo, Frank a Combinatorics 06A07 (Primary) 06A06 (Secondary) For a connected finite poset $P$, let $E(P)$ be the poset induced by the extremal points of $P$. We show that the fixed point property of $E(P)$ implies the fixed point property of $P$. On the other hand, we show that a homomorphism $f : E(P) \rightarrow Q$ can be extended to $P$ if $Q$ is a flat poset not containing a 4-crown. We conclude that every retract-crown of $E(P)$ with more than four points is a retract-crown of $P$, too. We see that for $P$ having the fixed point property but $E(P)$ not, every edge of every crown in $E(P)$ must belong to a so-called improper 4-crown, with additional specifications if $P$ has height two. The results provide several sufficient and necessary conditions for $P$ having the fixed point property, and these conditions refer to objects simpler than $P$. |
| title | The fixed point property of a poset and the fixed point property of the poset induced by its extremal points |
| topic | Combinatorics 06A07 (Primary) 06A06 (Secondary) |
| url | https://arxiv.org/abs/2107.03122 |