Abstract:An upper homogeneous (upho) poset is a poset whose every principal filter is isomorphic to the whole poset. Fu--Peng--Zhang conjectured that every finitary upho poset admits a compatible left-cancellative, invertible-free monoid structure whose left-divisibility order coincides with the given order. We disprove this conjecture. For every vertex-transitive graph $G$, we construct a finitary upho poset $P(G,v_0)$ from walks starting at a fixed vertex $v_0$. Applying this construction to the Petersen graph, we show that multiplicability of $P(G,v_0)$ would force the automorphism group of $G$ to contain a regular subgroup. This would imply that $G$ is a Cayley graph, contradicting the fact that the Petersen graph is not Cayley. Hence $P(G,v_0)$ is a non-multiplicable finitary upho poset. We also show that the analogous poset associated with the line graph of the Petersen graph is multiplicable, demonstrating that non-Cayleyness of the underlying graph alone does not determine multiplicability.
From: Ryunosuke Matsuoka [view email]
[v1]
Tue, 16 Jun 2026 05:52:25 UTC (16 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。