A Non-Multiplicable Upho Poset Constructed from the Petersen Graph

Mathematics > Combinatorics [Submitted on 16 Jun 2026] Title:A Non-Multiplicable Upho Poset Constructed from the Petersen Graph View PDF HTML (experimental)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. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.