A note on M\"{o}bius functions of upho posets

TL;DR

This paper explores the properties of upho posets, revealing that their rank and characteristic generating functions are multiplicative inverses, which offers new insights into their algebraic structure.

Contribution

It introduces the relationship between rank and characteristic generating functions of upho posets, highlighting their multiplicative inverse connection.

Findings

Rank and characteristic generating functions are multiplicative inverses.

Upho posets have a unique algebraic structure.

The paper provides foundational observations for further research.

Abstract

A poset is called upper homogeneous (or "upho") if every principal order filter of the poset is isomorphic to the whole poset. We observe that the rank and characteristic generating functions of upho posets are multiplicative inverses of one another.