A non-associative incidence near-ring with a generalized M\"obius function

TL;DR

This paper introduces a generalized M"obius function on partial flag functions of posets, establishing its algebraic properties and applications to matroid theory, including polynomial computations and valuation properties.

Contribution

It defines a new non-associative incidence near-ring with a generalized M"obius function, extending classical combinatorial theorems to this broader context.

Findings

The generalized M"obius function is a one-sided inverse of the zeta function.

Analogues of classical theorems like Rota's Crosscut are proved for this setting.

The generalized M"obius polynomial has -1 as a root for modular matroids.

Abstract

There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized M\"obius function. Under the product this generalized M\"obius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall's Theorem on the M\"obius function as an alternating sum of chain counts, Weisner's theorem, and Rota's Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical M\"obius functions. Using this generalized M\"obius function we define analogues of the characteristic polynomial and M\"obius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized M\"obius polynomial has -1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.