Incidence Monoids: Automorphisms and Complexity

TL;DR

This paper explores the algebraic structure of incidence monoids, revealing how their automorphisms are determined by multiplicative structure, and relates their complexity to poset properties, including classifications and covering relations.

Contribution

It provides a new formula linking incidence monoid complexity to the zeta polynomial and characterizes posets with low complexity incidence monoids.

Findings

Automorphisms are determined by the multiplicative structure.

Complexity is expressed via the zeta polynomial of the poset.

Finite posets with complexity ≤ 1 are characterized.

Abstract

The algebraic monoid structure of an incidence algebra is investigated. We show that the multiplicative structure alone determines the algebra automorphisms of the incidence algebra. We present a formula that expresses the complexity of the incidence monoid with respect to the two sided action of its maximal torus in terms of the zeta polynomial of the poset. In addition, we characterize the finite (connected) posets whose incidence monoids have complexity $\leq 1$. Finally, we determine the covering relations of the adherence order on the incidence monoid of a star poset.