Hopf algebra structure of incidence algebras

TL;DR

This paper explores the Hopf algebra structure of incidence algebras of posets, revealing a rich theory where the algebra supports a coalgebra structure, forming a weak Hopf algebra with the Möbius function as antipode.

Contribution

It introduces the concept of m-weak bialgebra and Hopf algebra structures in incidence algebras, expanding the theoretical framework of algebraic structures on posets.

Findings

Incidence algebra supports a natural coalgebra structure.

Möbius function acts as the antipode in the weak Hopf algebra.

Theory developed using equivalence relations on intervals in the poset.

Abstract

The incidence algebra of a partially ordered set (poset) supports in a natural way also a coalgebra structure, so that it becomes a m-weak bialgebra even a m-weak Hopf algebra with M\"obius function as antipode. Here m-weak means that multiplication and comultiplication are not required to be coalgebra- or algebra-morphisms, respectively. A rich theory is obtained in computing modulo an equivalence relation on the set of intervals in the poset.