The incidence algebras of posets and acyclic categories

TL;DR

This paper explores the algebraic structures of incidence algebras for posets and acyclic categories, establishing conditions for their quadratic Groebner bases and linking them to lex-shellability.

Contribution

It introduces a new characterization of incidence algebras of acyclic categories via quadratic Groebner bases and connects this to lex-shellability.

Findings

Quadratic Groebner basis exists if and only if the category is lex-shellable.

Incidence algebra is a quotient of a path algebra by the parallel ideal.

Provides algebraic criteria for topological and combinatorial properties.

Abstract

Acyclic categories were introduced by Kozlov and can be viewed as generalised posets. Similar to posets, one can define their incidence algebras and a related topological complex. We consider the incidence algebra of either a poset or acyclic category as the quotient of a path algebra by the parallel ideal. We show that this ideal has a quadratic Groeobner basis with a lexicographic monomial order if and only if the poset or acyclic category is lex-shellable.