Finitary incidence algebras

TL;DR

This paper generalizes incidence algebras to functions on arbitrary posets with convolution, exploring their algebraic properties and solving the isomorphism problem.

Contribution

It introduces a generalized incidence algebra framework for arbitrary posets and characterizes its algebraic structure and invertibility properties.

Findings

Describes properties like invertibility, radical, idempotents, regular elements

Provides a positive solution to the isomorphism problem for these algebras

Generalizes classical incidence algebra concepts to broader poset contexts

Abstract

We consider the functions in two variables on an arbitrary poset, for which the convolution operation is defined. We obtain the generalization of incidence algebra and describe its properties: invertibility, the Jackobson radical, idempotents, regular elements. As a consequence a positive solution of the isomorphism problem for such algebras is obtained.