Additive Biderivations of Incidence Algebras

TL;DR

This paper characterizes all additive biderivations of incidence algebras over a commutative ring, showing they decompose into sums of inner and extremal biderivations, especially when maximal chains are infinite.

Contribution

It provides an explicit description of additive biderivations of incidence algebras, revealing their structure as sums of inner and extremal biderivations, with special results for infinite chains.

Findings

Additive biderivations are sums of inner and extremal biderivations.

If maximal chains are infinite, all additive biderivations are sums of inner biderivations.

Explicit characterization of biderivations in incidence algebras.

Abstract

Let $\mathcal{R}$ be a commutative ring with unity, and let $P$ be a locally finite poset. The aim of the paper is to provide an explicit description of the additive biderivations of the incidence algebra $I(P, \mathcal{R})$. We demonstrate that every additive biderivation is the sum of several inner biderivations and extremal biderivations. Furthermore, if the number of elements in any maximal chain in $P$ is infinite, every additive biderivation of $I(P,\mathcal{R})$ is the sum of several inner biderivations.