Higher Derivations of Finitary Incidence Algebras

TL;DR

This paper characterizes all $R$-linear higher derivations of finitary incidence algebras over a poset, showing they decompose into inner derivations and those induced by higher transitive maps, revealing their structural nature.

Contribution

It provides a complete decomposition of higher derivations of finitary incidence algebras into inner and transitive map-induced parts, a novel structural insight.

Findings

Every $R$-linear higher derivation decomposes into an inner derivation and one induced by a higher transitive map.

The structure of higher derivations is fully characterized in terms of these two components.

This decomposition advances understanding of the algebraic structure of finitary incidence algebras.

Abstract

Let $P$ be a partially ordered set, $R$ a commutative unital ring and $FI(P,R)$ the finitary incidence algebra of $P$ over $R$. We prove that each $R$-linear higher derivation of $FI(P,R)$ decomposes into the product of an inner higher derivation of $FI(P,R)$ and the higher derivation of $FI(P,R)$ induced by a higher transitive map on the set of segments of $P$.