Identities and derived lengths of finitary incidence algebras and their group of units

TL;DR

This paper investigates the algebraic identities and structural properties of finitary incidence algebras over posets, focusing on polynomial identities, group identities, and derived lengths of the algebra and its unit group.

Contribution

It establishes conditions under which finitary incidence algebras satisfy polynomial and group identities and determines their Lie and group derived lengths.

Findings

FI(X,K) satisfies a polynomial identity under certain conditions

The group of units of FI(X,K) satisfies a group identity in specific cases

The Lie derived length of FI(X,K) and the derived length of its unit group are explicitly determined

Abstract

Let $FI(X,K)$ be the finitary incidence algebra of a poset $X$ over a field $K$. In this short note we establish when $FI(X,K)$ satisfies a polynomial identity and when its group of units $\mathcal{U}(FI(X,K))$ satisfies a group identity. The Lie derived length of $FI(X,K)$ and the derived length of $\mathcal{U}(FI(X,K))$ are also determined.