Classification of involutions on finitary incidence algebras of non-connected posets

TL;DR

This paper characterizes when two involutions on finitary incidence algebras of non-connected posets are equivalent, under the condition that all automorphisms are inner, providing a detailed algebraic classification.

Contribution

It offers necessary and sufficient conditions for involution equivalence on finitary incidence algebras of non-connected posets, assuming all automorphisms are inner.

Findings

Conditions for involution equivalence are established.

The study applies to algebras over fields with characteristic not 2.

Provides a classification framework for involutions in this algebraic setting.

Abstract

Let $FI(X,K)$ be the finitary incidence algebra of a non-connected partially ordered set $X$ over a field $K$ of characteristic different from $2$. For the case where every multiplicative automorphism of $FI(X,K)$ is inner, we present necessary and sufficient conditions for two involutions on $FI(X,K)$ to be equivalent.