Involutions of the second kind on finitary incidence algebras

TL;DR

This paper characterizes involutions of the second kind on finitary incidence algebras over a field, linking their existence to involutions on the underlying poset and automorphisms of the field, and explores conditions for their equivalence.

Contribution

It provides a complete characterization of involutions of the second kind on finitary incidence algebras and conditions for their equivalence under specific algebraic assumptions.

Findings

Involutions of the second kind exist iff the poset has an involution and the field has an automorphism of order 2.

Characterization of involutions of the second kind on finitary incidence algebras.

Necessary and sufficient conditions for involution equivalence when char K ≠ 2 and automorphisms are inner.

Abstract

Let $K$ be a field and $X$ a connected partially ordered set. In the first part of this paper, we show that the finitary incidence algebra $FI(X,K)$ of $X$ over $K$ has an involution of the second kind if and only if $X$ has an involution and $K$ has an automorphism of order $2$. We also give a characterization of the involutions of the second kind on $FI(X,K)$. In the second part, we give necessary and sufficient conditions for two involutions of the second kind on $FI(X,K)$ to be equivalent in the case where $char K\neq 2$ and every multiplicative automorphism of $FI(X,K)$ is inner.