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Let $K$ be a field and $X$ a connected partially ordered set. In the firstpart 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 aninvolution and $K$ has an automorphism of order $2$. We also give acharacterization of the involutions of the second kind on $FI(X,K)$. In thesecond part, we give necessary and sufficient conditions for two involutions ofthe 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.

Involutions of the second kind on finitary incidence algebras