Anti-isomorphisms and involutions on the idealization of the incidence space over the finitary incidence algebra

Let $K$ be a field and $X$ a partially ordered set (poset). Let $FI(X,K)$ and $I(X,K)$ be the finitary incidence algebra and the incidence space of $X$ over $K$, respectively, and let $D(X,K)=FI(X,K)(+)I(X,K)$ be the idealization of the $FI(X,K)$-bimodule $I(X,K)$. In the first part of this paper, we show that $D(X,K)$ has an anti-automorphism (involution) if and only if $X$ has an anti-automorphism (involution). We also present a characterization of the anti-automorphisms and involutions on $D(X,K)$. In the second part, we obtain the classification of involutions on $D(X,K)$ to the case when characteristic of $K$ is different from 2 and $X$ is a connected poset such that every multiplicative automorphism of $FI(X,K)$ is inner and every derivation from $FI(X,K)$ to $I(X,K)$ is inner (in particular, when $X$ has an element that is comparable with all its elements).