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

Let K be a field and P a partially ordered set (poset). Let F I(P, K) and I(P, K) be the finitary incidence algebra and the incidence space of P over K, respectively, and let D(P, K) = F I(P, K)circle plus I(P, K) be the idealization of the F I(P, K)-bimodule I(P, K). In the first part of this paper, we show that D(P, K) has an anti-automorphism (involution) if and only if P has an anti-automorphism (involution). We also present a characterization of the anti-automorphisms and involutions on D(P, K). In the second part, we obtain the classification of involutions on D(P, K) to the case when char K (sic) 2 and P is a connected poset such that every multiplicative automorphism of F I(P, K) is inner and every derivation from F I(P, K) to I(P, K) is inner (in particular, when P has an element that is comparable with all its elements). (c) 2021 Elsevier Inc. All rights reserved.