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

Abstract

Let K be a field and P a partially ordered set (poset). Let FI(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)=FI(P,K)⊕I(P,K) be the idealization of the FI(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 charK≠2 and P is a connected poset such that every multiplicative automorphism of FI(P,K) is inner and every derivation from FI(P,K) to I(P,K) is inner (in particular, when P has an element that is comparable with all its elements).