Anti-automorphisms and involutions on (finitary) incidence algebras

In the first part of this article, we show that the finitary incidence algebra of an arbitrary poset X over a field K has an anti-automorphism (involution) if and only if X has an anti-automorphism (involution), which generalizes the Theorem 2 of Spiegel [E. Spiegel, Involutions in incidence algebras, Linear Algebra Appl. 405 (2005), pp. 155–162]. In the second part of this article we generalize the decomposition theorem of Brusamarello and Lewis [R. Brusamarello and D.W. Lewis, Automorphisms and involutions on incidence algebras, Linear Multilinear Algebra (to appear)] and the results on classification of involutions of Brusamarello et al. [R. Brusamarello, E.Z. Fornaroli, and E.A. Santulo Jr, Classification of involutions on incidence algebras, Comm. Algebra (to appear)] to the case when X is locally finite.