  Paraorthomodular posets are bounded partially ordered set with an antitone
involution induced by quantum structures arising from the logico-algebraic
approach to quantum mechanics. The aim of the present work is starting a
systematic inquiry into paraorthomodular posets theory both from an algebraic
and order-theoretic perspective. On the one hand, we show that paraorthomodular
posets are amenable of an algebraic treatment by means of a smooth
representation in terms of bounded directoids with antitone involution. On the
other, we investigate their order-theoretical features in terms of forbidden
configurations. Moreover, sufficient and necessary conditions characterizing
bounded posets with an antitone involution whose Dedekind-MacNeille completion
is paraorthomodular are provided.
