  Generalized orthomodular posets were introduced recently by D. Fazio, A.
Ledda and the first author of the present paper in order to establish a useful
tool for studying the logic of quantum mechanics. They investigated structural
properties of these posets. In the present paper we study logical and algebraic
properties of these posets. In particular, we investigate conditions under
which they can be converted into operator residuated structures. Further, we
study their representation by means of algebras (directoids) with everywhere
defined operations. We prove congruence properties for the class of algebras
assigned to generalized orthomodular posets and, in particular, for a
subvariety of this class determined by a simple identity. Finally, in contrast
to the fact that the Dedekind-MacNeille completion of an orthomodular poset
need not be an orthomodular lattice we show that the Dedekind-MacNeille
completion of a stronger version of a generalized orthomodular poset is nearly
an orthomodular lattice.
