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Generalized orthomodular posets were introduced recently by D. Fazio, A.Ledda and the first author of the present paper in order to establish a usefultool for studying the logic of quantum mechanics. They investigated structuralproperties of these posets. In the present paper we study logical and algebraicproperties of these posets. In particular, we investigate conditions underwhich they can be converted into operator residuated structures. Further, westudy their representation by means of algebras (directoids) with everywheredefined operations. We prove congruence properties for the class of algebrasassigned to generalized orthomodular posets and, in particular, for asubvariety of this class determined by a simple identity. Finally, in contrastto the fact that the Dedekind-MacNeille completion of an orthomodular posetneed not be an orthomodular lattice we show that the Dedekind-MacNeillecompletion of a stronger version of a generalized orthomodular poset is nearlyan orthomodular lattice.

Logical and algebraic properties of generalized orthomodular posets