  We develop a direct method to recover an orthoalgebra from its poset of
Boolean subalgebras. For this a new notion of direction is introduced.
Directions are also used to characterize in purely order-theoretic terms those
posets that are isomorphic to the poset of Boolean subalgebras of an
orthoalgebra. These posets are characterized by simple conditions defining
orthodomains and the additional requirement of having enough directions.
Excepting pathologies involving maximal Boolean subalgebras of four elements,
it is shown that there is an equivalence between the category of orthoalgebras
and the category of orthodomains with enough directions with morphisms suitably
defined. Furthermore, we develop a representation of orthodomains with enough
directions, and hence of orthoalgebras, as certain hypergraphs. This hypergraph
approach extends the technique of Greechie diagrams and resembles projective
geometry. Using such hypergraphs, every orthomodular poset can be represented
by a set of points and lines where each line contains exactly three points.
