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.

中文翻译：

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正代数的布尔子代数

我们开发了一种直接方法来从布尔子代数的偏序中恢复正代数。为此，引入了新的方向概念。方向也用于以纯序论术语来表征与正代数的布尔子代数的偏序同构的偏序。这些偏序集的特点是定义正交域的简单条件和具有足够方向的额外要求。除了涉及四个元素的最大布尔子代数的病理学之外，它表明在正代数的范畴和具有适当定义的态射的足够方向的正域范畴之间存在等价。此外，我们开发了具有足够方向的正交域的表示，因此将正交代数作为某些超图。这种超图方法扩展了 Greechie 图的技术并类似于投影几何。使用这样的超图，每个正模偏集都可以由一组点和线表示，其中每条线恰好包含三个点。