Boolean subalgebras of orthoalgebras

TL;DR

This paper introduces a new method to reconstruct orthoalgebras from their Boolean subalgebras using the concept of directions, and characterizes their structure through order-theoretic and hypergraph representations.

Contribution

It develops a direct order-theoretic approach to recover orthoalgebras from Boolean subalgebras and introduces hypergraph representations extending Greechie diagrams.

Findings

Reconstruction of orthoalgebras from Boolean subalgebras using directions

Characterization of posets isomorphic to Boolean subalgebras of orthoalgebras

Representation of orthomodular posets with points and lines

Abstract

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.