Definitions
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 adj. Describing a lattice that is both orthocomplemented and modular
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Examples

V with L (V) orthomodular, is necessarily complete.

Conversely, an orthocomplemented poset is orthomodular iff

The lemma tells us that every orthocoherent orthoalgebra is, inter alia, an orthomodular poset.

Let us call a complete orthomodular lattice satisfying the hypotheses of Piron's theorem a Piron lattice.

In other words, in such cases we have only one logic, which is a complete orthomodular lattice.

This is not generally an orderisomorphism, but when it is, we obtain a complete orthomodular lattice, and thus come a step closer to the projection lattice of a Hilbert space.

Mackey presents a sequence of six axioms, framing a very conservative generalized probability theory, that underwrite the construction of a ˜logic™ of experimental propositions, or, in his terminology, ˜questions™, having the structure of a sigmaorthomodular poset.

Thus, orthomodular posets (the framework for Mackey's version of quantum logic) are equivalent to orthocoherent orthoalgebras.

Let L be a complete, atomic, irreducible orthomodular lattice satisfying the atomic covering law.

Î will inherit from L the structure of a complete lattice, which will then automatically be orthomodular by Lemma 4.3.
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