Reconstructing an atomic orthomodular lattice from the poset of its Boolean sublattices

从正交偏序集到正交模偏序集

From orthoposets to orthomodular posets

本文证明正交模偏序集范畴是强正交偏序集范畴的全核心反射子范畴，通过修改序结构构造核心反射函子，并限制为正交格范畴到正交模偏序集范畴的右伴随函子。

Effect algebras with the maximality property

The maximality property was introduced in in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete orthomodular posets. We show that various conditions used in the theory of effect algebras are stronger than the maximality property, clear up the connections between them and show some consequences of these conditions. In particular, we prove that a Jauch--Piron effect algebra with a countable unital set of states is an orthomodular lattice and that a unital set of Jauch--Piron states on an effect algebra with the maximality property is strongly order determin…

Subalgebras of orthomodular lattices

Sachs showed that a Boolean algebra is determined by its lattice of subalgebras. We establish the corresponding result for orthomodular lattices. We show that an orthomodular lattice L is determined by its lattice of subalgebras Sub(L), as well as by its poset of Boolean subalgebras BSub(L). The domain BSub(L) has recently found use in an approach to the foundations of quantum mechanics initiated by Butterfield and Isham, at least in the case where L is the orthomodular lattice of projections of a Hilbert space, or von Neumann algebra. The results here may add some additional perspective to th…

Quantum logics close to Boolean algebras

We consider orthomodular posets endowed with a symmetric difference. We call them ODPs. Expressed in the quantum logic language, we consider quantum logics with an XOR-type connective. We study three classes of "almost Boolean" ODPs, two of them defined by requiring rather specific behaviour of infima and the third by a Boolean-like behaviour of Frink ideals. We establish a (rather surprising) inclusion between the three classes, shadding thus light on their intrinsic properties. (More details can be found in the Introduction that follows.) Let us only note that the orthomodular posets pursued…