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 this line of work.
Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation
Δ and a binary operator
d(x,y) that are generalizations of the binary relation
C and the commutator
c(x,y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations
Δ and
↔ (the latter was introduced by P. Pták and S. Pulmannová) and the operator
d(x,y). In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of
Δ-blocks with respect to Boolean subalgebras and distributive subposets they include.
We present several known and one new description of orthomodular structures (orthomodular lattices, orthomodular posets and orthoalgebras). Originally, orthomodular structures were viewed as pasted families of Boolean algebras. Here we introduce semipasted families of Boolean algebras as an alternative description which is not as detailed, but substantially simplex. Semipasted families of Boolean algebras correspond to orthomodular structures in such a way that states and evaluation functionals are preserved. As semipasted families of Boolean algebras are quite general, they allow an easy construction of orthomodular structures with given state space properties. Based on this technique, we give a simplified proof of Shultz's Theorem on characterization of spaces of finitely additive states on orthomodular lattices. We also put some other results into the new context. We give a detailed exposition of the construction techniques as a tool for further applications, especially for finding counterexamples to questions about states on orthomodular structures.
Following Birkhoff and von Neumann, quantum logic has traditionally been based\non the lattice of closed linear subspaces of some Hilbert space, or, more generally,\non the lattice of projections in a von Neumann algebra A. Unfortunately, the logical\ninterpretation of these lattices is impaired by their nondistributivity and by various\nother problems. We show that a possible resolution of these difficulties, suggested by\nthe ideas of Bohr, emerges if instead of single projections one considers elementary\npropositions to be families of projections indexed by a partially ordered set C(A) of\nappropriate commutative subalgebras of A. In fact, to achieve both maximal generality\nand ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra\nand that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such\nfamilies of projections form a Heyting algebra in a natural way, so that the associated\npropositional logic is intuitionistic: distributivity is recovered at the expense of the\nlaw of the excluded middle.\nSubsequently, generalizing an earlier computation for n×n matrices, we prove that\nthe Heyting algebra thus associated to A arises as a basis for the internal Gelfand\nspectrum (in the sense of Banaschewski–Mulvey) of the "Bohrification" A of A, which\nis a commutative Rickart C*-algebra in the topos of functors from C(A) to the category\nof sets. We explain the relationship of this construction to partial Boolean algebras\nand Bruns–Lakser completions. Finally, we establish a connection between probability\nmeasure on the lattice of projections on a Hilbert space H and probability valuations\non the internal Gelfand spectrum of A for A = B(H).
The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset. In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa. Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following: (1) the greatest lower bound of a and Na exists and is 0, (2) a ≤ b implies Nb ≤ Na, (3) NNa = a. Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfying where a ≤ b means a ∧ b = a, as usual. Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following: An orthomodular lattice gives a Finch implication algebra by defining → by A Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ by The orthocomplementation is unchanged in both cases.