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 here, though close to Boolean algebras (i.e., close to standard quantum logics), still have a potential for an arbitrarily high degree of non-compatibility and hence they may enrich the studies of mathematical foundations of quantum mechanics.
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.
Abstract With many physical processes in which quantum mechanical phenomena can occur, it is essential to take into account a decision mechanism based on measurement data. This can be achieved by means of so-called numerical events, which are specified as follows: Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state
s∈S s ∈ S . A function
p:S→[0,1] p : S → [ 0 , 1 ] is called a numerical event or alternatively, an S-probability. If a set P of S-probabilities is ordered by the order of real functions, it becomes a poset which can be considered as a quantum logic. In case the logic P is a Boolean algebra, this will indicate that the underlying physical system is a classical one. The goal of this paper is to study sets of S-probabilities which are not far from being Boolean algebras by means of the addition and comparison of functions that occur in these sets. In particular, certain classes of so-called Boolean posets of S-probabilities are characterized and related to each other and descriptions based on sets of states are derived.