Remark on a paper of Mņczyński

In his paper, Boolean Properties of Observables in Axiomatic Quantum Mechanics,M$\text{n}\text{?}$czyński formulates the notion of a Boolean representation of the magnitudes of a physical theory, and proves the theorem (4.1) that a family P of magnitudes has a Boolean representation if and only if there is a homomorphism from $\text{L}$into a Boolean algebra, where $\text{L}$ is the orthomodular poset associated with P. Exploiting Finch's representation theorem for orthomodular posets, M$\text{n}\text{?}$czyński is able to show that this condition is satisfied if and only if the Boolean subalgebras of $\text{L}$ have a nondegenerate direct limit. The direct limit is then a maximal Boolean representation of P. Homomorphic relations were introduced by Kochen and Specker as a weakening of the concept of an imbedding between partial algebras. The purpose of this remark is to show that the direct limit of the Boolean subalgebras of $\text{L}$ has a natural characterization in terms of the notion of a homomorphic relation between $\text{L}$ and the direct product of its Boolean subalgebras.     

Testing for Classicality of a Physical System

Often quantum logics are algebraically modelled by orthomodular posets. The physical system described by such a quantum logic is classical if and only if the corresponding orthomodular poset is a Boolean algebra. We provide an easy testing procedure for this case. Moreover, we characterize orthomodular posets which are lattices and consider orthomodular posets which admit a full set of states and hence represent so-called spaces of numerical events. This way further test procedures are obtained.     

States On Orthocomplemented Difference Posets (Extensions)

We continue the investigation of orthocomplemented posets that are endowed with a symmetric difference (ODPs). The ODPs are orthomodular and, therefore, can be viewed as “enriched” quantum logics. In this note, we introduced states on ODPs. We derive their basic properties and study the possibility of extending them over larger ODPs. We show that there are extensions of states from Boolean algebras over unital ODPs. Since unital ODPs do not, in general, have to be set-representable, this result can be applied to a rather large class of ODPs. We then ask the same question after replacing Boolean algebras with “nearly Boolean” ODPs (the pseudocomplemented ODPs). Making use of a few results on ODPs, some known and some new, we construct a pseudocomplemented ODP, P, and a state on P that does not allow for extensions over larger ODPs.     

Regularity in Quantum Logic

In 1996, Harding showed that the binarydecompositions of any algebraic, relational, ortopological structure X form an orthomodular poset FactX. Here, we begin an investigation of the structuralproperties of such orthomodular posets of decompositions.We show that a finite set S of binary decompositions inFact X is compatible if and only if all the binarydecompositions in S can be built from a common n-arydecomposition of X. This characterization ofcompatibility is used to show that for any algebraic,relational, or topological structure X, the orthomodularposet Fact X is regular. Special cases of this result include the known facts that theorthomodular posets of splitting subspaces of an innerproduct space are regular, and that the orthomodularposets constructed from the idempotents of a ring are regular. This result also establishes theregularity of the orthomodular posets that Mushtariconstructs from bounded modular lattices, theorthomodular posets one constructs from the subgroups ofa group, and the orthomodular posets oneconstructs from a normed group with operators. Moreover,all these orthomodular posets are regular for the samereason. The characterization of compatibility is also used to show that for any structure X, thefinite Boolean subalgebras of Fact X correspond tofinitary direct product decompositions of the structureX. For algebraic and relational structures X, this result is extended to show that the Booleansubalgebras of Fact X correspond to representations ofthe structure X as the global sections of a sheaf ofstructures over a Boolean space. The above results can be given a physical interpretation as well.Assume that the true or false questions of a quantum mechanical system correspond tobinary direct product decompositions of the state spaceof the system, as is the case with the usual von Neumanninterpretation of quantum mechanics. Suppose S is asubset of . Then a necessary andsufficient condition that all questions in S can beanswered simultaneously is that any two questions in S can be answeredsimultaneously. Thus, regularity in quantum mechanicsfollows from the assumption that questions correspond todecompositions.     

Representations of empirical set theories

Any manual of Boolean locales in the strong sense, namely a subcategory of the opposite category of the category of complete Boolean algebras and complete Boolean homomorphisms satisfying not only conditions (3.1)–(3.10) of our previous paper [International Journal of Theoretical Physics,32, 1293 (1993b)], but also conditions (4.1)–(4.4) of that paper, is shown to be representable as the second-class orthomodular manual of Boolean locales on an orthomodular poset In this sense the study on manuals of Boolean locales in the strong sense is tantamount to the study on a special class of orthomodular posets, though our viewpoint is radically different from the conventional one in the traditional approach to orthomodular posets. Then the notion of a manual of Hilbert spaces or exactly what is called a manual of Hilbert locales is introduced, over which a variant of the celebrated Gelfand-Naimark-Segal theorem for a manual of Boolean locales in the strong sense is established.     

Spaces of Abstract Events

We generalize the concept of a space of numerical events in such a way that this generalization corresponds to arbitrary orthomodular posets whereas spaces of numerical events correspond to orthomodular posets having a full set of states. Moreover, we show that there is a natural one-to-one correspondence between orthomodular posets and certain posets with sectionally antitone involutions. Finally, we characterize orthomodular lattices among orthomodular posets.     

Empirical set theory

We share with Foulis and Randall the evangel that it is not orthomodular posets or the like, but manuals of operations that are of primary importance in the foundations of the empirical sciences. In sharp contrast to them, we regard an operation not as a set of possible outcomes, but as a complete Boolean algebra of observable events, which we adopt, following the lines of Davis and of Takeuti, as a building block of our empirical set theory. Just as a smooth manifold is covered by open subsets of a Euclidean space interconnected by smooth mappings, our empirical set theory is covered by the Scott-Solovay universesV ^(B) over complete Boolean algebrasB interconnected by geometric morphisms. Using the nomenclature of topos theory, our empirical set theory is a subcategory of the categoryBIop of Boolean localic toposes and geometric morphisms. It is shown that in this set theory observables can be identified with real numbers. This is the first step of formal development of Davis' ambitious program.     

Disjunctivity and Alternativity in Projection Logics

We study the notions of disjunctivity and alternativity of orthomodular posets inthe context of orthoprojections or skew projections in C ^*-algebras.     

Representation Systems and Quantum Structures

Two important classes of quantum structures, namely orthomodular posets and orthomodular lattices, can be characterized in a classical context, using notions like partial information and points of view. Using the formalism of representation systems, we show that these quantum structures can be obtained by expressing conditions on the existence of particular points of view, of particular ways to observe a system. PACS: 02.70.Wz, 03.67.Lx.     

Properties of Boolean orthoposets

A Boolean orthoposet is the orthoposetP fulfilling the following condition: Ifa, b P anda b = 0, thena b. This condition seems to be a sound generalization of distributivity in orthoposets. Also, the class of (orthomodular) Boolean orthoposets may play an interesting role in quantum logic theory. This class is wide enough and, on the other hand, enjoys some properties of Boolean algebras. In this paper we summarize results on Boolean orthoposets involving distributivity, set representation, properties of the state space, existence of Jauch-Piron states, and results concerning orthocompleteness and completion.     

On (3,3)-Homogeneous Greechie Orthomodular Posets

We describe (3,3)-homogeneous orthomodular posets for some cardinality of their sets of atoms. We examine a state space and a set of two-valued states of such logics. Particular homogeneous OMPs with exactly k pure states (k = 1,...,7, 10,11) have been constructed.     

Atoms in Boolean powers

We show that the Boolean power of an orthomodular latticeL is atomic if and only if bothL and the underlying Boolean algebraB are atomic. We describe the set of all atoms in the Boolean power.     

Bell-type inequalities in orthomodular lattices. I. Inequalities of order 2

We study Bell-type inequalities of ordern with emphasis on the casen = 2 in the framework of the structure of an orthomodular lattice, which is a logicoalgebraic model of quantum mechanics. We give necessary and sufficient conditions for the validity of Bell-type inequalities of order 2. In particular, we study Bell-type inequalities in various structures connected with a Hilert space, and we give a characterization of Boolean algebras via the validity of certain Bell-type inequalities.     

Partial and unsharp quantum logics

The total and the sharp character of orthodox quantum logic has been put in question in different contexts. This paper presents the basic ideas for a unified approach to partial and unsharp forms of quantum logic. We prove a completeness theorem for some partial logics based on orthoalgebras and orthomodular posets. We introduce the notion of unsharp orthoalgebra and of generalized MV algebra. The class of all effects of any Hilbert space gives rise to particular examples of these structures. Finally, we investigate the relationship between unsharp orthoalgebras, generalized MV algebras, and orthomodular lattices.     

A particular non-atomistic orthomodular poset

Every atomic orthomodular lattice is atomistic. We show that the corresponding statement for orthomodular posets fails. The result is of interest in the study of the Algebraic Structure of Quantum Mechanics, see [5].     

Jauch-Piron logics with finiteness conditions

We show that there are no non-Boolean block-finite orthomodular posets possessing a unital set of Jauch-Piron states. Thus, an orthomodular poset representing a quantum physical system must have infinitely many blocks.     

Direct Limits of Effect Algebras

In this paper, we prove that direct limits exist in the category of effect algebrasand effect algebra-morphisms. Then, as a consequence, we obtain similar knownresults for the categories of orthomodular posets and orthomodular lattices.     

Constructions of Some Quantum Structures and Fuzzy Effect Space

Quantum structures like effect algebras, -effect algebras, orthoalgebras, orthomodular posets, and -orthomodular posets are constructed by use of special fuzzy sets on posets. The concept of fuzzy effect space is introduced and a representation of a lattice effect algebra with a strong order determining system of states by means of fuzzy effect space is established.     

Small Quantum Structures with Small State Spaces

We summarize and extend results about “small” quantum structures with small dimensions of state spaces. These constructions have contributed to the theory of orthomodular lattices. More general quantum structures (orthomodular posets, orthoalgebras, and effect algebras) admit sometimes simplifications, but there are problems where no progress has been achieved.     

Tensor product of difference posets and effect algebras

A tensor product of difference posets and/or, equivalently, of effect algebras, which generalize orthoalgebras and orthomodular posets, is defined, and an equivalent condition is presented. The proof uses the notion of D-test spaces generalizing test spaces of Randall and Foulis. In particular, we show that a tensor product for difference posets with a nonempty system of probability measures exists.