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.     

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.     

Hilbert lattices: New results and unsolved problems

The class of Hilbert lattices that derive from orthomodular spaces containing infinite orthonormal sets (normal Hilbert lattices) is investigated. Relevant open problems are listed. Comments on form-topological orthomodular spaces and results on arbitrary orthomodular spaces are appended.Deceased (October 29, 1989).     

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.     

Commutativity in orthomodular posets

An abstract characterization of the commutation relation in orthomodular posets is given. This characterization is a generalization of Guz's result. In particular, if an orthomodular poset P is Boolean, then aCb iff a ∧ b exists in P.A method of constructing nonregular Boolean orthomodular posets is presented.     

Logical Approach for Two-Valued States on Quantum Systems

In this paper we develop a logical system associated to two-valued states on orthomodular lattices. An completeness theorem with respect to a variety of orthomodular lattices enriched with an unary operation that represents two-valued states is given.     

Orthomodular Lattices and a Quantum Algebra

We show that one can formulate an algebra with lattice ordering so as to contain one quantum and five classical operations as opposed to the standard formulation of the Hilbert space subspace algebra. The standard orthomodular lattice is embeddable into the algebra. To obtain this result we devised algorithms and computer programs for obtaining expressions of all quantum and classical operations within an orthomodular lattice in terms of each other, many of which are presented in the paper. For quantum disjunction and conjunction we prove their associativity in an orthomodular lattice for any triple in which one of the elements commutes with the other two and their distributivity for any triple in which a particular element commutes with the other two. We also prove that the distributivity of symmetric identity holds in Hilbert space, although whether or not it holds in all orthomodular lattices remains an open problem, as it does not fail in any of over 50 million Greechie diagrams we tested.     

Orthomodularity of Decompositions in a Categorical Setting

We provide a method to construct a type of orthomodular structure known as an orthoalgebra from the direct product decompositions of an object in a category that has finite products and whose ternary product diagrams give rise to certain pushouts. This generalizes a method to construct an orthomodular poset from the direct product decompositions of familiar mathematical structures such as non-empty sets, groups, and topological spaces, as well as a method to construct an orthomodular poset from the complementary pairs of elements of a bounded modular lattice. Mathematics Subject Classifications (2000): 06C15, 81P10, 03G12, 18A30     

Structure of the set of annihilators

We investigate the right annihilator lattice of a^*-ring and ask whether it is orthomodular with respect to a naturally given involution. In particular, we introduce a new class of^*-rings with orthomodular right annihilator lattice.     

Type-Decomposition of an Effect Algebra

Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.     

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].     

Coreflections in Algebraic Quantum Logic

Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular po-sets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.     

Sites and tours in orthoalgebras and orthomodular lattices

A block of an orthoalgebra (or of an orthomodular lattice) is a maximal Boolean subalgebra. A site is the intersection of two distinct blocks. L is block (site)-finite if there are only finitely many blocks (sites). We introduce a certain type of subalgebra of an orthoalgebra which is a subortholattice if the orthoalgebra is an ortholattice (and therefore an orthomodular lattice) and which is block finite if the orthoalgebra is site finite. The construction yields a cover of a site-finite orthoalgebra or orthomodular lattice L by block-finite substructures of the same type and having the same center as L. Every site-finite orthomodular lattice is commutator finite.In memory of Charles H. Randall.     

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.     

Reichenbach’s Common Cause in an Atomless and Complete Orthomodular Lattice

Hofer-Szabo, Redei and Szabo (Int. J. Theor. Phys. 39:913–919, 2000) defined Reichenbach’s common cause of two correlated events in an orthomodular lattice. In the present paper it is shown that if logical independent elements in an atomless and complete orthomodular lattice correlate, a common cause of the correlated elements always exists.     

Equations Holding in Hilbert Lattices

We produce and study several sequences of equations, in the language of orthomodular lattices, which hold in the ortholattice of closed subspaces of any classical Hilbert space, but not in all orthomodular lattices. Most of these equations hold in any orthomodular lattice admitting a strong set of states whose values are in a real Hilbert space. For some of these equations, we give conditions under which they hold in the ortholattice of closed subspaces of a generalised Hilbert space. These conditions are relative to the dimension of the Hilbert space and to the characteristic of its division ring of scalars. In some cases, we show that these equations cannot be deduced from the already known equations, and we study their mutual independence. To conclude, we suggest a new method for obtaining such equations, using the tensorial product. PACS numbers: 02.10, 03.65, 03.67     

States on Orthomodular Posets of Decompositions

In Harding (Transactions of American Mathematical Society (1996) 348(5), 1839–1862), it was shown that the direct product decompositions of a set X naturally form an orthomodular poset Fact X. Here it is shown that Fact X has a state if and only if X is finite. An example is also given of a finite orthomodular poset that can be embedded into Fact X for X countable, but not for X finite.     

An Intrinsic Topology for Orthomodular Lattices

We present a general way to define a topology on orthomodular lattices. We show that in the case of a Hilbert lattice, this topology is equivalent to that induced by the metrics of the corresponding Hilbert space. Moreover, we show that in the case of a boolean algebra, the obtained topology is the discrete one. Thus, our construction provides a general tool for studying orthomodular lattices but also a way to distinguish classical and quantum logics.     

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.     

Sharp Elements in Effect Algebras

We show that if for an arbitrary pair of orthogonal sharp elements of an effect algebra E its join exists and is sharp, then the set E_S of all sharp elements of E is a subeffect algebra of E that is an orthomodular poset. Such effect algebras need not be sharply dominating but S-dominating. Further, we show that in every nonproper effect algebra E, E_S is a subeffect algebra that is an orthomodular poset. Moreover, a general theorem for E_S is proved.