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     

Filters and supports in orthoalgebras

An orthoalgebra, which is a natural generalization of an orthomodular lattice or poset, may be viewed as a logic or proposition system and, under a welldefined set of circumstances, its elements may be classified according to the Aristotelian modalities: necessary, impossible, possible, and contingent. The necessary propositions band together to form a local filter, that is, a set that intersects every Boolean subalgebra in a filter. In this paper, we give a coherent account of the basic theory of Orthoalgebras, define and study filters, local filters, and associated structures, and prove a version of the compactness theorem in classical algebraic logic.     

Orthomodular Bell-Kochen-Specker Theorem

Four elements in an orthomodular lattice of height four generate a partial Boolean subalgebra that contains a Bell-Kochen-Specker theorem. This result directly explains and generalizes the 4-dimensional Bell-Kochen-Specker theorems of various authors.     

Coverings of [Mon] and Minimal Orthomodular Lattices

A finite, nonmodular orthomodular lattice (OML)T is called minimal if all its proper subOMLs aremodular. For a finite, nonmodular OML T, T minimal isequivalent to the equational class [T], generated by T, covers the equational class [MOn] forsome n. The main result of this paper is that thereexist infinitely many minimal OMLs. They are obtainedfrom quadratic spaces on finite fields. The automorphism groups of such OMLs are given.     

Refinement and unique Mackey decomposition for manuals and orthalogebras

In the empirical logic approach to quantum mechanics, the physical system under consideration is given in terms of a manual of sample spaces. The resulting propositional structure has been shown to form an orthoalgebra, generalizing the structure of an orthomodular poset. An orthoalgebra satisfies the unique Mackey decomposition (UMD) property if, given two commuting propositions a and b, there is a unique jointly orthogonal triple (e, f, c) such that a=ec and b=fc. In a manual, E is refined by F if E is logically equivalent to some partition of F, making results from F at least as informative as those from E. The main result is a characterization of the UMD property in terms of the refinement structure of an underlying manual, provided the manual is event saturated and orthogonally additive.     

Solution of lattice models by successive linkage

A linkage algorithm is presented for evaluating the partition function of a union of finite lattice blocks in terms of the partition functions of the component blocks. This algorithm leads to: (i) A fast enumeration method for evaluating the partition function of a finite lattice (for Ising spins in two dimensions, the number of terms needed to evaluate the partition function for a block ofL spins if reduced from 2 ^L to ); (ii) a recursive factorization procedure that accelerates the rate at which quantities evaluated on a finite lattice converge to their thermodynamic limit, and (iii) a scaling procedure that further accelerates the convergence to the thermodynamic limit. The scaling procedure is similar to a method previously used in turbulence calculations.Supported in part by the Applied Mathematics Subprogram of the Office of Energy Research, U.S. Department of Energy, under contract DE-AC03-76SF0098, and in part by the Office of Naval Research under contract N00014-76-C-0316     

Physical and geometrical interpretation of the Jordan-Hahn and the Lebesgue decomposition property

The Jordan-Hahn decomposition and the Lebesgue decomposition, two basic notions of classical measure theory, are generalized for measures on orthomodular posets. The Jordan-Hahn decomposition property (JHDP) and the Lebesgue decomposition property (LDP) are defined for sections of probability measures on an orthomodular poset L. If L is finite, then these properties can be characterized geometrically in terms of two parallelity relations defined on the set of faces of . A section is shown to have the JHDP if and only if every pair of f-parallel faces is p-parallel; it is shown to have the LDP if and only if every pair of disjoint faces is p-parallel. It follows from these results that the LDP is stronger than the JHDP in the setting of finite orthomodular posets. Mielnik's convex scheme of quantum theory provides the frame for a physical interpretation of these results.     

Minimal orthomodular lattices

The lattices calledminimal orthomodular (MOL) arise in a special exclusion problem concerning the class of all orthomodular lattices (OML) and the subclass of all modular orthocomplemented lattices. This problem was given in G. Kalmbach's book,Orthomodular Lattices. We prove that an exclusion system necessarily must contain an infinite lattice. We prove that, except one, all the finite, irreducible MOLs have only blocks with eight elements. We characterize finite MOLs by a covering property related to equational classes generated by the modular ortholattices MOn.     

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.     

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.     

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 Link between Quantum Logic and Categorical Quantum Mechanics

Abramsky and Coecke (Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425, IEEE Comput. Soc., New York, 2004) have recently introduced an approach to finite dimensional quantum mechanics based on strongly compact closed categories with biproducts. In this note it is shown that the projections of any object A in such a category form an orthoalgebra Proj  A. Sufficient conditions are given to ensure this orthoalgebra is an orthomodular poset. A notion of a preparation for such an object is given by Abramsky and Coecke, and it is shown that each preparation induces a finitely additive map from Proj  A to the unit interval of the semiring of scalars for this category. The tensor product for the category is shown to induce an orthoalgebra bimorphism Proj  A×Proj  B→Proj (A ⊗ B) that shares some of the properties required of a tensor product of orthoalgebras. These results are established in a setting more general than that of strongly compact closed categories. Many are valid in dagger biproduct categories, others require also a symmetric monoidal tensor compatible with the dagger and biproducts. Examples are considered for several familiar strongly compact closed categories.     

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.     

Attempt of an axiomatic foundation of quantum mechanics and more general theories. IV

This contribution continues the series of papers on the same subject which has been treated byLudwig in [1–3]. Using the system of axioms as given in [3], we shall succeed in constructing an orthomodular lattice of linear operators on the real vector space generated by the physical decision effects. There results an isomorphism between the orthomodular lattice of all physical decision effects and the lattice to be constructed.     

Quantum Observables and Effect Algebras

We study observables on monotone σ-complete effect algebras. We find conditions when a spectral resolution implies existence of the corresponding observable. We characterize sharp observables of a monotone σ-complete homogeneous effect algebra using its orthoalgebraic skeleton. In addition, we study compatibility in orthoalgebras and we show that every orthoalgebra satisfying RIP is an orthomodular poset.     

Sparse Long Blocks and the Micro-structure of the Longuest Common Subsequences

Consider two random strings having the same length and generated by an iid sequence taking its values uniformly in a fixed finite alphabet. Artificially place a long constant block into one of the strings, where a constant block is a contiguous substring consisting only of one type of symbol. The long block replaces a segment of equal size and its length is smaller than the length of the strings, but larger than its square-root. We show that for sufficiently long strings the optimal alignment (OA) corresponding to a longest common subsequence (LCS) treats the inserted block very differently depending on the size of the alphabet. For two-letter alphabets, the long constant block gets mainly aligned with the same symbol from the other string, while for three or more letters the opposite is true and the block gets mainly aligned with gaps. We further provide simulation results on the proportion of gaps in blocks of various lengths. In our simulations, the blocks are “regular blocks” in an iid sequence, and are not artificially inserted. Nonetheless, we observe for these natural blocks a phenomenon similar to the one shown in case of artificially-inserted blocks: with two letters, the long blocks get aligned with a smaller proportion of gaps; for three or more letters, the opposite is true. It thus appears that the microscopic nature of two-letter OAs and three-letter OAs are entirely different from each other.     

Algebras of bounded operators on nonclassical orthomodular spaces

A Hermitian space is called orthomodular if the Projection Theorem holds: every orthogonally closed subspace is an orthogonal summand. Besides the familiar real or complex Hilbert spaces there are non-classical infinite dimensional examples constructed over certain non-Archimedeanly valued, complete fields. We study bounded linear operators on such spaces. In particular we construct an operator algebraA of von Neumann type that contains no orthogonal projections at all. For operators inA we establish a representation theorem from which we deduce thatA is commutative. We then focus on a subalgebra which turns out to be an integral domain with unique maximal ideal. Both analytic and topological characterizations of are given.     

An extension of a characterization of the automorphisms of Hilbert space effect algebras

The aim of this paper is to show that if an order preserving bijective transformation of the Hilbert space effect algebra also preserves the probability with respect to a fixed pair of mixed states, then it is an ortho-order automorphism. A similar result for the orthomodular lattice of all sharp effects (i.e., projections) is also presented.     

Embedding quantum logics in Hilbert space

We show that an orthomodular lattice is embeddable in a Hilbert space if and only if states of a certain kind exist. A physical motivation for the existence of such states is given and a connection is provided between the quantum logic, algebraic, and operational approaches to quantum mechanics.