The Covering Law in Orthomodular Lattices Generated by Graphs of Functions

In the paper [2] we introduced and investigated complete orthomodular lattices generated by graphs of continuous functions. A natural question arises: can such a lattice be represented by the lattice of projectors in a Hilbert space (the standard quantum logic)? The answer is no, because the covering law is not satisfied in this case.     

Finitely Generated Free Orthomodular Lattices. III

The finitely generated free algebras F _V(Lk)(n) (k 2, n 3) in the varieties V(L _k )of orthomodular lattices generated by the ortholattices L _k which are horizontalsums of one block 2³ and k – 1 blocks 2² are described as abstract algebras. Thisis a continuation of earlier work and indicates the complexity one must expectwhen describing the finitely generated free algebras in finitely generated varietiesof orthomodular lattices generated by ortholattices containing Boolean blockslarger than 2².     

Decidability in Orthomodular Lattices

We discuss the possibility of automatic simplification of formulas in orthomodular lattices. We describe the principles of a program which decides the validity of equalities and inequalities, as well as implications between them and other important relations significant in quantum mechanics. PACS: 02.10.-v, 02.10.Ab, 02.10.De, 03.65.Fd. AMS Subject classification: 06C15, 03G12, 06B10, 06B25, 81P10.     

Orthomodular Lattices and Quantales

Let L be a complete orthomodular lattice. There is a one to one correspondence between complete Boolean subalgebras of L contained in the center of L and endomorphisms j of L satisfying the Borceux–van den Bossche conditions. This article is dedicated to Raquel Hernández. SUBJCLASS: 0210.Ab, 0210.De, 03.65.-ca     

n-Orthodistributivity in Orthomodular Lattices

A useful generalization of distributivity in lattices n-distributivity, \(n \in \mathbb{N}\) , was introduced in Huhn (Acta Sci. Math. 33:297–305, 1972). In Mayet and Roddy (Contrib. Gen. Algebra 5:285–294, 1987), ‘orthogonalized’ versions, n-orthodistributivity, \(n \in \mathbb{N}\) , of these equations were introduced and discussed. The discussion and results of Mayet and Roddy (Contrib. Gen. Algebra 5:285–294, 1987) centered on the class of modular ortholattices. In this paper we discuss and present some preliminary results for these conditions in orthomodular lattices. In particular, we completely classify the n-(ortho)distributive orthomodular lattices arising from Greechie’s classical 1971 construction, and we prove that a certain simple atomless orthomodular lattice, presented in Roddy (Algebra Univers. 29:564–597, 1992), is 4-orthodistributive. It is not 3-orthodistributive.     

Three Classes of Orthomodular Lattices

Let denote the class of all orthomodular lattices and denote the class of those that are commutator-finite. Also, let denote the class of orthomodular lattices that satisfy the block extension property, those that satisfy the weak block extension property, and those that are locally finite. We show that the following strict containments hold: Dedicated to the memory of Günter Bruns.     

Remarks on Concrete Orthomodular Lattices

An orthomodular lattice (OML) is called concrete if it is isomorphic to a collection of subsets of a set with partial ordering given by set inclusion, orthocomplementation given by set complementation, and finite orthogonal joins given by disjoint unions. Interesting examples of concrete OMLs are obtained by applying Kalmbach's construction K(L) to an arbitrary bounded lattice L. This note provides several results regarding Kalmbach's construction, concrete OMLs, and the relationship between the notions. First, we provide order-theoretic and categorical characterizations of the OML K(L) in terms of the bounded lattice L. Second, we provide an identity satisfied by each OML K(L), but not valid in every concrete OML. This shows that the class of OMLs of the form K(L) do not generate the variety of all concrete OMLs. Finally, we show that every concrete OML can be embedded into a concrete OML in which every element is a join of two or fewer atoms.     

A Note on Orthomodular Lattices

We introduce a new identity equivalent to the orthomodular law in every ortholattice.     

Noncommutative Symmetric Differences in Orthomodular Lattices

We deal with the following question: What is the proper way to introduce symmetric difference in orthomodular lattices? Imposing two natural conditions on this operation, six possibilities remain: the two (commutative) normal forms of the symmetric difference in Boolean algebras and four noncommutative terms. It turns out that in many respects the noncommutative forms, though more complex with respect to the lattice operations, in their properties are much nearer to the symmetric difference in Boolean algebras than the commutative terms. As application we demonstrate the usefulness of noncommutative symmetric differences in the context of congruence relations.     

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.     

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.     

On Some New Operations on Orthomodular Lattices

Kotas conditionals are used to define six pairs of disjunction- andconjunction-like operations on orthomodular lattices. Although five of them necessarily differfrom the lattice operations on elements that are not compatible, they coincidewith the lattice operations on all compatible elements of the lattice and theydefine on the underlying set a partial order relation that coincides with the originalone. Some of the new operations are noncommutative on noncompatible elements,but this does not exclude the possibility to endow them with a physicalinterpretation. The new operations are in general nonassociative, but for someof them a Foulis—Holland-type theorem concerning associativity instead ofdistributivity holds. The obtained results suggest that these new operations canserve as alternative algebraic models for the logical operations of disjunctionand conjunction.     

The Orthomodular Poset of Projections of A Symmetric Lattices

In a Hilbert space, there exists a natural correspondence between continuous projections and particular pairs of closed subspaces. In this paper, we generalize this situation and associate to a symmetric lattice L a subset P(L) of L× L, called its projection poset. If L is the lattice of closed subspaces of a topological vector space then elements of P(L) correspond to continuous projections and we prove that automorphisms of P(L) are determined by automorphisms of the lattice L when this lattice satisfies some basic properties of lattices of closed subspaces. Primary: 06C15, Secondary: 03G12 81P10.     

Orthomodular Lattices of Subspaces Obtained from Quadratic Forms

Being given a field K of characteristic different from 2 and 3, a 3-dimensional vector space E over K, and a nonsingular symmetric bilinear form φ over E, we define a structure of orthomodular lattice T(E,φ) on the set of all nonisotropic subspaces of E. We give a structure Theorem about the irreducible and 3-homogeneous subalgebras of T(E,φ). In particular, these subalgebras are all of the form T(E',φ ') where E' is a 3-dimensional subspace of E, if E is regarded as a vector space over a subfield K' of K, and φ ' is induced by φ. This structure Theorem allows us to achieve an old project, concerning minimal orthomodular lattices (an orthomodular lattice L is called minimal if it is nonmodular and if all its proper subalgebras are either modular, or isomorphic to L).     

Automorphism Groups of Orthomodular Lattices Obtained from Quadratic Spaces

We study the automorphism group of some orthomodular lattices, obtained from a quadratic space over a field K. We show how this group is linked to the semi-orthogonal group and with the group of all similarity transformations of the quadratic space. When the field K is finite, the cardinality of the automorphism group is given. AMS subject classification (1991): 06C15, 15A63, 20D45.     

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.     

Minimal Orthomodular Lattices from Quadratic Spaces over Finite Fields

If T is a finite, nonmodular, orthomodular lattice (OML), T is called minimal ifall its proper subOMLs are modular. In this paper we give a new infinite list ofminimal OMLs. They are obtained from quadratic spaces over a finite field Kof cardinality q 3 (mod 4). Their Greechie diagrams for q = 7 and q = 11are presented in a new way.     

Equivalencies, Identities, Symmetric Differences, and Congruencies in Orthomodular Lattices

It is shown that operations of equivalence cannot serve for building algebras which would induce orthomodular lattices as the operations of implication can. Several properties of equivalence operations have been investigated. Distributivity of equivalence terms and several other 3-variable expressions involving equivalence terms have been proved to hold in any orthomodular lattice. Symmetric differences have been shown to reduce to complements of equivalence terms. Some congruence relations related to equivalence operations and symmetric differences have been considered.