Orthomodular Lattices Research Articles

Central elements in pseudoeffect algebras

Abstract We introduce the definition of pseudoorthoalgebras and discuss some relationships between orthomodular lattices and pseudoorthoalgebras. Then we study the conditions that a pseudoeffect algebra is isomorphic to an “internal direct product” of ideals generated by orthogonal principal elements. At last, we give some characterizations of central elements in pseudoeffect algebras.

Characterization of Spaces of Filtering States

Filtering states on orthomodular lattices have been introduced by G.T. Ruttimann as an “opposite” of completely additive states. He proved that they form a face of the state space. The question which faces can be the sets of filtering states remained open. Here we prove that, for any semi-exposed face F of a compact convex set C, there is an orthomodular lattice L and an affine homeomorphism φ of C onto the state space of L such that φ(F) is the space of filtering states.

Exhaustive generation of orthomodular lattices with exactly one nonquantum state

We propose a kind of reverse Kochen-Specker theorem that amounts to generating orthomodular lattices with exactly one state that do not admit properties of the Hilbert space. We apply MMP algorithms to obtain smallest orthomodular lattices with 35 atoms and 35 blocks (35–35) and all other ones up to 38–38. We find out that all but one of them admit exactly one state and discover several other properties of them. Previously known such orthomodular lattices have 44 atoms and 44 blocks or more. We present some of them in our notation.

Orthomodular Lattices Induced by the Concurrency Relation

We apply to locally finite partially ordered sets a construction which associates a complete lattice to a given poset; the elements of the lattice are the subsets of a closure operator, defined starting from the concurrency relation. We show that, if the partially ordered set satisfies a property of local density, i.e.: N-density, then the associated lattice is also orthomodular. We then consider occurrence nets, introduced by C.A. Petri as models of concurrent computations, and define a family of subsets of the elements of an occurrence net; we call those subsets closed because they can be seen as subprocesses of the whole net which are, intuitively, with respect to the forward and backward local state changes. We show that, when the net is K-dense, the causally sets coincide with the sets induced by the closure operator defined starting from the concurrency relation. K-density is a property of partially ordered sets introduced by Petri, on the basis of former axiomatizations of special relativity theory.

PATH-CONNECTED AND NON PATH-CONNECTED ORTHOMODULAR LATTICES

A block of an orthomodular lattice L is a maximal Boolean subalgebra of L. A site is a subalgebra of an orthomodular lattice L of the form S = A <TEX>$\cap$</TEX> B, where A and B are distinct blocks of L. An orthomodular lattice L is called with finite sites if |A <TEX>$\cap$</TEX> B| < <TEX>$\infty$</TEX> for all distinct blocks A, B of L. We prove that there exists a weakly path-connected orthomodular lattice with finite sites which is not path-connected and if L is an orthomodular lattice such that the height of the join-semilattice [ComL]<TEX>$\vee$</TEX> generated by the commutators of L is finite, then L is pathconnected.

Effect algebras with the maximality property

The maximality property was introduced in orthomodular posets as a common generalization of orthomodular lattices and orthocomplete orthomodular posets. We show that various conditions used in the theory of effect algebras are stronger than the maximality property, clear up the connections between them and show some consequences of these conditions. In particular, we prove that a Jauch–Piron effect algebra with a countable unital set of states is an orthomodular lattice and that a unital set of Jauch–Piron states on an effect algebra with the maximality property is strongly order determining.

On Definition of Skew Frames

Skew frames represent a common generalization of frames and orthomodular lattices. They could serve as Lindenbaum algebras of quantum intuitionistic logic as well as invariants of noncommutative C*-algebras. It is shown that lattices of open projections with skew (partial) operations are complete invariants of C*-algebras and that these operations are preserved by morphims of C*-algebras.

Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras

We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.

Orthocomplemented lattices with a symmetric difference

Modelling an abstract version of the set-theoretic operation of symmetric difference, we first introduce the class of orthocomplemented difference lattices ($$\mathcal {ODL}$$). We then exhibit examples of ODLs and investigate their basic properties finding, for instance, that any ODL induces an orthomodular lattice (OML) but not all OMLs can be converted to ODLs. We then analyse an appropriate version of ideals and valuations in ODLs and show that the set-representable ODLs form a variety. We finally investigate the question of constructing ODLs from Boolean algebras and obtain, as a by-product, examples of ODLs that are not set-representable but that �live� on set-representable OMLs.

Terms that are self-dual in Boolean algebras but not in MO2

An absorption law is an identity of the form p = x. The ternary function x+y+z (ring addition) in Boolean algebras satisfies three absorption laws in two variables. If a term satisfies these three identities in a variety, it is called a minority term for that variety. We construct a minority term p for orthomodular lattices such the identity \(p = \tilde{p}\) defines Boolean algebras modulo orthomodular lattices. (The dual of p is denoted by \(\tilde{p}\).) Consequently, having a unique minority term function characterizes Boolean algebras among orthomodular lattices. Our main result generalizes this example to arbitrary arity and arbitrary consistent sets of 2-variable absorption laws.

On the reducibility of hypotheses and consequences

This paper only tries to shed some additional light on the concepts of hypotheses and consequences, introduced by Trillas et al. [E. Trillas, S. Cubillo, and E. Castiñeira, On conjectures in orthocomplemented lattices, Artificial Intelligence, 117 (2) (2000) 255–275] as particular cases of the more general concept of conjecture. The definitions of reducible hypothesis and reducible consequence are presented, and it is shown that in orthomodular lattices all hypotheses and consequences are reducible, but that this is not necessarily the case in proper ortholattices. In addition, the sets of reducible hypotheses and consequences are characterized, both for general ortholattices as well as for the particular case of orthomodular lattices.

States and Boolean Algebra

We investigate subadditive measures on orthomodular lattices. We show as the main result that the Boolean algebra, the special metric orthomodular lattice and the orthomodular lattice which is unital with respect to subadditive states are equivalent. This result may find an application in the foundation of quantum theories and mathematical logic.

Generalized Hermitian Algebras

We refer to the real Jordan Banach algebra of bounded Hermitian operators on a Hilbert space as a Hermitian algebra. In this paper we define and launch a study of a class of generalized Hermitian (GH) algebras. Among the examples of GH-algebras are ordered special Jordan algebras, JW-algebras, and AJW-algebras, but unlike these more restricted cases, a GH-algebra is not necessarily a Banach space and its lattice of projections is not necessarily complete. In this paper we develop the basic theory of GH-algebras, identify their unit intervals as effect algebras, and observe that their projection lattices are sigma-complete orthomodular lattices. We show that GH-algebras are spectral order-unit spaces and that they admit a substantial spectral theory.

Markov property in quantum logic: A reflection

A possible way of the Markov property introduction within the framework of the orthomodular quantum logic, which is commonly used as the calculus model for quantum mechanics is presented in this paper. The presented work follows the logical line rather than any physical interpretation in the framework of quantum mechanics. The basic algebraic structure, which is used as a model for noncompatible random events is an orthomodular lattice. On the orthomodular lattice, a dynamical structure is introduced coupled with mappings which have similar properties as sn-maps on the orthomodular lattice. This construction leads to the definition of an L-process with the Markov property on the orthomodular lattice.

Standard Logics Are Valuation-Nonmonotonic

It has recently been discovered that both quantum and classical propositional logics can be modelled by classes of non-orthomodular and thus non-distributive lattices that properly contain standard orthomodular and Boolean classes, respectively. In this article we prove that these logics are complete even for those classes of the former lattices from which the standard orthomodular lattices and Boolean algebras are excluded. We also show that neither quantum nor classical computers can be founded on the latter models. It follows that logics are valuation-nonmonotonic in the sense that their possible models (corresponding to their possible hardware implementations) and the valuations for them drastically change when we add new conditions to their defining conditions. These valuations can even be completely separated by putting them into disjoint lattice classes by a technique presented in the paper.

Entropy of Quantum Dynamical Systems and Sufficient Families in Orthomodular Lattices with Bayessian State

The purpose of the present paper is to study the entropy hs(Φ) of a quantum dynamical systems Φ = (L,s,ϕ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy hs(ϕ,\U0001d49c) of partition \U0001d49c of a Boolean algebra B with respect to a state s and a state preserving homomorphism ϕ, we prove a few results on that, define the entropy of a dynamical system hs(Φ), and show its invariance. The concept of sufficient families is also given and we establish that hs(Φ) comes out to be equal to the supremum of hs(ϕ,\U0001d49c), where \U0001d49c varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,ϕ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B,s0,ϕ), where B is a Boolean algebra and s0 is a state on B.

Note on ideals of effect algebras

We show that in a lattice effect algebra, each lattice ideal is an effect algebra ideal if and only if the lattice effect algebra is an orthomodular lattice.

Coverings of [MO n ] and minimal orthomodular lattices

If T is an orthomodular lattice (OML), we denote by [T] the equational class generated by T. In this paper we characterize the finite OMLs T such that [T] covers some [MO n ]. These OMLs T are the non-modular OMLs such that all proper sub-OMLs of T are modular. An OML satisfying that last property is called minimal. There exist infinitely many minimal OMLs provided by quadratic spaces over finite fields. We describe them and give a new way to represent their Greechie diagrams in two separate parts. Other methods to obtain finite minimal OMLs are given.

Many-valued quantum algebras

We deal with algebras \({\bf A} = (A, \oplus, \neg, 0)\) of the same signature as MV-algebras which are a common extension of MV-algebras and orthomodular lattices, in the sense that (i) A bears a natural lattice structure, (ii) the elements a for which \(\neg a\) is a complement in the lattice form an orthomodular sublattice, and (iii) subalgebras whose elements commute are MV-algebras. We also discuss the connections with lattice-ordered effect algebras and prove that they form a variety.

Existence of states on quantum structures

Orthomodular lattices occurred as generalized event structures in the models of probability for quantum mechanics. Here we contribute to the question of existence of states (=probability measures) on orthomodular lattices. We prove that known techniques do not allow to find examples with less than 19 blocks (=maximal Boolean subalgebras). This bound is achieved by the example by Mayet [R. Mayet, Personal communication, 1993]. Although we do not finally exclude the existence of other techniques breaking this bound, existence of smaller examples is highly unexpected.