Richard J. Greechie

Orthomodular lattices admitting no states

The purpose of this paper is to construct a class of orthomodular lattices which admit no bounded measures.

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.

The center of an effect algebra

An effect algebra is a partial algebra modeled on the standard effect algebra of positive self-adjoint operators dominated by the identity on a Hilbert space. Every effect algebra is partially ordered in a natural way, as suggested by the partial order on the standard effect algebra. An effect algebra is said to be distributive if, as a poset, it forms a distributive lattice. We define and study the center of an effect algebra, relate it to cartesian-product factorizations, determine the center of the standard effect algebra, and characterize all finite distributive effect algebras as products of chains and diamonds.

Sequential products on effect algebras

Abstract A sequential effect algebra (SEA) is an effect algebra on which a sequential product with natural properties is defined. The properties of sequential products on Hilbert space effect algebras are discussed. For a general SEA, relationships between sequential independence, coexistence and compatibility are given. It is shown that the sharp elements of a SEA form an orthomodular poset. The sequential center of a SEA is discussed and a characterization of when the sequential center is isomorphic to a fuzzy set system is presented. It is shown that the existence, of a sequential product is a strong restriction that eliminates many effect algebras from being SEAs. For example, there are no finite nonboolean SEAs, A measure of sharpness called the sharpness index is studied. The existence of horizontal sums of SEAs is characterized and examples of horizontal sums and tensor products are presented.

On the Structure of Orthomodular Lattices Satisfying the Chain Condition

Abstract Beginning with the external point of view we show how orthomodular lattices may be “pasted” together to yield a new orthomodular lattice. Changing to the internal point of view we show that any two blocks (maximal Boolean suborthomodular lattices) of an orthomodular lattice satisfying the chain condition can be “connected” by blocks which intersect in a specific fashion. Returning to our initial point of view we obtain a method of constructing a given orthomodular lattice from Boolean lattices.

Sums and products of interval algebras

Aninterval algebra is an interval from zero to some positive element in a partially ordered Abelian group, which, under the restriction of the group operation to the interval, is a partial algebra. In this paper we study interval algebras from a categorical point of view, and show that Cartesian products and horizontal sums are effective as categorical products and coproducts, respectively. We show that the category of interval algebras admits a tensor product, and introduce a new class of interval algebras, which are in fact orthoalgebras, calledχ-algebras.

A Non-Standard Quantum Logic with a Strong Set of States

Since the discovery of non-Euclidian geometries the academic community has widely recognized the importance of non-standard models of axiomatic systems. This paper presents a non-standard quantum logic, call it (L44, M22). Previous examples5,6 of non-standard quantum logics were non-standard by reason of the fact that the states were not strongly order determining. M22 is strongly order determining on L44. The property violated in L44 but satisfied in Hilbert space is a variant of Desargues’ Theorem. It is called the ortho- Arguesian law and was first formulated by Alan Day.

Transition to effect algebras

An account is given of the recent development of the theory of effect algebras, their connection with partially ordered abelian groups, and their use for the mathematical representation of fuzzy or unsharp events. We submit an annotated list of important open problems, appropriate research projects, and unresolved philosophical issues engendered by the developing theory.

Completions of orthomodular lattices II

If K is a variety of orthomodular lattices generated by a finite orthomodular lattice the MacNeille completion of every algebra in K again belongs to K.

Commutator-finite orthomodular lattices

AbstractThe class