, Volume 31, Issue 5, pp 843-854

Concrete quantum logics with covering properties

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Abstract

LetL be a concrete (=set-representable) quantum logic. Letn be a natural number (or, more generally, a cardinal). We say thatL admits intrinsic coverings of the ordern, and writeL∈ C n , if for any pairA, B∈L we can find a collection {C i ∶ i∈I}, where cardI<n andC i ∈L for anyi∈I, such thatAB=∪ i∈l C i . Thus, in a certain sense, ifLC n , then “the rate of noncompatibility” of an arbitrary pairA,BL is less than a given numbern. In this paper we first consider general and combinatorial properties of logics ofC n and exhibit typical examples. In particular, for a givenn we construct examples ofL∈ C n+1\C n . Further, we discuss the relation of the classesC n to other classes of logics important within the quantum theories (e.g., we discover the interesting relation to the class of logics which have an abundance of Jauch-Piron states). We then consider conditions on which a class of concrete logics reduce to Boolean algebras. We conclude with some open questions.