International Journal of Theoretical Physics

, Volume 31, Issue 5, pp 843–854

Concrete quantum logics with covering properties

  • Vladimir Müller
  • Pavel Pták
  • Josef Tkadlec
Article

DOI: 10.1007/BF00678549

Cite this article as:
Müller, V., Pták, P. & Tkadlec, J. Int J Theor Phys (1992) 31: 843. doi:10.1007/BF00678549

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∈Cn, if for any pairA, B∈L we can find a collection {Ci∶ i∈I}, where cardI<n andCi∈L for anyi∈I, such thatAB=∪i∈lCi. Thus, in a certain sense, ifLCn, 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 ofCn and exhibit typical examples. In particular, for a givenn we construct examples ofL∈Cn+1\Cn. Further, we discuss the relation of the classesCn 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.

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • Vladimir Müller
    • 1
  • Pavel Pták
    • 2
  • Josef Tkadlec
    • 2
  1. 1.Institute of MathematicsCzechoslovak Academy of SciencesPragueCzechoslovakia
  2. 2.Department of MathematicsTechnical University of PraguePragueCzechoslovakia