Concrete quantum logics with covering properties

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 writeLC_n, if for any pairA, BL we can find a collection {C_i iI}, where cardI<n andC_iL for anyiI, such thatA B=_ilC_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 ofLC_n+1C_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.