, Volume 31, Issue 5, pp 881-888

Modular almost orthogonal quantum logics

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Abstract

Almost orthogonal quantum logics, i.e., atomic orthomodular lattices in which to every atom there exist only finitely many nonorthogonal atoms, are studied. It is shown that an almost orthogonal quantum logic is modular if and only if it has the exchange property if and only if it can be embedded into a direct product of finite modular quantum logics. The class of almost orthogonal modular OMLs is the largest subclass of the class of atomic modular OMLs in which the conditions commutator-finite and block-finite are equivalent. A finite faithful valuation on an almost orthogonal quantum logicL exists if and only ifL is modular and the set of all atoms ofL is at most countable.