Lattice properties of ring-like quantum logics

Generalized Boolean quasirings (GBQRs) are extensions of partial algebras that are in one-to-one correspondence to bounded lattices with an involutory antiautomorphism. This correspondence generalizes the bijection between Boolean rings and Boolean algebras and provides for a large variety of presumptive presumptive quantum logics (including logics which can be defined by means of Mackey's probability function). It is shown how properties of the corresponding lattices are reflected in GBQRs and what the implications are of the associativity of the +-operation of GBQRs, which can be interpreted as some kind of an "exclusive or"-operation. We prove that under very weak conditions, which, however, seem to he essential for experimental verifications, the associativity of; implies the classicality of the considered quantum mechanical system.