A physical interpretation of the orthomodularity in a lattice-theoretical axiomatics is given in terms of the compatibility relations defined in the paper. A connection between compatibility and commutativity has also been established
A class of quantum logics (orthomodular atomic lattices) is defined and studied. The obtained results suggest that the covering property acts as a selection criterion for theories in the sense that, given a set of orthomodular atomic lattices, those which may be theories for a fixed set of systems may be selected by verifyng if they have the covering property or not. A relation between physical meanings of the covering property, lattice - algebraic operations and the commutativity relation is also suggested
The classical theories, represented by a special type of Boolean orthomodular lattices, are defined. By using the notion of classical theory, several reasons are given, permitting to assume that any orthomodular lattice which may be considered as a physical theory is «constructed» from a set of «classical components». This assumption leads to an interpretation of the atomicity of physical theories/orthomodular lattices. Arguments for the existence of nonclassical theories are examined by employing nonclassical observables, which are also defined in the paper
An axiomatic mathematical structure is presented in terms of which arbitrary mathematical models of a physical system can be rigorously formulated and studied. Physical systems, observables, and states all arise in a natural way from the primitive notion of a physical operation. In particular, the structure is independent of the special mathematical completion defined by the quantum mechanical model, since no lattice theoretical or Hilbert space assumptions are used. The precise relationship between the present structure and the usual quantum mechanical model is investigated in a succeeding paper.