Independence of automorphism group, center, and state space of quantum logics

Abstract

We prove that quantum logics (=orthomodular posets) admit full independence of the attributes important within the foundations of quantum mechanics. Namely, we present the construction of quantum logics with given sublogics (=physical subsystems), automorphism groups, centers (=“classical parts” of the systems), and state spaces. Thus, all these “parameters” are independent. Our result is rooted in the line of investigation carried out by Greechie; Kallus and Trnkova; Kalmbach; and Navara and Ptak; and considerably enriches the known algebraic methods in orthomodular posets.