The Lattice of Effective Quantum Logic

Abstract

In this chapter, we investigate the algebraic structure of the calculus Q _eff of effective quantum logic. Since Q _eff can be shown to possess the property of syntactic completeness (Section 5.1) the calculus can be completely replaced by a lattice. This lattice will be called quasi-implicative and denoted by L _qi. In Section 5.2, some important properties of the lattice L _qi will be mentioned. Furthermore, we investigate the relation between the commensurability and implicative sublattices and show that the lattice L _qi is a relaxation of the implicative lattice L _i (Section 5.3). Finally, it will be shown that L _qi is also a relaxation of the orthocomplemented quasimodular lattice from which it differs by only the ‘tertium non datur’ law.