The Algebraic Structure Of Quantum Logic And Operational Continuity

The theory of quantum logic, which has developed during the course of themathematical axiomatization of the quantum theory, is copious, with a historyof 80 years. This thesis focuses on the algebraic structures of partial Abeliansemigroups and difference sets, which were introduced ten years before, and theoperation continuity of a class of important quantum logics. The main contentsand results can be summarized as follows:1. Partial Abelian semigroup is one of the generalizations of effect alge-bra which is no longer bounded. Special congruences are introduced in partialAbelian semigroups and it is shown that under some condition the quotient in-duced by a special congruence is also a partial Abelian semigroup.2. Special ideals are introduced in partial Abelian monoid. It is proved thatunder certain condition, special congruence are just induced by the special ideals.Moreover, it is also shown that a class of special ideals named S-ideals have somelattice properties.3. Filter is a dual notion of ideal. There are three filters in orthoalgebra:CR-filter, FGR-filter and local filter and there are two filters in lattice effectalgebra: effect algebra filter and lattice filter. It is shown that in orthoalgebraslocal filters are equivalent to CR-filters and it is also shown that in lattice effectalgebras FGR-filters are equivalent to local filters while FGR-filters are strictlystronger than CR-filters.4. Difference set is one kind of unbounded difference poset. The existenceof the tensor products of two cancellative difference sets are investigated. It isproved that the tensor product of a Boolean algebra and a cancellative differenceset always exists.5. It is an important and difficult topic to study the topology on quantumlogics. The difficulty is that the quantum logic operations must be continuouswith respect to the ideal topology. It is proved that for some kind of effect algebras two variable operations⊕and (?) are both continuous with respect tothe ideal topology and it is also shown that under some condition two variablelattice operations∧and∨are both continuous with respect to the ideal topology.