On the lattice structure of quantum logics

Krystyna Bugajska, Slawomir Bugajski

Hao Zeng

In a classicial world people use Boolean logic,but in a quantum world must use quantum logic,the two logic's algebra structures are Boolean lattice and orthmodular lattice respectively. It is described the two algebra structures,interpreted the problem uncompatibility and analysed,compared two solutions to the problem.

Peter D. Finch

A weak logical structure is defined as a set of boolean propositional logics in which one can define common operations of negation and implication. The set union of the boolean components of a weak logical structure is a logic of propositions which is an orthocomplemented poset, where orthocomplementation is interpreted as negation and the partial order as implication. It is shown that if one can define on this logic an operation of logical conjunction which has certain plausible properties, then the logic has the structure of an orthomodular lattice. Conversely, if the logic is an orthomodular lattice then the conjunction operation may be defined on it.

Peter D. Finch

В.Л. Васюков

The paper is the contribution to quantum toposophy focusing on the abstract orthomodular structures (following Dunn-Moss-Wang terminology). Early quantum toposophical approach to "abstract quantum logic" was proposed based on the topos of functors $\mathsf{[E,Sets]}$ where $\mathsf{E}$ is a so-called orthomodular preorder category – a modification of categorically rewritten orthomodular lattice (taking into account that like any lattice it will be a finite co-complete preorder category). In the paper another kind of categorical semantics of quantum logic is discussed which is based on the modification of the topos construction itself – so called $quantos$ – which would be evaluated as a non-classical modification of topos with some extra structure allowing to take into consideration the peculiarity of negation in orthomodular quantum logic. The algebra of subobjects of quantos is not the Heyting algebra but an orthomodular lattice. Quantoses might be apprehended as an abstract reflection of Landsman's proposal of "Bohrification", i.e., the mathematical interpretation of Bohr's classical concepts by commutative $C^*$-algebras, which in turn are studied in their quantum habitat of noncommutative $C^*$-algebras – more fundamental structures than commutative $C^*$-algebras. The Bohrification suggests that topos-theoretic approach also should be modified. Since topos by its nature is an intuitionistic construction then Bohrification in abstract case should be transformed in an application of categorical structure based on an orthomodular lattice which is more general construction than Heyting algebra – orthomodular lattices are non-distributive while Heyting algebras are distributive ones. Toposes thus should be studied in their quantum habitat of "orthomodular" categories i.e. of quntoses. Also an interpretation of some well-known systems of orthomodular quantum logic in quantos of functors $\mathsf{[E,QSets]}$ is constructed where $\mathsf{QSets}$ is a quantos (not a topos) of quantum sets. The completeness of those systems in respect to the semantics proposed is proved.

Peter Mittelstaedt