Concrete quantum logics with covering properties
A concrete logic is a nonempty family \(L\) of subsets of a set \(X\) which is closed under complements and disjoint unions. For each cardinal \(\alpha\), the authors define a class \({\mathcal C}_ \alpha\) of concrete logics \(L\) such that, for each \(A,B\in L\), \(A\neq B\), there is a collection \(\{C_ i: i\in I\}\subset L\) satisfying \(\text{card }I<\alpha\) and \(A\cap B=\bigcup_{i\in I} C_ i\) \(({\mathcal C}_ 2\) is the class of Boolean algebras). Highly nontrivial examples are presented which show that \({\mathcal C}_ \beta\subsetneq{\mathcal C}_ \alpha\) for \(\beta<\alpha\). Moreover, \(\bigcup_{\beta<\alpha}{\mathcal C}_ \beta\subsetneq{\mathcal C}_ \alpha\). For \(\alpha\) finite, this answers questions posed by the reviewer and \textit{P. Pták} [J. Pure Appl. Algebra 60, 105-111 (1989; Zbl 0691.03045)]. A state on a concrete logic is a (finitely additive) probability measure. A Jauch-Piron state is a state \(s\) on \(L\) such that \(s(A)=s(B)=1\) implies the existence of a \(C\in L\), \(C\subset A\cap B\), such that \(s(C)=1\). The authors clarify the relations between the classes \({\mathcal C}_ \alpha\) and the classes of concrete logics on which (i) all states, resp. (ii) all two-valued states, resp. (iii) all states carried by a point, are Jauch-Piron. Sufficient conditions are given for concrete logics from these classes to be Boolean algebras. The paper presents a detailed and precise study of the topic. A list of open questions is included. One of them has been solved recently by the first author [``Jauch-Piron states on concrete quantum logics'', Int. J. Theor. Phys. (to appear)]: There is a concrete logic \(L\) which is not a Boolean algebra and all states on \(L\) are Jauch-Piron.
scientific article; zbMATH DE number 95358
scientific article