D-test spaces and difference posets (Q1805287)

The authors introduce the notion of a D-test space generalizing the Kolmogorov probability model, orthomodular posets, and the test spaces of Foulis and Randall. They study a tensor product of D-test spaces. Let \({\mathcal T} \subseteq \{F \in X^ I : I \in {\mathcal I} \}\) be nonempty, where \(X \neq \emptyset\) and \({\mathcal I}\) is a nonvoid family of index sets. The pair \((X, {\mathcal T})\) is said to be a D-test space if the following three conditions are fulfilled: (i) \(T^{-1} (x)\) is of finite cardinality for any \(T \in {\mathcal T}\) and any \(x \in X\); (ii) for every \(x \in X\) there is a \(T \in {\mathcal T}\) such that \(x\) is in the range of \(T\); (iii) if \(S,T \in {\mathcal T}\) and if \(\sigma : I \to I\) is an injection with \(S = T \circ \sigma\), then \(\sigma\) is a bijection. Any element of \({\mathcal T}\) is said to be a D-test.