Maps on a quantum logic (Q989682)

Let \(L\) be an orthomodular lattice. The author introduces a class \(\Gamma\) of functions \(G: L \times L \to L\) and studies their properties. \(\Gamma\) splits into eight subclasses \(\Gamma_i\) according to the values of each \(G\) in the points \((O,O)\), \((O,I)\) and \((I,I)\). Three of them, \(\Gamma_2\), \(\Gamma_3\) and \(\Gamma_4\), are of special interest, for they consist, respectively, of the so-called maps for simultaneous measurement, join maps and difference maps, which have already been considered by the first author in earlier papers. More specifically, every function from \(\Gamma_2 \cup \Gamma_3 \cup \Gamma_4\) gives rise to a state \(m\) on \(L\): if \(G \in \Gamma_2\), then \(m(x) := G(x,I)\) and \(G(a,b) = m(a \wedge b)\); if \(G \in \Gamma_3\), then \(m(x) := G(x,x)\) and \(G(a,b) = m(a \vee b)\); if \(G \in \Gamma_4\), then \(m(x) := G(x,O)\) and \(G(a,b) = m(a \triangle b)\) (\(\triangle\) stands for the symmetric difference; \(a\) and \(b\) in these identities are assumed to be compatible).