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On representation theorems for observables in weakly complemented posets. (English) Zbl 0767.03032
A weakly orthocomplemented \(\sigma\)-poset is a poset \(M\) with an orthocomplementation \('\) such that (i) \(a \leq b\) implies \(b' \leq a'\), (ii) \((a')' \geq a\), (iii) \(a' \not= a\), and (iv) \(\bigvee_{i \in N} a_ i\) exists whenever \(a_ i \leq a_ j'\) for \(i \not= j\). An observable on \(M\) is a \(\sigma\)-homomorphism of the Borel \(\sigma\)- algebra into \(M\) (despite some mistakes in the definition of a \(\sigma\)- homomorphism, this notion is clear). The authors prove that if \(z,y\) are observables on \(M\) and the range of \(y\) contains the range of \(z\), then \(z = y \circ T^{-1}\) for some Borel function \(T\). Generalizations of this result for other \(\sigma\)-algebras of subsets are discussed. Numerous examples clarify the connections to more special structures, especially to \(\sigma\)-orthomodular posets and F-quantum spaces.
Reviewer: M.Navara (Praha)

03G12 Quantum logic
06C15 Complemented lattices, orthocomplemented lattices and posets
28A60 Measures on Boolean rings, measure algebras
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)