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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)

##### MSC:
 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)