State spaces of orthomodular structures
We present several known and one new description of orthomodular structures (orthomodular lattices, orthomodular posets and orthoalgebras). Originally, orthomodular structures were viewed as pasted families of Boolean algebras. Here we introduce semipasted families of Boolean algebras as an alternative description which is not as detailed, but substantially simplex. Semipasted families of Boolean algebras correspond to orthomodular structures in such a way that states and evaluation functionals are preserved. As semipasted families of Boolean algebras are quite general, they allow an easy construction of orthomodular structures with given state space properties. Based on this technique, we give a simplified proof of Shultz's Theorem on characterization of spaces of finitely additive states on orthomodular lattices. We also put some other results into the new context. We give a detailed exposition of the construction techniques as a tool for further applications, especially for finding counterexamples to questions about states on orthomodular structures.
Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
31 (2000) suppl.1
Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Mirko Navara, "State spaces of orthomodular structures", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 31 (2000) suppl.1, pp. 143-201.