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On interval homogeneous orthomodular lattices. (English) Zbl 1052.06007
Summary: An orthomodular lattice \(L\) is said to be interval homogeneous (respectively centrally interval homogeneous) if it is \(\sigma \)-complete and satisfies the following property: Whenever \(L\) is isomorphic to an interval, \([a,b]\), in \(L\) then \(L\) is isomorphic to each interval \([c,d]\) with \(c\leq a\) and \(d\geq b\) (respectively the same condition as above only under the assumption that all elements \(a\), \(b\), \(c\), \(d\) are central in \(L\)).
Let us denote by {Inthom} (respectively Inthom\(_c\)) the class of all interval homogeneous orthomodular lattices (respectively centrally interval homogeneous orthomodular lattices). We first show that the class Inthom is considerably large – it contains any Boolean \(\sigma \)-algebra, any block-finite \(\sigma \)-complete orthomodular lattice, any Hilbert space projection lattice and several other examples. Then we prove that \(L\) belongs to Inthom exactly when the Cantor-Bernstein-Tarski theorem holds in \(L\). This makes it desirable to know whether there exist \(\sigma \)-complete orthomodular lattices which do not belong to Inthom. Such examples indeed exist as we then establish. At the end we consider the class Inthom\(_c\). We find that each \(\sigma \)-complete orthomodular lattice belongs to Inthom\(_c\), establishing an orthomodular version of the Cantor-Bernstein-Tarski theorem. With the help of this result, we settle the Tarski cube problem for the \(\sigma \)-complete orthomodular lattices.

MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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