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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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