On interval homogeneous orthomodular lattices

An orthomodular lattice L is said to be interval homogeneous (resp. centrally interval homogeneous) if it is σ-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 ≤ a and d ≥ b (resp. 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 (resp. Inthomc) the class of all interval homogeneous orthomodular lattices (resp. centrally interval homogeneous orthomodular lattices). We first show that the class Inthom is considerably large — it contains any Boolean σ-algebra, any block-finite σ-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 σ-complete orthomodular lattices which do not belong to Inthom. Such examples indeed exist as we than establish. At the end we consider the class Inthomc . We find that each σ-complete orthomodular lattice belongs to Inthomc, establishing an orthomodular version of Cantor-Bernstein-Tarski theorem. With the help of this result, we settle the Tarski cube problem for the σ-complete orthomodular lattices.