interval in a $\sigma$-complete orthomodular lattice; center; Boolean $\sigma$-algebra; Cantor-Bernstein-Tarski theorem
An orthomodular lattice $L$ is said to be interval homogeneous (resp. 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 $d\geq b$ (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. Inthom$_c$) 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 $\sigma$-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 than 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 Cantor-Bernstein-Tarski theorem. With the help of this result, we settle the Tarski cube problem for the $\sigma$-complete orthomodular lattices.
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