A. De Simone, M. Navara, P. Pt\'ak
On interval homogeneous orthomodular lattices

Comment.Math.Univ.Carolinae 42,1 (2001) 23-30.

Abstract: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 $L$). \par 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 $-complete orthomodular lattices which do \underbar {not} belong to {Inthom}. Such examples indeed exist as we than establish. At the end we consider the class {Inthom$_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.

Keywords: interval in a $\sigma $-complete orthomodular lattice, center, Boolean $\sigma $-algebra, {Cantor-Bernstein-Tarski} theorem
AMS Subject Classification: 06C15, 06E05, 81P10