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.
@article{119221,
author = {Anna de Simone and Mirko Navara and Pavel Pt\'ak},
title = {On interval homogeneous orthomodular lattices},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {42},
year = {2001},
pages = {23-30},
zbl = {1077.06005},
mrnumber = {1825370},
language = {en},
url = {http://dml.mathdoc.fr/item/119221}
}
de Simone, Anna; Navara, Mirko; Pták, Pavel. On interval homogeneous orthomodular lattices. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 23-30. http://gdmltest.u-ga.fr/item/119221/
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