On interval homogeneous orthomodular lattices

de Simone, Anna, Navara, Mirko, and Pták, Pavel. "On interval homogeneous orthomodular lattices." Commentationes Mathematicae Universitatis Carolinae 42.1 (2001): 23-30. <http://eudml.org/doc/261971>.

@article{deSimone2001,
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 $c\le a$ and $d\ge b$ (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.},
author = {de Simone, Anna, Navara, Mirko, Pták, Pavel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {interval in a $\sigma $-complete orthomodular lattice; center; Boolean $\sigma $-algebra; Cantor-Bernstein-Tarski theorem; orthomodular lattice; -completeness; interval homogeneity; Cantor-Bernstein theorem},
language = {eng},
number = {1},
pages = {23-30},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On interval homogeneous orthomodular lattices},
url = {http://eudml.org/doc/261971},
volume = {42},
year = {2001},
}

TY - JOUR
AU - de Simone, Anna
AU - Navara, Mirko
AU - Pták, Pavel
TI - On interval homogeneous orthomodular lattices
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 1
SP - 23
EP - 30
AB - 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 $c\le a$ and $d\ge b$ (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.
LA - eng
KW - interval in a $\sigma $-complete orthomodular lattice; center; Boolean $\sigma $-algebra; Cantor-Bernstein-Tarski theorem; orthomodular lattice; -completeness; interval homogeneity; Cantor-Bernstein theorem
UR - http://eudml.org/doc/261971
ER -