On interval homogeneous orthomodular lattices

Anna de Simone; Mirko Navara; Pavel Pták

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 1, page 23-30
  • ISSN: 0010-2628

Abstract

top
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. 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 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 Inthom c . We find that each σ -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 σ -complete orthomodular lattices.

How to cite

top

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 -

References

top
  1. Beran L., Orthomodular Lattices. Algebraic Approach, Academia, Praha and D. Reidel, Dordrecht, 1984. Zbl0558.06008MR0785005
  2. Bruns G., Greechie R., Orthomodular lattices which can be covered by finitely many blocks, Canadian J. Math. 34 (1982), 696-699. (1982) Zbl0493.06008MR0663312
  3. Halmos P.R., Lectures on Boolean Algebras, Van Nostrand, Princeton, 1963. Zbl0285.06010MR0167440
  4. Kalmbach G., Orthomodular Lattices, Academic Press, London, 1983. Zbl0554.06009MR0716496
  5. Ketonen J., The structure of countable Boolean algebras, Ann. Math. 108 (1978), 41-89. (1978) Zbl0418.06006MR0491391
  6. Monk J.D., Bonnet R., Handbook of Boolean Algebras I., North Holland Elsevier Science Publisher B.V., 1989. 
  7. Navara M., Pták P., Rogalewicz V., Enlargements of quantum logics, Pacific J. Math. 135 (1988), 361-369. (1988) MR0968618
  8. Pták P., Pulmannová S., Orthomodular Structures as Quantum Logics, Kluwer, DordrechtBoston-London, 1991. MR1176314
  9. Sikorski R., Boolean Algebras, 3rd ed., Springer Verlag, Berlin/Heidelberg/New York, 1969. Zbl0191.31505MR0242724

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.