# On the distributive radical of an Archimedean lattice-ordered group

• Volume: 59, Issue: 3, page 687-693
• ISSN: 0011-4642

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## Abstract

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Let $G$ be an Archimedean $\ell$-group. We denote by ${G}^{d}$ and ${R}_{D}\left(G\right)$ the divisible hull of $G$ and the distributive radical of $G$, respectively. In the present note we prove the relation ${\left({R}_{D}\left(G\right)\right)}^{d}={R}_{D}\left({G}^{d}\right)$. As an application, we show that if $G$ is Archimedean, then it is completely distributive if and only if it can be regularly embedded into a completely distributive vector lattice.

## How to cite

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Jakubík, Ján. "On the distributive radical of an Archimedean lattice-ordered group." Czechoslovak Mathematical Journal 59.3 (2009): 687-693. <http://eudml.org/doc/37951>.

@article{Jakubík2009,
abstract = {Let $G$ be an Archimedean $\ell$-group. We denote by $G^d$ and $R_D(G)$ the divisible hull of $G$ and the distributive radical of $G$, respectively. In the present note we prove the relation $(R_D(G))^d=R_D(G^d)$. As an application, we show that if $G$ is Archimedean, then it is completely distributive if and only if it can be regularly embedded into a completely distributive vector lattice.},
author = {Jakubík, Ján},
journal = {Czechoslovak Mathematical Journal},
keywords = {Archimedean $\ell$-group; divisible hull; distributive radical; complete distributivity; Archimedean -group; divisible hull; distributive radical; complete distributivity},
language = {eng},
number = {3},
pages = {687-693},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the distributive radical of an Archimedean lattice-ordered group},
url = {http://eudml.org/doc/37951},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Jakubík, Ján
TI - On the distributive radical of an Archimedean lattice-ordered group
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 3
SP - 687
EP - 693
AB - Let $G$ be an Archimedean $\ell$-group. We denote by $G^d$ and $R_D(G)$ the divisible hull of $G$ and the distributive radical of $G$, respectively. In the present note we prove the relation $(R_D(G))^d=R_D(G^d)$. As an application, we show that if $G$ is Archimedean, then it is completely distributive if and only if it can be regularly embedded into a completely distributive vector lattice.
LA - eng
KW - Archimedean $\ell$-group; divisible hull; distributive radical; complete distributivity; Archimedean -group; divisible hull; distributive radical; complete distributivity
UR - http://eudml.org/doc/37951
ER -

## References

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3. Darnel, M. R., Theory of Lattice-Ordered Groups, M. Dekker, Inc. New York-Basel- Hong Kong (1995). (1995) Zbl0810.06016MR1304052
4. Jakubík, J., Representation and extension of $\ell$-groups, Czech. Math. J. 13 (1963), 267-283 Russian. (1963) MR0171865
5. Jakubík, J., Distributivity in lattice ordered groups, Czech. Math. J. 22 (1972), 108-125. (1972) MR0325487
6. Jakubík, J., 10.1023/A:1013781300217, Czech. Math. J. 51 (2001), 889-896. (2001) MR1864049DOI10.1023/A:1013781300217
7. Lapellere, M. A., Valente, A., Embedding of Archimedean $\ell$-groups in Riesz spaces, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 249-254. (1998) MR1628633
8. Sikorski, R., Boolean Algebras, Second Edition Springer Verlag Berlin (1964). (1964) Zbl0123.01303MR0126393

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