| Title:
|
On the distributive radical of an Archimedean lattice-ordered group (English) |
| Author:
|
Jakubík, Ján |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
59 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
687-693 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Archimedean $\ell $-group |
| Keyword:
|
divisible hull |
| Keyword:
|
distributive radical |
| Keyword:
|
complete distributivity |
| MSC:
|
06F20 |
| MSC:
|
46A40 |
| idZBL:
|
Zbl 1224.06033 |
| idMR:
|
MR2545649 |
| . |
| Date available:
|
2010-07-20T15:33:24Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140509 |
| . |
| Reference:
|
[1] Birkhoff, G.: Lattice Theory.Revised Edition Providence (1948). Zbl 0033.10103, MR 0029876 |
| Reference:
|
[2] Byrd, R. D., Lloyd, J. T.: Closed subgroups and complete distributivity in lattice-ordered groups.Math. Z. 101 (1967), 123-130. Zbl 0178.02902, MR 0218284, 10.1007/BF01136029 |
| Reference:
|
[3] Darnel, M. R.: Theory of Lattice-Ordered Groups.M. Dekker, Inc. New York-Basel- Hong Kong (1995). Zbl 0810.06016, MR 1304052 |
| Reference:
|
[4] Jakubík, J.: Representation and extension of $\ell$-groups.Czech. Math. J. 13 (1963), 267-283 Russian. MR 0171865 |
| Reference:
|
[5] Jakubík, J.: Distributivity in lattice ordered groups.Czech. Math. J. 22 (1972), 108-125. MR 0325487 |
| Reference:
|
[6] Jakubík, J.: Complete distributivity of lattice ordered groups and of vector lattices.Czech. Math. J. 51 (2001), 889-896. MR 1864049, 10.1023/A:1013781300217 |
| Reference:
|
[7] Lapellere, M. A., Valente, A.: Embedding of Archimedean $\ell$-groups in Riesz spaces.Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 249-254. MR 1628633 |
| Reference:
|
[8] Sikorski, R.: Boolean Algebras.Second Edition Springer Verlag Berlin (1964). Zbl 0123.01303, MR 0126393 |
| . |
