Previous |  Up |  Next


Archimedean $\ell $-group; divisible hull; distributive radical; complete distributivity
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.
[1] Birkhoff, G.: Lattice Theory. Revised Edition Providence (1948). MR 0029876 | Zbl 0033.10103
[2] Byrd, R. D., Lloyd, J. T.: Closed subgroups and complete distributivity in lattice-ordered groups. Math. Z. 101 (1967), 123-130. DOI 10.1007/BF01136029 | MR 0218284 | Zbl 0178.02902
[3] Darnel, M. R.: Theory of Lattice-Ordered Groups. M. Dekker, Inc. New York-Basel- Hong Kong (1995). MR 1304052 | Zbl 0810.06016
[4] Jakubík, J.: Representation and extension of $\ell$-groups. Czech. Math. J. 13 (1963), 267-283 Russian. MR 0171865
[5] Jakubík, J.: Distributivity in lattice ordered groups. Czech. Math. J. 22 (1972), 108-125. MR 0325487
[6] Jakubík, J.: Complete distributivity of lattice ordered groups and of vector lattices. Czech. Math. J. 51 (2001), 889-896. DOI 10.1023/A:1013781300217 | MR 1864049
[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
[8] Sikorski, R.: Boolean Algebras. Second Edition Springer Verlag Berlin (1964). MR 0126393 | Zbl 0123.01303
Partner of
EuDML logo