Locally solid topological lattice-ordered groups

Hong, Liang

Archivum mathematicum, Tome 51 (2015) no. 2, pp. 107-128

Abstract (VO)

Locally solid Riesz spaces have been widely investigated in the past several decades; but locally solid topological lattice-ordered groups seem to be largely unexplored. The paper is an attempt to initiate a relatively systematic study of locally solid topological lattice-ordered groups. We give both Roberts-Namioka-type characterization and Fremlin-type characterization of locally solid topological lattice-ordered groups. In particular, we show that a group topology on a lattice-ordered group is locally solid if and only if it is generated by a family of translation-invariant lattice pseudometrics. We also investigate (1) the basic properties of lattice group homomorphism on locally solid topological lattice-ordered groups; (2) the relationship between order-bounded subsets and topologically bounded subsets in locally solid topological lattice-ordered groups; (3) the Hausdorff completion of locally solid topological lattice-ordered groups.

Détail

DOI : 10.5817/AM2015-2-107

Classification : 06B35, 06F15, 06F20, 06F30, 20F60, 22A26
Keywords: characterization; Hausdorff completion; lattice homomorphisms; locally solid topological $l$-groups; neighborhood theorem; order-bounded subsets

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