Article
MSC:
06B35,
06F15,
06F20,
06F30,
20F60,
22A26 | MR 3367096 | Zbl 06487024 | DOI: 10.5817/AM2015-2-107
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Keywords:
characterization; Hausdorff completion; lattice homomorphisms; locally solid topological $l$-groups; neighborhood theorem; order-bounded subsets
characterization; Hausdorff completion; lattice homomorphisms; locally solid topological $l$-groups; neighborhood theorem; order-bounded subsets
Summary:
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.
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