2014/04/23 by Liang Hong, Hong, Liang
Mathematics · #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #math.GN #math.GR
paper · pdf · doi:10.48550/arxiv.1404.6155
arxiv created 2015/06/02 · arxiv updated 2015/06/04
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 Robert-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.