Locally solid topological lattice-ordered groups

Abstract

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.