# Locally solid topological lattice-ordered groups

## Liang Hong

Address: Department of Mathematics, Robert Morris University, Moon, PA 15108, USA

E-mail: hong@rmu.edu

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 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.

AMSclassification: primary 06B35; secondary 06F15, 06F20, 06F30, 22A26, 20F60.

Keywords: characterization, Hausdorff completion, lattice homomorphisms, locally solid topological l-groups, neighborhood theorem, order-bounded subsets.

DOI: 10.5817/AM2015-2-107