Locally solid topological lattice-ordered groups

Liang Hong

Archivum Mathematicum (2015)

  • Volume: 051, Issue: 2, page 107-128
  • ISSN: 0044-8753

Abstract

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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.

How to cite

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Hong, Liang. "Locally solid topological lattice-ordered groups." Archivum Mathematicum 051.2 (2015): 107-128. <http://eudml.org/doc/270117>.

@article{Hong2015,
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.},
author = {Hong, Liang},
journal = {Archivum Mathematicum},
keywords = {characterization; Hausdorff completion; lattice homomorphisms; locally solid topological $l$-groups; neighborhood theorem; order-bounded subsets},
language = {eng},
number = {2},
pages = {107-128},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Locally solid topological lattice-ordered groups},
url = {http://eudml.org/doc/270117},
volume = {051},
year = {2015},
}

TY - JOUR
AU - Hong, Liang
TI - Locally solid topological lattice-ordered groups
JO - Archivum Mathematicum
PY - 2015
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 051
IS - 2
SP - 107
EP - 128
AB - 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.
LA - eng
KW - characterization; Hausdorff completion; lattice homomorphisms; locally solid topological $l$-groups; neighborhood theorem; order-bounded subsets
UR - http://eudml.org/doc/270117
ER -

References

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