On locally solid topological lattice groups

Khan, Abdul Rahim, and Rowlands, Keith. "On locally solid topological lattice groups." Czechoslovak Mathematical Journal 57.3 (2007): 963-973. <http://eudml.org/doc/31175>.

@article{Khan2007,
abstract = {Let $(G,\tau )$ be a commutative Hausdorff locally solid lattice group. In this paper we prove the following: (1) If $(G,\tau )$ has the $A$(iii)-property, then its completion $(\widehat\{G\},\hat\{\tau \})$ is an order-complete locally solid lattice group. (2) If $G$ is order-complete and $\tau $ has the Fatou property, then the order intervals of $G$ are $\tau $-complete. (3) If $(G,\tau )$ has the Fatou property, then $G$ is order-dense in $\widehat\{G\}$ and $(\widehat\{G\},\hat\{\tau \})$ has the Fatou property. (4) The order-bound topology on any commutative lattice group is the finest locally solid topology on it. As an application, a version of the Nikodym boundedness theorem for set functions with values in a class of locally solid topological groups is established.},
author = {Khan, Abdul Rahim, Rowlands, Keith},
journal = {Czechoslovak Mathematical Journal},
keywords = {topological completion; locally solid $\ell $-group; topological continuity; Fatou property; order-bound topology; topological completion; locally solid -group; topological continuity; Fatou property; order-bound topology},
language = {eng},
number = {3},
pages = {963-973},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On locally solid topological lattice groups},
url = {http://eudml.org/doc/31175},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Khan, Abdul Rahim
AU - Rowlands, Keith
TI - On locally solid topological lattice groups
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 3
SP - 963
EP - 973
AB - Let $(G,\tau )$ be a commutative Hausdorff locally solid lattice group. In this paper we prove the following: (1) If $(G,\tau )$ has the $A$(iii)-property, then its completion $(\widehat{G},\hat{\tau })$ is an order-complete locally solid lattice group. (2) If $G$ is order-complete and $\tau $ has the Fatou property, then the order intervals of $G$ are $\tau $-complete. (3) If $(G,\tau )$ has the Fatou property, then $G$ is order-dense in $\widehat{G}$ and $(\widehat{G},\hat{\tau })$ has the Fatou property. (4) The order-bound topology on any commutative lattice group is the finest locally solid topology on it. As an application, a version of the Nikodym boundedness theorem for set functions with values in a class of locally solid topological groups is established.
LA - eng
KW - topological completion; locally solid $\ell $-group; topological continuity; Fatou property; order-bound topology; topological completion; locally solid -group; topological continuity; Fatou property; order-bound topology
UR - http://eudml.org/doc/31175
ER -