Abstract
This paper considers metrics valued in abelian l-groups and their induced topologies. In addition to a metric into an l-group, one needs a filter in the positive cone to determine which balls are neighborhoods of their center. As a key special case, we discuss a topology on a lattice ordered abelian group from the metric dG and the positive filter consisting of the weak units of G; in the case of \({\mathbb R^{n}}\) , this is the Euclidean topology. We also show that there are many Nachbin convex topologies on an l-group which are not induced by any positive filter of the l-group.
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