Quantales and continuity spaces
The paper under review attempts to generalize the notion of a metric using as distance sets the so-called value quantales. Such a generalized metric space is called a continuity space. In the first part of the paper the author introduces the notion of a value distributive lattice as a completely distributive lattice such that the set of all elements well above the bottom element is a filter. In the second part the notion of a value quantale as a quantale that is a value distributive lattice is defined. He proves that any element \(p\) in a value quantale is the meet of all elements \(p+c\), \(c\) well above 0. In the next part the notion of a \(V\)-continuity space is introduced. Natural examples of \(V\)-continuity spaces are posets, (probabilistic) metric spaces and structure spaces. Since we have a generalized metric, the author defines the notion of an open ball and a closed ball. So we have an induced topology, the dual topology and the symmetric topology. The author proves that the symmetric topology is the join of the induced topology and the dual one. The last part establishes the connection between the value semigroups and the value quantales in the context of continuity spaces.
scientific article; zbMATH DE number 1226205