Uniformities and a quantale structure on localic groups

Abstract

of their size), the idea of shifting neighbourhoods of the group unit via the translation homeomorphisms do not work. Despite this difficulty, the paper proves that there are again induced uniformities (both of Tukey or Weil type) and the two are equivalent. The major tool in the paper is that of a special type of involutive quantale (called G -quantale ), which is borne out of the group structure of the locale. Such quantales are investigated and characterised paving the path to formulating proper-ties of the enriched quantales from which the group structure can be reconstructed. The localic group homomorphisms then turn out to be in one to one correspondence with quantale homomorphisms. Finally, the G -quantales are also ordered semigroups and hence can be used as a set of values for a generalised metric. The paper shows the natural group uniformities are metric uniformities of thus generalised metrics.