Uniformities and a quantale structure on localic groups

Abstract

Similarly like classical topological groups, the point-free counterparts, localic groups, possess natural uniformities (see e.g. [4,2,10]) obtained from an involutive binary operation on $L$ (roughly corresponding to the algebra of subsets of a classical group). It is the operation that naturally induces the uniformities (even if it would not result from a group) and a study of this aspect of the construction is the main topic of this article. We have here a functor associating with the localic groups quantales of a special type (and with frame group homomorphisms quantale homomorphisms) which are shown to create the uniformities, in fact as a special case of the natural uniformities connected with metric structures. Also, we present a condition under which the quantale allows a reconstruction of the localic group.