The quantale of Galois connections

Abstract

Galois connections were originally expressed in a contravariant form with transformations that reverse (rather than preserve) order. Nowadays its covariant form (as residuated maps) is more often used since it is more convenient; namely compositions of residuated maps are handled more easily. In this paper we show that this is not a serious disadvantage of the contravariant form (at least in the natural context for uniform structures, where we need it), by introducing an operation of composition in the complete lattice Gal(L, L) of all (contravariant) Galois connections in a complete lattice L, that allows us to work with Galois connections in the same way as one usually works with residuated maps. This operation endows Gal(L, L) with a structure of quantale whenever L is a locale, allowing the description of uniform structures in terms of Galois connections.