## The quantale of Galois connections.(English)Zbl 1084.06003

Summary: 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 the structure of a quantale whenever $$L$$ is a locale, allowing the description of uniform structures in terms of Galois connections.

### MSC:

 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06F07 Quantales 54E15 Uniform structures and generalizations
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