Squares and rectangles in relation categories. Three cases: semilattice, distributive lattice and Boolean non-unitary

In this paper we investigate rectangle morphisms of Dedekind categories (or semilattice relation categories, or unitary allegories) which appear to be a basic structure for relation categories. In order to reconstruct the Dedekind category generated by all the rectangle morphisms, we are obliged to clarify the role of both the square morphisms and the functions s which assign to every square morphism c the ideal morphism cs = 1c1. We translate this observations into adapted new objects that we call the “MP-families”. In this way, the reconstruction problem becomes an adjunction problem. To each Dedekind category we associate a MP-family; to each functor between two Dedekind categories we associate a morphism between two MP-families. This association is a functor. This functor has a left adjoint. This result is also true for other universes: distributive lattices, Boolean algebras.