Cartouche Writing of Galois and Adjoint Pairs and Their Applications for Geometrically Depicting Consequence Relations in Logics and in Relational Data Bases

Jacques Riguet

Séminaire lotharingien de combinatoire, Tome 17 (1987)

Résumé

Four fundamental remarks are giving the motivation and are explaining the developments of this paper:

1. In some real dynamical systems one can find action pairs whose properties can be described as Galois or adjoint pairs between ordered sets or, more generally, as jj pairs between sets (Galois or adjoint pairs being jj pairs with special properties, corresponding to some basic order relations.)
2. As it is well known, Galois and adjoint pairs between ordered sets are but a special case of henceforth classical notions of Galois and adjoint pairs between categories.
3. The cartouche writing (as we call it) of jĵ pairs supplies a suggestive geometrical way of depicting them and also of depicting adjoint pairs between categories.
4. The algebraic modelisation of a real system is not a category, but a more general structure that we call an actegory.

As a consequence of these remarks, we are led

• to build foundations for an actegory-theory and to show Galois and adjoint pairs in ordered sets or in categories can be formulated and studied in the framework of actegory-theory.
• to show the expressive power of the cartouche writing by examples coming from algebra, logic and theory of relational data bases.