Extended relational Calculus, Galois Diconnections and DiChu Categories
by
Jacques Riguet
University Paris V

In the first part of this lecture , we express by operational (eq & ineq)uations the relational properties of pairs of mappings having the property of being generalised inverses or more generally forming a Galois connection between (partially) ordered sets. The operational expressions are those generated from some basic ones in a way quite similar to that one developped by Richard Bird and Oege de Moor in [1]. Moreover, the operator transforming a binary relation into its “exact part” as defined in [2] plays a leading role.

In the second part, we extend the notion of pairs of generalised inverse functional relations to pairs of difunctional relations or more generally to those forming a “Galois Diconnection” between preordered sets. We express the definitions and properties of this generalisation in the operational relational language. We give apllications of this generalisation in various fields: universal algebra, computer science programming ( concurrency, abstract interpretations ; etc. )

In the third part, we describe how “Galois Diconnection” can be considered as a special case of mporphisms of “ DiChu Categories”

[1] Bird R. & O. de Moor ; Algebra of Programming ; Prentice Hall ; 1997

[2] Riguet J. ; Deductibility and exactness ; Logic Journal of the IGPL ; Vol. 6 n°2 ; 1998 ; 341-347

Date received: January 28, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagi-13.