Equivalence relations and operators on ordered algebraic structures with difference.

This work concerns algebraic models of fuzzy and many-valued propositional logics, in particular Boolean Algebras, Heyting algebras, GBL-algebras and their dual structures, and partial algebras. The central idea is the representation of complex structures through simpler structures and equivalence relations on them: in order to achieve this, a structure is often considered under two points of view, as total algebra and partial algebra. The equivalence relations which allow the representation are congruences of partial algebras. The first chapter introduces D-posets, the partial algebraic structures used for this representation, which generalize Boolean algebras and MV-algebras. The second chapter is a study of congruences on D-posets and the structure of the quotients, in particular for congruences induced by some kinds of idempotent operators, here called S-operators. The case of Boolean algebras and MV-algebras is studied more in detail. The third chapter introduces GBL-algebras and their dual, and shows how the interplay of an S-operator with a closure operator gives rise to a dual GBL-algebra. Other results about the representation of finite GBL-algebras and GBL*algebras (GBL-algebras with monoidal sum), part of two papers previously published, are summarized and put in relation with the other results of this work.