On étudie les propriétés des théories de Jonsson et des conditions nécessaires afin que des théories soient de Jonsson ; on caractérise les théories de Jonsson cosémantiques c’est-à-dire ayant un modèle homogène universel commun ainsi que certaines théories complètes contenant des théories de Jonsson. On donne des exemples de certaines situations.

We prove that for every simple theory T there is a compact abstract theory T.

We define elementary extension and elementary equivalence in Positive Logic.

We continue [2], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones use in first order theories. With these modified tools we obtain more or less classical behaviour: simplicity is characterized by the existence of a certain notion of independence, stability is characterized by simplicity and bounded multiplicity, and hyperimaginary canonical bases exist.

We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such.

We continue the analysis of foundations of positive model theory as introduced by Ben Yaacov and Poizat. The objects of this analysis are $h$-inductive theories and their models, especially the “positively” existentially closed ones. We analyze topological properties of spaces of types, introduce forms of quantifier elimination, and characterize minimal completions of arbitrary $h$-inductive theories. The main technical tools consist of various forms of amalgamations in special classes of structures.

We revisit the foundations of positive model theory, introducing h-inductive sentences. These allow a considerably simplified presentation of positive model theory, as well as a characterisation of Hausdorff cats by an amalgamation property of their h-inductive theory.

La Logique positive a été introduite au début de ce troisième millénaire par Itaï Ben Yaacov, qui y a été conduit par une nécessité interne à la Théorie des modèles. Dans ce contexte de validité du Théorème de compacité, l’absence de négation provoque des situations inhabituelles, comme celle des structures infinies qui ont une extension élémentaire maximale, que nous étudions ici.

We prove that under reasonable assumptions, every cat (compact abstract theory) is metric, and develop some of the theory of metric cats. We generalise Morley's theorem: if a countable Hausdorff cat T has a unique complete model of density character Λ ≥ ω₁, then it has a unique complete model of density character Λ for every Λ ≥ ω₁.

We revisit the foundations of positive model theory, introducing h-inductive sentences. These allow a considerably simplified presentation of positive model theory, as well as a characterisation of Hausdorff cats by an amalgamation property of their h-inductive theory.