arXiv subjects

Jean Berthet

Publications and source records attributed to Jean Berthet.

3 recordsLinked to original sources

arXiv2015

Positive model theory and infinitary logic

We study the basic properties of a dual "spectral" topology on positive type spaces of h-inductive theories and its essential connection to infinitary logic. The topology is Hausdorff, has the Baire property, and its compactness characterises positive model completeness; it also has a basis of clopen sets and is described by the formulas of geometric logic. The "geometric types" are closed in the type spaces under all the operations of infinitary logic, and we introduce a positive analogue of existentially universal structures, through which we interpret the full first order logic in positive type spaces. This shows how "positive $ω$-saturation" is a fundamental connection between positive and infinitary logic, and we suggest a geometric analogue of positive Morleyisation.

math.LO↗

arXiv2013

Geometrically closed rings

We develop the basic theory of geometrically closed rings as a generalisation of algebraically closed fields, on the grounds of notions coming from positive model theory and affine algebraic geometry. For this purpose we consider several connections between finitely presented rings and ultraproducts, affine varieties and definable sets, and we introduce the key notion of an arithmetic theory as a purely algebraic version of coherent logic for rings.

math.RA↗

arXiv2012

Prime types and geometric completeness

The geometric form of Hilbert's Nullstellensatz may be understood as a property of "geometric saturation" in algebraically closed fields. We conceptualise this property in the language of first order logic, following previous approaches and borrowing ideas from classical model theory, universal algebra and positive logic. This framework contains a logical equivalent of the algebraic theory of prime and radical ideals, as well as the basics of an "affine algebraic geometry" in quasivarieties. Hilbert's theorem may then be construed as a model-theoretical property, weaker than and equivalent in certain cases to positive model-completeness, and this enables us to geometrically reinterpret model-completeness itself. The three notions coincide in the theories of (pure) fields and we apply our results to group-based algebras, which supply a way of dealing with certain functional field expansions.

math.LO↗