Online research in philosophy

Non-perfect separably closed fields are stable, and not superstable. As a result, not all types can be ranked. We develop here a new tool, a "semi-rank", which takes values in the non-negative reals, and gives a sufficient condition for forking of types. This semi-rank is built up from a transcendence function, analogous to the one considered by Kolchin in the context of differentially closed fields. It yields some orthogonality and stratification results. /// Un corps séparablement clos non algébriquement clos est (...) stable sans être superstable. Cela signifie que seuls certains de ses types sont rangés. Nous développons un autre outil, un "semi-rang" à valeurs réelles, qui donne un critère de déviation des types. Ce semi-rang est construit à partir d'un analogue de la fonction de Kolchin associée à un type au-dessus d'un corps différentiel. Il produit des résultats de stratification des modèles et des résultats d'orthogonalité. (shrink)

We consider separably closed fields of characteristic p > 0 and fixed imperfection degree as modules over a skew polynomial ring. We axiomatize the corresponding theory and we show that it is complete and that it admits quantifier elimination in the usual module language augmented with additive functions which are the analog of the p-component functions.

An Ax-Kochen-Ershov principle for intermediate structures between valued groups and valued fields. We will consider structures that we call valued B-groups and which are of the form $\langle G, B, *, v\rangle$ where - G is an abelian group, - B is an ordered group, - v is a valuation defined on G taking its values in B, - * is an action of B on G satisfying: ∀ x ∈ G ∀ b ∈ B v(x * b) = v(x) (...) · b. The analysis of Kaplanski for valued fields can be adapted to our context and allows us to formulate an Ax-Kochen-Ershov principle for valued B-groups: we axiomatise those which are in some sense existentially closed and also obtain many of their model-theoretical properties. Let us mention some applications: 1. Assume that v(x) = v(nx) for every integer n ≠ 0 and x ∈ G, B is solvable and acts on G in such a way that, for the induced action, $\mathbb{Z}[B] \setminus \{0\}$ embeds in the automorphism group of G. Then $\langle G, B, *, v\rangle$ is decidable if and only if B is decidable as an ordered group. 2. Given a field k and an ordered group B, we consider the generalised power series field k((B)) endowed with its canonical valuation. We consider also the following structure: $\mathbf{M} = \langle k((B))_+, S, v, \times \upharpoonright_{k((B))\times S}\rangle,$ where k((B)) + is the additive group of k((B)), S is a unary predicate interpreting ${T^b| b \in B}$ , and $\times \upharpoonright_{k((B))\times S}$ is the multiplication restricted to k((B))× S, structure which is a reduct of the valued field k((B)) with its canonical cross section. Then our result implies that if B is solvable and decidable as an ordered group, then M is decidable. 3. A valued B-group has a residual group and our Ax-Kochen-Ershov principle remains valid in the context of expansions of residual group and value group. In particular, by adding a residual order we obtain new examples of solvable ordered groups having a decidable theory. (shrink)

We study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the (...) definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k. (shrink)

The theory of immediate pairs of Henselian valued fields, with a given residual theory (of characteristic zero) and a given theory of valuation group (nonzero), is undecidable and has 2ℵ0 completions.

We study the structures $(K \subset K^\mathrm{r})$ , where K is an ordered field and Kr its real closure, in the language of ordered fields with an additional unary predicate for the subfield K. Two such structures $(K \subset K^\mathrm{r})$ and $(L \subset L^\mathrm{r})$ are not necessarily elementary equivalent when K and L are. But with some saturation assumption on K and L, then the two structures become equivalent, and we give a description of the complete theory.

A chain-closed field is defined as a chainable field (i.e. a real field such that, for all n ∈ N, Σ K2n+1 ≠ Σ K2n) which does not admit any "faithful" algebraic extension, and can also be seen as a field having a Henselian valuation ν such that the residue field K/ν is real closed and the value group ν K is odd divisible with |ν K/2ν K| = 2. If K admits only one such valuation, we show that f (...) ∈ K(X) is in $\mathbf{\Sigma} K(X)^{2n} \operatorname{iff}$ for any real algebraic extension L of $K, "f(L) \subseteq \mathbf{\Sigma}L^{2n}"$ holds. The conclusion is also true for K = R((t)) (a chainable but not chain-closed field), and in the case n = 1 it holds for several variables and any real field K. (shrink)

Separated and immediate extensions of valued fields. The notion of separated extension of valued fields was introduced by Baur. He showed that extensions of maximal fields are separated. We prove that, when (K, v) is Henselian with residual characteristic 0, then $(K, v) \subset (L, w)$ is separated iff L is linearly disjoint over K from each immediate extension of K.

Consider k((G)) in the language of valued fields enriched with a unary predicate for the set of constants and another one for the cross-section. For perfect k, this structure is undecidable if it does not satisfy Kaplansky's conditions.

L. van den Dries proved that the theory of n-valued rings has a model companion. We show here that this result is still true when the valuation rings are required to satisfy given inclusion relations (we restrict ourselves to the case of residual characteristic zero).