We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. The notion is named b-minimality and is based on definable families of points and balls. We develop a dimension theory and prove a cell decomposition theorem for b-minimal structures. We show that b-minimality applies to the theory of Henselian valued fields of characteristic zero, generalizing work by Denef–Pas [25, 26]. Structures which are o-minimal, v-minimal, or p-minimal and which satisfy some slight extra conditions are (...) also b-minimal, but b-minimality leaves more room for nontrivial expansions. The b-minimal setting is intended to be a natural framework for the construction of Euler characteristics and motivic or p-adic integrals. The b-minimal cell decomposition is a generalization of concepts of Cohen [11], Denef [15], and the link between cell decomposition and integration was first made by Denef [13]. (shrink)

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger (...) language. (shrink)

C-minimality is a variant of o-minimality in which structures carry, instead of a linear ordering, a ternary relation interpretable in a natural way on set of maximal chains of a tree. This notion is discussed, a cell-decomposition theorem for C-minimal structures is proved, and a notion of dimension is introduced. It is shown that C-minimal fields are precisely valued algebraically closed fields. It is also shown that, if certain specific ‘bad’ functions are not definable, then algebraic closure has the exchange (...) property, and for definable sets dimension coincides with the rank obtained from algebraic closure. (shrink)

The notion of a strongly determined type over A extending p is introduced, where p .S. A strongly determined extension of p over A assigns, for any model M )- A, a type q S extending p such that, if realises q, then any elementary partial map M → M which fixes acleq pointwise is elementary over . This gives a crude notion of independence which arises very frequently. Examples are provided of many different kinds of theories with strongly determined (...) types, and some without. We investigate a notion of multiplicity for strongly determined types with applications to ‘involved’ finite simple groups, and an analogue of the Finite Equivalence Relation Theorem. Lifting of strongly determined types to covers of a structure is discussed, and an application to finite covers is given. (shrink)

A theory is stable up to Δ if any Δ-type over a model has a few extensions up to complete types. We prove that a theory has no the independence property iff it is stable up to some Δ, where each equation image has no the independence property.

We will study some aspects of the local structure of models of certain C-minimal theories. We will prove (theorem 19) that, in a sufficiently saturated C-minimal structure in which the algebraic closure has the exchange property and which is locally modular, we can construct an infinite type-definable group around any non trivial point (a term to be defined later). On va étudier ici certains aspects de la structure locale des modèles de certaines théories C-minimales. On va prouver (théorème 19) que, (...) dans une structure C-minimale suffisamment saturée, localement modulaire et dans laquelle la clôture algébrique permet de définir une notion de dimension, on peut construire un groupe type-définissable infini autour de tout point non trivial (terme à définir par la suite). (shrink)

We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has non-empty interior, and any theory of pure tree is dp-minimal.

Recently, Cluckers et al. developed a new axiomatic framework for tame non-Archimedean geometry, called Hensel minimality. It was extended to mixed characteristic together with the author. Hensel minimality aims to mimic o-minimality in both strong consequences and wide applicability. In this paper, we continue the study of Hensel minimality, in particular focusing on [Formula: see text]-h-minimality and [Formula: see text]-h-minimality, for [Formula: see text] a positive integer. Our main results include an analytic criterion for [Formula: see text]-h-minimality, preservation of [Formula: (...) see text]-h-minimality under coarsening of the valuation and [Formula: see text]-dimensional geometry. (shrink)

Let T be a theory. If T eliminates \, it need not follow that \ eliminates \, as shown by the example of the p-adics. We give a criterion to determine whether \ eliminates \. Specifically, we show that \ eliminates \ if and only if \ is eliminated on all interpretable sets of “unary imaginaries.” This criterion can be applied in cases where a full description of \ is unknown. As an application, we show that \ eliminates \ when (...) T is a C-minimal expansion of ACVF. (shrink)

We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian (...) group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p -adic numbers is dp-minimal. (shrink)

Here we study properties of weakly circularly minimal cyclically ordered groups. The main result of the paper is that any weakly circularly minimal cyclically ordered group is abelian.

Macpherson and Steinhorn 165–209) introduce some variants of the notion of o-minimality. One of the most interesting is C-minimality, which provides a natural setting to study algebraically closed-valued fields and some valued groups. In this paper we go further in the study of the structure of C-minimal valued groups, giving a partial characterization in the abelian case. We obtain the following principle: for abelian valued groups G for which the valuation satisfies some kind of compatibility with the multiplication by any (...) prime number p, being C-minimal is equivalent to the o-minimality of the expansion of the chain of valuations by the maps induced by the multiplication by each prime number in G, and by some unary predicates controlling the cardinality of the residual structures. This result is quite nice, because the class considered contains all the natural examples of abelian C-minimal valued groups of 165–209), and allows us to find many more examples. (shrink)

We introduce a very weak language L M on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language L M are trivial functions. We also give a definitional expansion $L\begin{array}{*{20}{c}} ' \\ M \\ \end{array} $ of L M in which K has quantifier elimination, (...) and we obtain a cell decomposition result for L M -definable sets. Our language L M can serve as a p-adic analogue of the very weak language (<) on the real numbers, to define a notion of minimality on the field of p-adic numbers and on related valued fields. These fields are not necessarily Henselian and may have positive characteristic. (shrink)

We explore analogues of o-minimality and weak o-minimality for circularly ordered sets. Much of the theory goes through almost unchanged, since over a parameter the circular order yields a definable linear order. Working over ∅ there are differences. Our main result is a structure theory for ℵ0-categorical weakly circularly minimal structures. There is a 5-homogeneous example which is not 6-homogeneous, but any example which is k-homogeneous for some k ≥ 6 is k-homogeneous for all k.

We compute the domination monoid in the theory of dense meet‐trees. In order to show that this monoid is well‐defined, we prove weak binarity of and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory from [7]. We then describe the domination monoids of such expansions in terms of those of the expanding relations.

In this paper, we consider the question of definability of types in non‐stable theories. In order to do this we introduce a notion of a relatively stable theory: a theory is stable up to Δ if any Δ‐type over a model has few extensions up to complete types. We prove that an n‐type over a model of a theory that is stable up to Δ is definable if and only if its Δ‐part is definable.

We investigate the structure of C-minimal valued groups that are not abelian-by-finite. We prove among other things that they are nilpotent-by-finite.

We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.

Let E be a subset of. A linear extension operator is a linear map that sends a function on E to its extension on some superset of E. In this paper, we show that if E is a semi‐algebraic or subanalytic subset of, then there is a linear extension operator such that is semi‐algebraic (subanalytic) whenever f is semi‐algebraic (subanalytic).