50 research outputs found
o-Minimal cohomology: Finiteness and Invariance Results
No full textThe topology of definable sets in an o-minimal expansion of a group is not fully understood due to the lack of a triangulation theorem. Despite the general validity of the cell decomposition theorem, we do not know whether any definably compact set is a definable CW-complex. Moreover the closure of an o-minimal cell can have arbitrarily high Betti numbers. Nevertheless we prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language
Arithmetic of Dedekind cuts of ordered Abelian groups
No full textAbstractWe study Dedekind cuts on ordered Abelian groups. We introduce a monoid structure on them, and we characterise, via a suitable representation theorem, the universal part of the theory of such structures
E-ideals in exponential polynomial rings
No full textWe investigate exponential ideals within the context of exponential polynomial rings over exponential fields. We establish two distinct notions of maximality for exponential ideals and explore their relationship to primeness. These three concepts—prime, maximal, and E-maximal—are shown to be independent, in contrast to the classical scenario. Furthermore, we demonstrate that, over an algebraically closed field K, the correspondence between points of (Formula presented.) and maximal exponential ideals of the ring of exponential polynomials breaks down. Finally, we introduce and characterize exponential radical ideals
A weak version of the strong exponential closure
No full textAssuming Schanuel’s Conjecture we prove that for any irreducible variety V ⊆ C^n × (C*)^n over Q^alg, of dimension n, and with dominant projections on both the first n coordinates and the last n coordinates, there exists a generic point (overline a , e^overline n) ∈ V. We obtain in this way many instances of the Strong Exponential Closure axiom introduced by Zilber
The addition theorem for locally monotileable monoid actions
No full textWe prove an instance of the so-called Addition Theorem for the algebraic entropy of actions of cancellative right amenable monoids S on discrete abelian groups A by endomorphisms, under the hypothesis that S is locally monotileable (that is, S admits a right Følner sequence (Fn)n∈N such that Fn is a monotile of Fn+1 for every n∈N). We study in details the class of locally monotileable groups, also in relation with already existing notions of monotileability for groups, introduced by B. Weiss and developed further by other authors recently
Relative Pfaffian closure for Definably Complete Baire Structures
No full textSpeissegger proved that the Pfaffian closure of an o-
minimal expansion of the real field is o-minimal. Here we give a
first order version of this result: having introduced the notion of
definably complete Baire structure, we define the relative Pfaf-
fian closure of an o-minimal structure inside a definably complete
Baire structure, and we prove its o-minimality. We derive effec-
tive bounds on some topological invariants of sets definable in
the Pfaffian closure of an o-minimal expansion of the real field
Definably Complete Baire Structures
No full textWe consider definably complete Baire expansions of ordered
fields: every definable subset of the domain of the structure has a
supremum and the domain can not be written as the union of a definable
increasing family of nowhere dense sets. Every expansion
of the real field is definably complete and Baire, and so is every
o-minimal expansion of a field. Moreover, unlike the o-minimal
case, the structures considered form an axiomatizable class. In
this context we prove the following version of Wilkie’s Theorem
of the Complement: given a definably complete Baire expansion
K of an ordered field with a family of smooth functions, if there
are uniform bounds on the number of definably connected components
of quantifier free definable sets, then K is o-minimal. We
further generalize the above result, along the line of Speissegger’s
theorem, and prove the o-minimality of the relative Pfaffian closure
of an o-minimal structure inside a definably complete Baire
structure
Dimensions, matroids, and dense pairs of first-order structures
No full textAbstractA structure M is pregeometric if the algebraic closure is a pregeometry in all structures elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of Lascar U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding an integral domain, while not pregeometric in general, do have a unique existential matroid.Generalising previous results by van den Dries, we define dense elementary pairs of structures expanding an integral domain and with an existential matroid, and we show that the corresponding theories have natural completions, whose models also have a unique existential matroid. We also extend the above result to dense tuples of structures
