Searcharxiv⌕ Search

arXiv subjects

Pietro Freni

Publications and source records attributed to Pietro Freni.

7 recordsLinked to original sources
arXiv2025

Absorbed Types and Derivations in Exponential o-Minimal Theories

I analyze $\mathcal{O}$-weakly immediate and $\mathcal{O}$-residual types in an o-minimal expansion of an ordered field $\mathbb{E}$, where $\mathcal{O}$ is a convex valuation ring. The main result is a characterization of those exponential theories $T$ such that for all $(\mathbb{E}, \mathcal{O})\models T_{\mathrm{convex}}$ the image of any $\mathcal{O}$-weakly immediate type is given by some composition of translations, sign changes and exponential, of some \emph{possibly different} $\mathcal{O}$-weakly immediate type. I call these theories \emph{transserial} and they encompass simply exponential theories such as $T_{\exp}$ and $T_{an, \exp}$. A consequence of the analysis is that there are no counterexamples to \emph{Tressl's signature-alternative} (cf [15]) in models of transserial theories admitting an Archimedean prime model. The characterization has at its core some arguments that use very few but fundamental properties of the valued differential field of germs at a cut in an o-minimal structure. These are abstracted in some conditions of compatibility between the derivation and the order or the derivation and the valuation, both ultimately stemming from the mean-value-theorem in o-minimal structures. I develop some basic theory around these notions and observe that in the case of few constants (i.e.\ when the valuation ring contains the constants) these notions specialize to notions thoroughly studied in arXiv:1509.02588

math.LO↗
arXiv2025

Residually Constructible Extensions

Let $T$ be an o-minimal theory expanding $\mathrm{RCF}$ and $T_\mathrm{convex}$ be the common theory of its models expanded by predicate for a non-trivial $T$-convex valuation ring. We call an elementary extension $(\mathbb{E}, \mathcal{O}) \prec (\mathbb{E}_*, \mathcal{O}_*) \models T_{\mathrm{convex}}$ $\textit{res-constructible}$ if there is a tuple $\overline{s}$ in $\mathcal{O}_*$ such that $\mathbb{E}_* = \mathrm{dcl}(\mathbb{E},\overline{s})$, and the projection $\mathbf{res}(\overline{s})$ of $\overline{s}$ in the residue field sort is $\mathrm{dcl}$-independent over the residue field $\mathbf{res}(\mathbb{E}, \mathcal{O})$ of $(\mathbb{E}, \mathcal{O})$. We study factorization properties of res-constructible extensions. Our main result is that a res-constructible extension $(\mathbb{E}, \mathcal{O}) \prec (\mathbb{E}_*, \mathcal{O}_*)$ has the property that all $(\mathbb{E}_1, \mathcal{O}_1)$ with $(\mathbb{E}, \mathcal{O}) \prec (\mathbb{E}_1, \mathcal{O}_1) \prec (\mathbb{E}_*, \mathcal{O}_*)$ are res-constructible over $(\mathbb{E}, \mathcal{O})$, if and only if $\mathbb{E}_*$ has countable $\mathrm{dcl}$-dimension over $\mathbb{E}$ or the value group $\mathbf{val}(\mathbb{E}_*, \mathcal{O}_*)$ is $\textit{short}$ (i.e. contains no uncountable well-ordered subset). This analysis entails complete answers to [11, Problem 5.12].

math.LO↗
arXiv2024

Truncations in languages of generalized power series and the structure of $T$-$\lambda$-spherical completions of o-minimal fields

Let $T$ be the theory of an o-minimal field and $T_0$ a common reduct of $T$ and $T_{an}$. I adapt Mourgues' and Ressayre's constructions to deduce structure results for $T_0$-reducts of $T$-$\lambda$-spherical completion of models of $T_{\mathrm{convex}}$. These in particular entail that whenever $T$ is the theory of a reduct of $\mathbb{R}_{an,\exp}$ defining the exponentiation (e.g.\ $T=T_{\exp}$, the theory of the field of reals expanded by the exponential function), every model of $T$ has an initial elementary embedding in the field $\mathbf{No}$ of surreal numbers. This answers positively an open question in (arXiv:2002.07739). The main technical result is that expanding an integral domain of generalized series in the sense of Hahn-Higman-Ribenboim (such as a Hahn field) by a family of generalized power series interpreted as functions defined on certain infinitesimal elements, has the property that truncation closed subsets generate truncation closed substructures, provided that the family of generalized power series is itself closed under truncations and partial derivatives. It is also shown that the further closure of the generated set under solutions to certain equations is as well closed under truncations. The formal results on power series leave room for possible generalizations to the case in which $T_0$ is power bounded but not necessarily a reduct of $T_{an}$.

math.LO↗
arXiv2024

$T$-convexity, Weakly Immediate Types, and $T$-$\lambda$-Spherical Completions of o-minimal Structures

It is well known that ordered exponential fields with a compatible non-trivial valuation cannot be spherically complete, but there are some that are ``complete enough''. This paper gives analogues of Kaplansky's theorem on maximally valued fields that hold for a suitable class of elementary extensions of some ordered exponential fields with a compatible valuation. More precisely it does so for models of any theory $T_{\text{convex}}$ given by the expansion of a fixed complete o-minimal theory of ordered fields $T$, by a predicate $\mathcal{O}$ for a non-trivial $T$-convex valuation ring. For $\lambda$ an uncountable cardinal, say that a unary type $p(x)$ over a model of $T_{\text{convex}}$ is \emph{$\lambda$-bounded weakly immediate} if its cut is defined by an empty intersection of fewer than $\lambda$ many nested valuation balls. Call an elementary extension \emph{$\lambda$-bounded wim-constructible} if it is obtained as a transfinite composition of extensions each generated by one element whose type is $\lambda$-bounded weakly immediate. I show that $\lambda$-bounded wim-constructible extensions do not extend the residue-field sort and that any two wim-constructible extensions can be amalgamated in an extension which is again $\lambda$-bounded wim-constructible over both. A consequence of this is that given an uncountable cardinal $\lambda$, every model of $T_{\text{convex}}$ has a unique-up-to-isomorphism $\lambda$-spherically complete $\lambda$-bounded wim-constructible extension providing an analogue of Kaplansky's theorem. I call this extension the $T$-$\lambda$-spherical completion. Another consequence is that $T_{\mathrm{convex}}$ is \emph{definably spherically complete}. When $T$ is power bounded wim-constructible extensions are just the immediate extensions. I discuss the example of power bounded theories expanded by $\exp$ (\emph{simply exponential} theories).

math.LO↗
arXiv2023

On Vector Spaces with Formal Infinite Sums

I discuss possible definitions of categories of vector spaces enriched with a notion of formal infinite linear combination in the likes of the formal infinite linear combinations one has in the context of generalized power series, I call these \emph{reasonable categories of strong vector spaces} (r.c.s.v.s.). I show that, in a precise sense, the more general possible definition for a strong vector space is that of a small $\mathrm{Vect}$-enriched endofunctor of $\mathrm{Vect}$ that is right orthogonal for every cardinal $\lambda$, to the cokernel of the canonical inclusion of the $\lambda$-th copower in the $\lambda$-th power of the identity functor: these form the objects for a universal r.c.s.v.s. I call $\Sigma\mathrm{Vect}$. I show this is equivalent to the category of \emph{ultrafinite summability spaces} defined independently in arXiv:2403.05827. I relate this category to what could be understood to be the obvious category of strong vector spaces $B\Sigma\mathrm{Vect}$ and to the r.c.s.v.s. $K\mathrm{TVect}_s$ of separated linearly topologized spaces that are generated by linearly compact spaces. I analyze the monoidal closed structures on various r.s.v.s. induced by the natural one on $\mathrm{Ind}(\mathrm{Vect}^{\mathrm{op}})$. In particular with respect to the problem of closure under the tensor product of $\mathrm{Ind}(\mathrm{Vect}^{\mathrm{op}})$. Most of the technical results apply to a more general class of orthogonal subcategories of $\mathrm{Ind}(\mathrm{Vect}^{\mathrm{op}})$ and we work with that generality as it's cost-free.

math.CT↗