Searcharxiv⌕ Search

arXiv subjects

Rosario Mennuni

Publications and source records attributed to Rosario Mennuni.

15 recordsLinked to original sources
arXiv2026

The Amalgamation Property for automorphisms of ordered abelian groups

We prove that the category of ordered abelian groups equipped with an automorphism has the Amalgamation Property, deduce that their inductive theory is NIP in the sense of positive logic, and initiate a development of the latter framework. As byproducts of the proof, we obtain a generalised version of the Hahn Embedding Theorem which allows to lift each automorphism of an ordered abelian group to one of an ordered real vector space, and we show that, on existentially closed structures, linear combinations of iterates of the automorphism have the Intermediate Value Property.

math.LO↗
arXiv2026

Monochromatic sums and quotients in $\mathbb N$

We prove partition regularity of the configuration $x,y,x+y,y/x$ in a strong infinitary form that extends Hindman's Theorem. We study the related issue of partition regularity of configurations involving products of a degree one polynomial in $x$ with one in $y$, reducing the general problem to a handful of special cases.

math.LO↗
arXiv2026

Ramsey's witnesses

We introduce the notion of Ramsey partition regularity, a generalisation of partition regularity involving infinitary configurations. We provide characterisations of this notion in terms of certain ultrafilters related to tensor products and dubbed Ramsey's witnesses; and we also consider their nonstandard counterparts as pairs of hypernatural numbers, called Ramsey pairs. These characterisations are then used to determine whether various configurations involving polynomials and exponentials are Ramsey partition regular over the natural numbers. In particular, this provides negative answers to several questions recently posed by Kra, Moreira, Richter and Robertson.

math.LO↗
arXiv2025

Automorphisms of valued fields: amalgamation and existential closedness

We study valued fields equipped with an automorphism. We prove that all of them have an extension admitting an equivariant cross-section of the valuation. In residual characteristic zero, and in the presence of such a cross-section, we show that amalgamation problems are solvable precisely when the induced residual problem is, characterise the existentially closed objects of this category, and prove that its positive theory does not have the tree property of the second kind. We prove analogous results with cross-sections replaced by angular components. Along the way, we show that array modelling does not require thickness.

math.LO↗
arXiv2025

Extending orders to types

Given an ordered structure, we study a natural way to extend the order to preorders on type spaces. For definably complete, linearly ordered structures, we give a characterisation of the preorder on the space of 1-types. We apply these results to the divisibility preorder on the space of ultrafilters on the set of natural numbers, giving an independence result about the suborder consisting of ultrafilters with only one fixed prime divisor, as well as a classification of ultrafilters with finitely many prime divisors.

math.LO↗
arXiv2025

Orthogonality and domination in o-minimal expansions of ordered groups

We analyse domination between invariant types in o-minimal expansions of ordered groups, showing that the domination poset decomposes as the direct product of two posets: the domination poset of an o-minimal expansion of a real closed field, and one derived from a linear o-minimal structure. We prove that if the Morley product is well-defined on the former poset, then the same holds for the poset computed in the whole structure. We establish our results by employing the `short closure' pregeometry ($\mathrm{scl}$) in semi-bounded o-minimal structures, showing that types of $\mathrm{scl}$-independent tuples are weakly orthogonal to types of short tuples. As an application we prove that, in an o-minimal expansion of an ordered group, every definable type is domination-equivalent to a product of 1-types. Furthermore, there are precisely two or four classes of definable types up to domination-equivalence, depending on whether a global field is definable or not.

math.LO↗
arXiv2023

The domination monoid in henselian valued fields

We study the domination monoid in various classes of structures arising from the model theory of henselian valuations, including RV-expansions of henselian valued fields of residue characteristic 0 (and, more generally, of benign valued fields), p-adically closed fields, monotone D-henselian differential valued fields with many constants, regular ordered abelian groups, and pure short exact sequences of abelian structures. We obtain Ax-Kochen-Ershov type reductions to suitable fully embedded families of sorts in quite general settings, and full computations in concrete ones.

math.LO↗
arXiv2023

Vector spaces with a union of independent subspaces

Motivated by the theory of locally definable groups, we study the theory of $K$-vector spaces with a predicate for the union $X$ of an infinite family of independent subspaces. We show that if $K$ is infinite then the theory is complete and admits quantifier elimination in the language of $K$-vector spaces with predicates for the $n$-fold sums of $X$ with itself. If $K$ is finite this is no longer true, but we still have that a natural completion is near-model-complete.

math.LO↗
arXiv2023

Self-divisible ultrafilters and congruences in $β\mathbb{Z}$

We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation $\equiv_w$ introduced by Šobot is an equivalence relation on $β\mathbb{Z}$. We provide several examples and additional characterisations; notably we show that $w$ is self-divisible if and only if $\equiv_w$ coincides with the strong congruence relation $\equiv^{\mathrm{s}}_{w}$, if and only if the quotient $(β\mathbb{Z},\oplus)/\mathord{\equiv^{\mathrm{s}}_w}$ is a profinite group. We also construct an ultrafilter $w$ such that $\equiv_w$ fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion $\hat{\mathbb{Z}}$ of the integers.

math.LO↗
arXiv2022

Model-theoretic dividing lines via posets

We show that for each property $\mathsf{P}\in \{\mathsf{OP}, \mathsf{IP}, \mathsf{TP}_1, \mathsf{TP}_2, \mathsf{ATP}, \mathsf{SOP}_3\}$ there is a poset $Σ_{\mathsf{P}}$ such that a theory has property $\mathsf{P}$ if and only if some model interprets a poset in which $Σ_{\mathsf{P}}$ can be embedded. We also introduce a new property $\mathsf{SUP}$, consistent with $\mathsf{NIP}_2$ and implying $\mathsf{ATP}$ and $\mathsf{SOP}$.

math.LO↗
arXiv2020

The domination monoid in o-minimal theories

We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes of 1-types. We show this to hold in Real Closed Fields, where generators of this monoid correspond to invariant convex subrings of the monster model. Combined with arxiv:1702.06504, this allows us to compute the domination monoid in the weakly o-minimal theory of Real Closed Valued Fields.

math.LO↗
arXiv2020

Weakly binary expansions of dense meet-trees

We compute the domination monoid in the theory DMT of dense meet-trees. In order to show that this monoid is well-defined, we prove weak binarity of DMT and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory DTR from arXiv:1909.04626. We then describe the domination monoids of such expansions in terms of those of the expanding relations.

math.LO↗
arXiv2019

On double-membership graphs of models of Anti-Foundation

We answer some questions about graphs which are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.

math.LO↗
arXiv2019

Product of Invariant Types Modulo Domination-Equivalence

We investigate the interaction between the product of invariant types and domination-equivalence. We present a theory where the latter is not a congruence with respect to the former, provide sufficient conditions for it to be, and study the resulting quotient when it is.

math.LO↗