Research keywords: real algebraic geometry, model theory, spectral spaces, hyperstructures, lattice-ordered Abelian groups, real closed rings, C∞-rings.
Preprints
Cross-sections of divisible abelian o-groups via tame pairs. PDF. ArXiv.
(to be submitted)
It is shown that images of cross-sections of surjective morphisms f: Γ ↠ Δ of divisible abelian o-groups are exactly divisible, tame (equivalently, relative Dedekind complete) and cofinal subgroups of Γ compatible with f in a suitable sense. The note concludes with an application to real closed valued fields.
Model theory of local real closed SV-rings of finite rank. PDF. ArXiv.
(to be submitted)
This note begins the model-theoretic study of local real closed SV-rings of finite rank; to this end, a structure theorem for reduced local SV-rings of finite rank is given and branching ideals in local real closed rings of finite rank are analysed. The class of local real closed SV-rings of rank n ∈ ℕ≥2 is elementary in the language of rings ℒ := {+,−,·,0,1} and its ℒ-theory Tn has a model companion Tn,1; models of Tn,1 are n-fold fibre products (((V₁× k V₂)× k V₃) ...× k Vn−1)× k Vn of non-trivial real closed valuation rings Vi with isomorphic residue field k . The ℒ-theory Tn,1 is complete, decidable, and NIP. After enriching ℒ with a predicate for the maximal ideal, models of Tn have prime extensions in models of Tn,1, and Tn,1 is the model completion of Tn in this enriched language. A quantifier elimination result for Tn,1 is also given. The class of those local real closed SV-rings of rank n ∈ ℕ≥2 which are n-fold fibre products (((V₁×WV)×WV₃) ...×WVn-1)×WVn of non-trivial real closed valuation rings Vi along surjective morphisms Vi↠W onto a non-trivial domain W is elementary in the language of rings, and its ℒ-theory Tn,2 is also complete, decidable, and NIP; after enriching ℒ with predicates for the maximal ideal and the unique branching ideal, Tn,2 is model complete.
C∞-rings and real closed rings are two distinct classes of rings on which a set of functions
ℝn → ℝ (n ∈ {0, 1, 2, ...}) operates: smooth and continuous ℤ-semi-algebraic,
respectively. The former were introduced to construct topos-theoretic models for synthetic
differential geometry, while the latter were designed to serve as rings of global sections of
certain analogues of Grothendieck's affine schemes in real algebraic geometry. In this talk I
will introduce and compare these two classes of rings, highlighting along the way various open
research directions. I will also report on recent developments on the algebra of C∞-rings emanating from novel connections with (super) real closed rings in joint work with J. Cerqueira Berni (UFABC), H. L. Mariano (IME-USP) and M. Tressl (University of Manchester, UK).
Local real closed SV-rings of finite rank and their model theory (at OAL 2025). PDF.
A commutative unital ring is an SV-ring if all its residue domains (that is, all its quotients by prime ideals) are valuation rings. SV-rings were first introduced in the context of rings of continuous functions C(X), and later studied in the more general context of f-rings. Among all f-rings, Schwartz’s real closed rings form a subclass which play a central role in semi-algebraic geometry; examples of such rings are the rings C(X) and rings of continuous semi-algebraic functions on a semi-algebraic set over a real closed field. Residue domains of real closed SV-rings are real closed valuation rings (equivalently, convex subrings of real closed fields), and their model theory is well understood. The goal of this talk is to present an approach to the model-theoretic analysis of some classes of local real closed SV-rings in terms of their residue domains.
A local real closed SV-ring has rank n if it has exactly n minimal prime ideals, and the class of local real closed SV-rings of rank n is elementary in the language of rings; a geometric example of such a ring is the ring of germs G of continuous semi-algebraic functions at a point in a semi-algebraic curve with exactly n half-branches. First, I will spell out a representation theorem for local real closed SV-rings of rank n, which describes them as iterated fibre products of n non-trivial real closed valuation rings (that is, those real closed valuation rings which are not fields). I will then focus on the class of n-fold fibre products of non-trivial real closed valuation rings with isomorphic residue field along their residue field map; an example of such a ring is the ring of germs G. I will sketch proofs of the main model-theoretic properties of this class, in particular:
1. It has a recursive axiomatization in the language of rings and its theory is complete (hence
decidable); and
2. Its theory is model complete and it is the model companion of the theory of local real closed (SV-) rings of rank n.
The same results as in 1) hold for another class of n-fold fibre products of non-trivial real closed
valuation rings. As a consequence, the theory of local real closed SV-rings of rank 2 has exactly two
completions, both of which are decidable. I will conclude by showing how these results suggest an
elementary classification of all local real closed SV-rings of finite rank in terms of the isomorphism type
of their poset of branching prime ideals.
Relative quantifier elimination for the lattice-ordered module of continuous semi-algebraic functions on a curve (at OAL-RAG 2024). PDF.
In the late 1980s, Shen and Weispfenning proved, via relative quantifier elimination in a suitable 2-sorted language, that under a mild condition on a divisible abelian lattice-ordered group G of functions, the theory of G is completely determined by the theory of its lattice of zero sets. In this talk I will give the relevant context and details of their result, to then explain how the ideas in their proof can be adapted to lattice-ordered modules M of continuous semi-algebraic functions on a curve by enriching their 2-sorted language with a new sort for a real closed valuation ring; as a consequence of the method, decidability of M is obtained whenever the base field is a recursive real closed field.
