Mathematics

We survey discrete and continuous model-theoretic notions which have important connections to general topology. We present a self-contained exposition of several interactions between continuous logic and C p -theory which have applications to a classification problem involving Banach spaces not including c 0 or l p , following recent results obtained by P. Casazza and J. Iovino for compact continuous logics. Using C p -theoretic results involving Grothendieck spaces and double limit conditions, we extend their results to a broader family of logics, namely those with a first countable weakly Grothendieck space of types. We pose C p -theoretic problems which have model-theoretic implications.

We consider general structures where formulas have truth values in the real unit interval as in continuous model theory, but whose predicates and functions need not be uniformly continuous with respect to a distance predicate. Every general structure can be expanded to a pre-metric structure by adding a distance predicate that is a uniform limit of formulas. Moreover, that distance predicate is unique up to uniform equivalence. We use this to extend the central notions in the model theory of metric structures to general structures, and show that many model-theoretic results from the literature about metric structures have natural analogues for general structures.

This paper is a survey on model theory of adeles and applications to model theory, algebra, and number theory. Sections 1-12 concern model theory of adeles and the results are joint works with Angus Macintyre. The topics covered include quantifier elimination in enriched Boolean algebras, quantifier elimination in restricted products and in adeles and adele spaces of algebraic varieties in natural languages, definable subsets of adeles and their measures, solution to a problem of Ax from 1968 on decidability of the rings Z/mZ for all m>1 , definable sets of minimal idempotents (or "primes of the number field" ) in the adeles, stability-theoretic notions of stable embedding and tree property of the second kind, elementary equivalence and isomorphism for adele rings, axioms for rings elementarily equivalent to restricted products and for the adeles, converse to Feferman-Vaught theorems, a language for adeles relevant for Hilbert symbols in number theory, imaginaries in adeles, and the space adele classes. Sections 13-18 are concerned with connections to number theory around zeta integrals and L -functions. Inspired by our model theory of adeles, I propose a model-theoretic approach to automorphic forms on G L 1 (Tate's thesis) and G L 2 (work of Jacquet-Langlands), and formulate several notions, problems and questions. The main idea is to formulate notions of constructible adelic integrals and observe that the integrals of Tate and Jacquet-Langlands are constructible. These constructible integrals are related to the p -adic and motivic integrals in model theory.

Let T be the differential field of logarithmic-exponential transseries. We show that the expansion of T by its natural exponential function is model complete and locally o-minimal. We give an axiomatization of the theory of this expansion that is effective relative to the theory of the real exponential field. We adapt our results to show that the expansion of T by this exponential function and by its natural restricted sine and restricted cosine functions is also model complete and locally o-minimal.

Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to admit a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion implies that the variety has equationally definable principal congruences. This result is then used to provide necessary and sufficient conditions for the existence of a model completion for theories of Hamiltonian varieties of pointed residuated lattices, a broad family of varieties that includes lattice-ordered abelian groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of pointed residuated lattices admits a model completion, it must have equationally definable principal congruences. In particular, the theories of lattice-ordered abelian groups and MV-algebras do not have a model completion, as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is shown that certain varieties of pointed residuated lattices generated by their linearly ordered members, including lattice-ordered abelian groups and MV-algebras, can be extended with a binary operation in order to obtain theories that do have a model completion.

We show that the theory of Galois actions of a torsion Abelian group A is companionable if and only if for each prime p , the p -primary part of A is either finite or it coincides with the Prüfer p -group. We also provide a model-theoretic description of the model companions we obtain.

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse-Schmidt derivations and about Galois actions. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

In this paper we study the class of modules with fusion and implication based over distributive lattices, or FIDL-modules, for short. We introduce the concepts of FIDL-subalgebra and FIDL-congruence as well as the notions of simple and subdirectly irreducible FIDL-modules. We give a bi-sorted Priestley-like duality for FIDL-modules and moreover, as an application of such a duality, we provide a topological bi-spaced description of the FIDL-congruences. This result will allows us to characterize the simple and subdirectly irreducible FIDL-modules.

By combining classical results of Büchi, some elementary Tauberian theorems and some basic tools from logic and combinatorics we show that every ordinal α with ε 0 ≥α≥ ω ω satisfies a natural monadic second order limit law and that every ordinal α with ω ω >α≥ω satisfies a natural monadic second order Cesaro limit law. In both cases we identify as usual α with the class of substructures {β:β<α} . We work in an additive setting where the norm function N assigns to every ordinal α the number of occurrrences of the symbol ω in its Cantor normal form. This number is the same as the number of edges in the tree which is canonically associated with α . For a given α with ω≤α≤ ε 0 the asymptotic probability of a monadic second order formula φ from the language of linear orders is lim n→∞ #{β<α:Nβ=n∧β⊨Φ} #{β<α:Nβ=n} if this limit exists. If this limit exists only in the Cesaro sense we speak of the Cesaro asympotic probability of φ . Moreover we prove monadic second order limit laws for the ordinal segments below below Γ 0 (where the norm function is extended appropriately) and we indicate how this paper's results can be extended to larger ordinal segments and even to certain impredicative ordinal notation systems having notations for uncountable ordinals. We also briefly indicate how to prove the corresponding multiplicative results for which the setting is defined relative to the Matula coding. The results of this paper concerning ordinals not exceeding ε 0 have been obtained partly in joint work with Alan R. Woods.

In this paper we analyse some questions concerning trees on κ , both for the countable and the uncountable case, and the connections with Cohen reals. In particular, we provide a proof for one of the implications left open in \cite[Question 5.2]{FKK16} about the diagram for regularity properties.