Model Companion Research Articles

Model theory of fields with free operators in positive characteristic

We give algebraic conditions for a finite commutative algebra B B over a field of positive characteristic, which are equivalent to the companionability of the theory of fields with “ B B -operators” (i.e., the operators coming from homomorphisms into tensor products with B B ). We show that, in the most interesting case of a local B B , these model companions admit quantifier elimination in the “smallest possible” language, and they are strictly stable. We also describe the forking relation there.

Homogeneous universal $H$-fields

We consider derivations $\partial$ on Conway's field $\mathbf{No}$ of surreal numbers such that the ordered differential field $(\mathbf{No},\partial)$ has constant field $\mathbb{R}$ and is a model of the model companion of the theory of $H$-fields with small derivation. We show that this determines $(\mathbf{No},\partial)$ uniquely up to isomorphism, and that this structure is absolutely homogeneous universal for models of this theory with constant field $\mathbb{R}$.

Model theory of fields with virtually free group actions

For a group G, we define the notion of a G-kernel and show that the properties of G-kernels are closely related with the existence of a model companion of the theory of Galois actions of G. Using Bass–Serre theory, we show that this model companion exists for virtually free groups generalizing the existing results about free groups and finite groups. We show that the new theories we obtain are not simple and not even NTP 2 .

Correction to: The model companion of the class of pseudocomplemented semilattices is finitely axiomatizable

In the paper “The model companion of the class of pseudocomplemented semilattices is finitely axiomatizable” a list of nine axioms, which are claimed to axiomatize the class of existentially closed pseudocomplemented semilattices, is given. After its publication an error in a key lemma was found by the reviewer of a subsequent paper. The error was rectified by strengthening one of these nine axioms and then proving that the resulting nine axioms do axiomatize this class, thereby proving the statement of the title. Here we explain the error, outline the correction and refer the reader to arXiv.org for a full revision of the published paper.

Generic expansion and Skolemization in NSOP1 theories

We study expansions of NSOP1 theories that preserve NSOP1. We prove that if T is a model complete NSOP1 theory eliminating the quantifier ∃∞, then the generic expansion of T by arbitrary constant, function, and relation symbols is still NSOP1. We give a detailed analysis of the special case of the theory of the generic L-structure, the model companion of the empty theory in an arbitrary language L. Under the same hypotheses, we show that T may be generically expanded to an NSOP1 theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in NSOP1 theories, adding instances of algebraic independence to their conclusions.

Model-theoretic aspects of the Gurarij operator system

We establish some of the basic model theoretic facts about the Gurarij operator system GS recently constructed by the second-named author. In particular, we show: (1) GS is the unique separable 1-exact existentially closed operator system; (2) GS is the unique separable nuclear model of its theory; (3) every embedding of GS into its ultrapower is elementary; (4) GS is the prime model of its theory; and (5) GS does not have quantifier-elimination, whence the theory of operator systems does not have a model companion. We also show that, for any q ∈ N, the theories of M_q -spaces and M_q -systems do have a model companion, namely the Fraisse limit of the class of finite-dimensional M_q -spaces and M_q -systems respectively; moreover, we show that the model companion is separably categorical. We conclude the paper by showing that no C* algebra can be existentially closed as an operator system.

Embedded Picard–Vessiot extensions

ABSTRACTWe prove that if T is a theory of large, bounded, fields of characteristic 0 with almost quantifier elimination, and TD is the model companion of T∪{“∂ is a derivation”}, then for any model (𝒰,∂) of TD, differential subfield K of 𝒰 such that CK⊧T, and linear differential equation ∂Y = AY over K, there is a Picard-Vessiot extension L of K for the equation with K≤L≤𝒰, i.e. L can be embedded in 𝒰 over K, as a differential field. Moreover such L is unique to isomorphism over K as a differential field. Likewise for the analogue for strongly normal extensions for logarithmic differential equations in the sense of Kolchin.

Hilbert spaces with generic predicates

Estudiamos la teoría de modelos de expansiones de espacios de Hilbert mediante predicados genéricos. Primero demostramos la existencia de modelo-compañeras de expansiones genéricas de espacios de Hilbert mediante una función-distancia a una estructura aleatoria, y luego una distancia a un subconjunto aleatorio. La teoría obtenida con la subestructura aleatoria es ω-estable; la obtenida mediante la distancia a subconjunto aleatorio es TP2 y NSOP1. Este ejemplo es la primera estructura de esta clase de complejidad en lógica continua.

The countable existentially closed pseudocomplemented semilattice

As the class \(\mathcal {PCSL}\) of pseudocomplemented semilattices is a universal Horn class generated by a single finite structure it has a \(\aleph _0\)-categorical model companion \(\mathcal {PCSL}^*\). As \(\mathcal {PCSL}\) is inductive the models of \(\mathcal {PCSL}^*\) are exactly the existentially closed models of \(\mathcal {PCSL}\). We will construct the unique existentially closed countable model of \(\mathcal {PCSL}\) as a direct limit of algebraically closed pseudocomplemented semilattices.

A MODEL-THEORETIC CHARACTERIZATION OF MONADIC SECOND ORDER LOGIC ON INFINITE WORDS

AbstractMonadic second order logic and linear temporal logic are two logical formalisms that can be used to describe classes of infinite words, i.e., first-order models based on the natural numbers with order, successor, and finitely many unary predicate symbols.Monadic second order logic over infinite words (S1S) can alternatively be described as a first-order logic interpreted in${\cal P}\left( \omega \right)$, the power set Boolean algebra of the natural numbers, equipped with modal operators for ‘initial’, ‘next’, and ‘future’ states. We prove that the first-order theory of this structure is the model companion of a class of algebras corresponding to a version of linear temporal logic (LTL) without until.The proof makes crucial use of two classical, nontrivial results from the literature, namely the completeness of LTL with respect to the natural numbers, and the correspondence between S1S-formulas and Büchi automata.

Coherency, free inverse monoids and related free algebras

AbstractA monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is the non-additive notion corresponding to that for a ring R stating that every finitely generated submodule of every finitely presented right R-module is finitely presented. For monoids (and rings) right coherency is an important finitary property which determines, amongst other things, the existence of a model companion of the class of right S-acts (right R-modules) and hence that the class of existentially closed right S-acts (right R-modules) is axiomatisable.Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruškuc, have shown that (free) groups, free commutative monoids and free monoids have the same properties. It is then natural to ask whether other free algebras in varieties of monoids, possibly with an augmented signature, are right coherent. We demonstrate that free inverse monoids are not.Munn described the free inverse monoid FIM(Ω) on Ω as consisting of birooted finite connected subgraphs of the Cayley graph of the free group on Ω. Sitting within FIM(Ω) we have free algebras in other varieties and quasi-varieties, in particular the free left ample monoid FLA(Ω) and the free ample monoid FAM(Ω). The former is the free algebra in the variety of unary monoids corresponding to partial maps with distinguished domain; the latter is the two-sided dual. For example, FLA(Ω) is obtained from FIM(Ω) by considering only subgraphs with vertices labelled by elements of the free monoid on Ω.The main objective of the paper is to show that FLA(Ω) is right coherent. Furthermore, by making use of the same techniques we show that FIM(Ω), FLA(Ω) and FAM(Ω) satisfy (R), (r), (L) and (l), related conditions arising from the axiomatisability of certain classes of right S-acts and of left S-acts.

THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS

AbstractA model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Journal of Mathematical Logic 14(2), 2014]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel’s Lemma for local finite algebras over differentially closed fields.

Existentially closed fields withG-derivations

We prove that the theories of fields with Hasse–Schmidt derivations corresponding to actions of formal groups admit model companions. We also give geometric axiomatizations of these model companions.

Canonical and existential groups in universal classes of Abelian groups

Universal classes of Abelian groups are classified in terms of sets of finitely generated groups closed with respect to the discrimination operator. The notions of a principal universal class and a canonical group for such a class are introduced. For any universal class K, the class K ec of existentially closed groups generated by the universal theory of K is described. It is proved that K ec is axiomatizable and, therefore, the universal theory of K has a model companion.

On the model companion of partial differential fields with an automorphism

We prove that the class of partial differential fields of characteristic zero with an automorphism has a model companion. We then establish the basic model theoretic properties of this theory and prove that it satisfies the canonical base property, and consequently the Zilber dichotomy, for finite-dimensional types.

Model companions of theories of graphs

We study model companions of theories extending the graph axioms. First we prove general results concerning the existence of the model companion. Then, by applying these results to the case of graphs, we give a series of companionable and non‐companionable examples.

Model companion of ordered theories with an automorphism

Kikyo and Shelah showed that if T T is a theory with the Strict Order Property in some first-order language L \mathcal {L} , then in the expanded language L σ := L ∪ { σ } \mathcal {L}_\sigma := \mathcal {L}\cup \{\sigma \} with a new unary function symbol σ \sigma , the bigger theory T σ := T ∪ { “ σ is an L -automorphism } T_\sigma := T\cup \{“\sigma \mbox { is an } \mathcal {L}\mbox {-automorphism}\} does not have a model companion. We show in this paper that if, however, we restrict the automorphism and consider the theory T σ T_\sigma as the base theory T T together with a “restricted” class of automorphisms, then T σ T_\sigma can have a model companion in L σ \mathcal {L}_\sigma . We show this in the context of linear orders and ordered abelian groups.

Model theory of fields with free operators in characteristic zero

Generalizing and unifying the known theorems for difference and differential fields, it is shown that for every finite free algebra scheme 𝒟 over a field A of characteristic zero, the theory of 𝒟-fields has a model companion 𝒟-CF0 which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.

MODEL THEORETIC PROPERTIES OF METRIC VALUED FIELDS

We study model theoretic properties of valued fields (equipped with a real-valued multiplicative valuation), viewed as metric structures in continuous first order logic.For technical reasons we prefer to consider not the valued field (K, |·|) directly, but rather the associated projective spaces KPn, as bounded metric structures.We show that the class of (projective spaces over) metric valued fields is elementary, with theory MVF, and that the projective spaces Pn and are Pm biinterpretable for every n, m ≥ 1. The theory MVF admits a model completion ACMVF, the theory of algebraically closed metric valued fields (with a nontrivial valuation). This theory is strictly stable (even up to perturbation).Similarly, we show that the theory of real closed metric valued fields, RCMVF, is the model companion of the theory of formally real metric valued fields, and that it is dependent.

The model companion of the class of pseudocomplemented semilattices is finitely axiomatizable

It is shown that the class $${\mathcal{PCSL}^{ec}}$$ of existentially closed pseudocomplemented semilattices is finitely axiomatizable by appropriately extending a finite axiomatization of the class $${\mathcal{PCSL}^{ac}}$$ of algebraically closed pseudocomplemented semilattices. Because $${\mathcal{PCSL}^{ec}}$$ coincides with the model companion of the class $${\mathcal{PCSL}}$$ of pseudocomplemented semilattices, this answers the question asked by Albert and Burris in a paper in 1986: “Does the class of pseudocomplemented semilattices have a finitely axiomatizable model companion?"