Model Companion Research Articles

Model companion properties of some theories

Model companion properties of some theories

Preservation of NATP

We prove the preservation theorems for NATP; many of them extend the previously established preservation results for other model-theoretic tree properties. Using them, we also furnish proper examples of NATP theories which are simultaneously TP2 and SOP. First, we show that NATP is preserved by the parametrization and sum of the theories of Fraïssé limits of Fraïssé classes satisfying strong amalgamation property. Second, the preservation of NATP for two kinds of dense/co-dense expansions, i.e. the theories of lovely pairs and of [Formula: see text]-structures for geometric theories, and dense/co-dense expansion on vector spaces is proved. Next, we prove the preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also the preservation of NTP1 and NTP2 is considered. Finally, we present some proper examples of NATP theories using the results proved in this paper, including the parametrization of DLO and the expansion of an algebraically closed field by adding a generic linear order. In particular, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP.

Galois actions of finitely generated groups rarely have model companions

AbstractWe show that if is a finitely generated group such that its profinite completion is “far from being projective” (i.e., the kernel of the universal Frattini cover of is not a small profinite group), then the class of existentially closed ‐actions on fields is not elementary. Since any infinite, finitely generated, virtually free, and not free group is “far from being projective,” the main result of this paper corrects an error in our paper, Beyarslan and Kowalski (Proc. London Math. Soc., (2) 118 (2019), 221–256), by showing the negation of Theorem 3.26 in that paper.

Valued fields with a total residue map

When [Formula: see text] is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in Model Theory of Algebra and Arithmetic, Proceedings Karpacz, Poland, Lecture Notes in Mathematics, Vol. 834 (Springer, Berlin, 1979)] observed that the total residue map [Formula: see text], which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for [Formula: see text]. Driven by this observation, we study the theory [Formula: see text] of valued fields equipped with a linear form [Formula: see text] which restricts to the residue map on the valuation ring. We prove that [Formula: see text] does not admit a model companion. In addition, we show that [Formula: see text] is undecidable whenever [Formula: see text] is an infinite field. As a consequence, we get that [Formula: see text] is undecidable, where [Formula: see text] maps [Formula: see text] to its complex residue at [Formula: see text].

Companionability characterization for the expansion of an o-minimal theory by a dense subgroup

This paper provides a full characterization for when the expansion of a complete o-minimal theory, one that extends the theory of ordered divisible abelian groups, by a unary predicate that picks out a divisible, dense and codense group has a model companion. This result is motivated by criteria and questions introduced in the recent works [14] and [10] concerning the existence of model companions, as well as preservation results for some neostability properties when passing to the model companion. Examples are included both in which the predicate is an additive subgroup of a real ordered vector space, and where it is a multiplicative subgroup of the nonzero elements of an o-minimal expansion of a real closed field. The paper concludes with a brief discussion of neostability properties and examples that illustrate the lack of preservation (from the base o-minimal theory to the model companion of the expansion we define) for properties such as strong, NIP, and NTP2, though there are also examples for which some or all three of those properties are preserved.

An AEC framework for fields with commuting automorphisms

In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields with commuting automorphisms are closely related to difference fields. Some authors define a difference ring (or field) as a ring (or field) together with several commuting endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, the model companion of difference fields with one automorphism. Our fields with commuting automorphisms generalize this setting. We have several automorphisms and they are required to commute. Hrushovski has proved that in the case of fields with two or more commuting automorphisms, the existentially closed models do not necessarily form a first order model class. In the present paper, we introduce FCA-classes, an AEC framework for studying the existentially closed models of the theory of fields with commuting automorphisms. We prove that an FCA-class has AP and JEP and thus a monster model, that Galois types coincide with existential types in existentially closed models, that the class is homogeneous, and that there is a version of type amalgamation theorem that allows to combine three types under certain conditions. Finally, we use these results to show that our monster model is a simple homogeneous structure in the sense of S. Buechler and O. Lessman (this is a non-elementary analogue for the classification theoretic notion of a simple first order theory).

NICMOS Kernel-phase Interferometry. II. Demographics of Nearby Brown Dwarfs

Star formation theories have struggled to reproduce binary brown dwarf population demographics (e.g., frequency, separation, and mass ratio). Kernel-phase interferometry is sensitive to companions at separations inaccessible to classical imaging, enabling tests of these theories at new physical scales below the hydrogen burning limit. We analyze the detections and sensitivity limits from our previous kernel-phase analysis of archival HST/NICMOS surveys of field brown dwarfs. After estimating physical properties of the 105 late-M to T dwarfs using Gaia distances and evolutionary models, we use a Bayesian framework to compare these results to a model companion population defined by log-normal separation and power-law mass-ratio distributions. When correcting for Malmquist bias, we find a companion fraction of and a separation distribution centered at au, smaller and tighter than seen in previous studies. We also find a mass-ratio power-law index that strongly favors equal-mass systems: depending on the assumed age of the field population (0.9–3.1 Gyr). We attribute the change in values to our use of kernel-phase interferometry, which enables us to resolve the peak of the semimajor axis distribution with significant sensitivity to low-mass companions. We confirm the previously seen trends of decreasing binary fraction with decreasing mass and a strong preference for tight and equal-mass systems in the field-age substellar regime; only % of systems are wider than 20 au and % of systems have a mass ratio q < 0.6. We attribute this to turbulent fragmentation setting the initial conditions followed by a brief period of dynamical evolution, removing the widest and lowest-mass companions, before the birth cluster dissolves.

MODEL THEORY OF GALOIS ACTIONS OF TORSION ABELIAN GROUPS

AbstractWe 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.

Second order arithmetic as the model companion of set theory

Second order arithmetic as the model companion of set theory

Models of Bounded Arithmetic Theories and Some Related Complexity Questions

In this paper, we study bounded versions of some model-theoretic notions and results. We apply these results to the context of models of bounded arithmetic theories as well as some related complexity questions. As an example, we show that if the theory \(\rm S_2 ^1(PV)\) has bounded model companion then \(\rm NP=coNP\). We also study bounded versions of some other related notions such as Stone topology.

Connection between the amalgam and joint embedding properties

The paper aims to study the model-theoretic properties of differentially closed fields of zero and positive characteristics in framework of study of Jonsson theories. The main attention is paid to the amalgam and joint embedding properties of DCF theory as specific features of Jonsson theories, namely, the implication of JEP from AP. The necessity is identified and justified by importance of information about the mentioned properties for certain theories to obtain their detailed model-theoretic description. At the same time, the current apparatus for studying incomplete theories (Jonsson theories are generally incomplete) is not sufficiently developed. The following results have been obtained: The subclasses of Jonsson theories are determined from the point of view of joint embedding and amalgam properties. Within the exploration of one of these classes, namely the AP-theories, that the theories of differential and differentially closed fields of characteristic 0, differentially perfect and differentially closed fields of fixed positive characteristic are shown to be Jonsson and perfect. Along with this, the theory of differential fields of positive characteristic is considered as an example of an AP-theory that is not Jonsson, but has the model companion, which is perfect Jonsson theory, and the sufficient condition for the theory of differential fields is formulated in the context of being Jonsson.

MODEL THEORY OF FIELDS WITH FINITE GROUP SCHEME ACTIONS

AbstractWe 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 [13] and about Galois actions [14]. 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.

MODEL THEORY OF DERIVATIONS OF THE FROBENIUS MAP REVISITED

AbstractWe prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. Along the way we also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators.

New mid-infrared imaging constraints on companions and protoplanetary disks around six young stars

Context. Mid-infrared (mid-IR) imaging traces the sub-micron and micron-sized dust grains in protoplanetary disks and it offers constraints on the geometrical properties of the disks and potential companions, particularly if those companions have circumplanetary disks. Aims. We use the VISIR instrument and its upgrade NEAR on the VLT to take new mid-IR images of five (pre-)transition disks and one circumstellar disk with proposed planets and obtain the deepest resolved mid-IR observations to date in order to put new constraints on the sizes of the emitting regions of the disks and the presence of possible companions. Methods. We derotated and stacked the data to find the disk properties. Where available, we compare the data to PRODIMO (Protoplanetary Disk Model) radiation thermo-chemical models to achieve a deeper understanding of the underlying physical processes within the disks. We applied the circularised point spread function subtraction method to find upper limits on the fluxes of possible companions and model companions with circumplanetary disks. Results. We resolved three of the six disks and calculated position angles, inclinations, and (upper limits to) sizes of emission regions in the disks, improving upper limits on two of the unresolved disks. In all cases the majority of the mid-IR emission comes from small inner disks or the hot inner rims of outer disks. We refined the existing PRODIMO HD 100546 model spectral energy distribution (SED) fit in the mid-IR by increasing the PAH abundance relative to the ISM, adopting coronene as the representative PAH, and increasing the outer cavity radius to 22.3 AU. We produced flux estimates for putative planetary-mass companions and circumplanetary disks, ruling out the presence of planetary-mass companions with L &gt; 0.0028 L⊙ for a &gt; 180 AU in the HD 100546 system. Upper limits of 0.5–30 mJy are obtained at 8–12 μm for potential companions in the different disks. We rule out companions with L &gt; 10−2 L⊙ for a &gt; 60 AU in TW Hydra, a &gt; 110 AU in HD 169142, a &gt; 150 AU in HD 163296, and a &gt; 160 AU in HD 36112. Conclusions. The mid-IR emission comes from the central regions and traces the inner areas of the disks, including inner disks and inner rims of outer disks. Planets with mid-IR luminosities corresponding to a runaway accretion phase can be excluded from the HD 100546, HD 169142, TW Hydra, and HD 36112 systems at separations &gt;1′′. We calculated an upper limit to the occurrence rate of wide-orbit massive planets with circumplanetary disks of 6.2% (68% confidence). Future observations with METIS on the ELT will be able to achieve a factor of 10 better sensitivity with a factor of five better spatial resolution. MIRI on JWST will be able to achieve 250 times better sensitivity. Both will possibly detect the known companions to all six targets.

Permutation groups with small orbit growth

Abstract Let 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} be the class of all structures 𝔄 {\mathfrak{A}} such that the automorphism group of 𝔄 {\mathfrak{A}} has at most c ⁢ n d ⁢ n {cn^{dn}} orbits in its componentwise action on the set of n-tuples with pairwise distinct entries, for some constants c , d {c,d} with d &lt; 1 {d&lt;1} . We show that 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} is precisely the class of finite covers of first-order reducts of unary structures, and also that 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} . We also show that Thomas’ conjecture holds for 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} : all structures in 𝒦 exp + {\mathcal{K}_{{\operatorname{exp}}{+}}} have finitely many first-order reducts up to first-order interdefinability.

Generic expansions by a reduct

Consider the expansion [Formula: see text] of a theory [Formula: see text] by a predicate for a submodel of a reduct [Formula: see text] of [Formula: see text]. We present a setup in which this expansion admits a model companion [Formula: see text]. We show that some of the nice features of the theory [Formula: see text] transfer to [Formula: see text]. In particular, we study conditions for which this expansion preserves the [Formula: see text]-ness, the simplicity or the stability of the starting theory [Formula: see text]. We give concrete examples of new [Formula: see text] not simple theories obtained by this process, among them the expansion of a perfect [Formula: see text]-free [Formula: see text] field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of any characteristic by a generic multiplicative subgroup.

Model theory of R-trees

We show the theory of pointed $\R$-trees with radius at most $r$ is axiomatizable in a suitable continuous signature. We identify the model companion $\rbRT_r$ of this theory and study its properties. In particular, the model companion is complete and has quantifier elimination; it is stable but not superstable. We identify its independence relation and find built-in canonical bases for non-algebraic types. Among the models of $\rbRT_r$ are $\R$-trees that arise naturally in geometric group theory. In every infinite cardinal, we construct the maximum possible number of pairwise non-isomorphic models of $\rbRT_r$; indeed, the models we construct are pairwise non-homeomorphic. We give detailed information about the type spaces of $\rbRT_r$. Among other things, we show that the space of $2$-types over the empty set is nonseparable. Also, we characterize the principal types of finite tuples (over the empty set) and use this information to conclude that $\rbRT_r$ has no atomic model.

Interpolative fusions

We define the interpolative fusion [Formula: see text] of a family [Formula: see text] of first-order theories over a common reduct [Formula: see text], a notion that generalizes many examples of random or generic structures in the model-theoretic literature. When each [Formula: see text] is model-complete, [Formula: see text] coincides with the model companion of [Formula: see text]. By obtaining sufficient conditions for the existence of [Formula: see text], we develop new tools to show that theories of interest have model companions.

О СУЩЕСТВОВАНИИ МОДЕЛЬНОГО КОМПАНЬОНА

Found syntactic conditions sufficient for the existence model companion inductive theory.

Fields with automorphism and valuation

Abraham Robinson's method for finding model completions is refined and and generalized for model companions and is applied to the theory of fields equipped with both a valuation and an automorphism.