We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group ${\mathrm {UT}}_3({\mathbb {Z}})$, the continuous Heisenberg group ${\mathrm {UT}}_3({\mathbb {R}})$, and, more generally, groups of upper unitriangular and invertible upper triangular (...) matrices over unital rings. (shrink)

Fisher [10] and Baur [6] showed independently in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In [25, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq _p)$ such that K is a class of modules and $\leq _p$ is the pure submodule relation. In this paper we give some instances where this is true:Theorem.Assume (...) R is an associative ring with unity. Let $(K, \leq _p)$ be an AEC such that $K \subseteq R\text {-Mod}$ and K is closed under finite direct sums, then: •If K is closed under pure-injective envelopes, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda \geq \operatorname {LS}(\mathbf {K})$ such that $\lambda ^{|R| + \aleph _0}= \lambda $.•If K is closed under pure submodules and pure epimorphic images, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $.•Assume R is Von Neumann regular. If $\mathbf {K}$ is closed under submodules and has arbitrarily large models, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $.As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, Dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy.Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of $\mathfrak {s}$ -torsion modules. (shrink)

We introduce a family of local ranks $D_Q$ depending on a finite set Q of pairs of the form $(\varphi (x,y),q(y)),$ where $\varphi (x,y)$ is a formula and $q(y)$ is a global type. We prove that in any NSOP $_1$ theory these ranks satisfy some desirable properties; in particular, $D_Q(x=x)<\omega $ for any finite tuple of variables x and any Q, if $q\supseteq p$ is a Kim-forking extension of types, then $D_Q(q) (...) Kim-non-forking extension, then $D_Q(q)=D_Q(p)$ for every Q that involves only invariant types whose Morley powers are -stationary. We give natural examples of families of invariant types satisfying this property in some NSOP $_1$ theories. We also answer a question of Granger about equivalence of dividing and dividing finitely in the theory $T_\infty $ of vector spaces with a generic bilinear form. We conclude that forking equals dividing in $T_\infty $, strengthening an earlier observation that $T_\infty $ satisfies the existence axiom for forking independence. Finally, we slightly modify our definitions and go beyond NSOP $_1$ to find out that our local ranks are bounded by the well-known ranks: the inp-rank (burden), and hence, in particular, by the dp-rank. Therefore, our local ranks are finite provided that the dp-rank is finite, for example, if T is dp-minimal. Hence, our notion of rank identifies a non-trivial class of theories containing all NSOP $_1$ and NTP $_2$ theories. (shrink)

The following strong form of density of definable types is introduced for theoriesTadmitting a fibered dimension functiond: given a modelMofTand a definable setX⊆Mn, there is a definable typepinX, definable over a code forXand of the samed-dimension asX. Both o-minimal theories and the theory of closed ordered differential fields are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

Assuming the existence of a monster model, tameness, and continuity of nonsplitting in an abstract elementary class (AEC), we extend known superstability results: let $\mu>\operatorname {LS}(\mathbf {K})$ be a regular stability cardinal and let $\chi $ be the local character of $\mu $ -nonsplitting. The following holds: 1.When $\mu $ -nonforking is restricted to $(\mu,\geq \chi )$ -limit models ordered by universal extensions, it enjoys invariance, monotonicity, uniqueness, existence, extension, and continuity. It also has local character $\chi $. This generalizes (...) Vasey’s result [37, Corollary 13.16] which assumed $\mu $ -superstability to obtain same properties but with local character $\aleph _0$.2.There is $\lambda \in [\mu,h(\mu ))$ such that if $\mathbf {K}$ is stable in every cardinal between $\mu $ and $\lambda $, then $\mathbf {K}$ has $\mu $ -symmetry while $\mu $ -nonforking in (1) has symmetry. In this case:(a) $\mathbf {K}$ has the uniqueness of $(\mu,\geq \chi )$ -limit models: if $M_1,M_2$ are both $(\mu,\geq \chi )$ -limit over some $M_0\in K_{\mu }$, then $M_1\cong _{M_0}M_2$ ;(b)any increasing chain of $\mu ^+$ -saturated models of length $\geq \chi $ has a $\mu ^+$ -saturated union. These generalize [31] and remove the symmetry assumption in [10, 38].Under $(<\mu )$ -tameness, the conclusions of (1), (2)(a)(b) are equivalent to $\mathbf {K}$ having the $\chi $ -local character of $\mu $ -nonsplitting.Grossberg and Vasey [18, 38] gave eventual superstability criteria for tame AECs with a monster model. We remove the high cardinal threshold and reduce the cardinal jump between equivalent superstability criteria. We also add two new superstability criteria to the list: a weaker version of solvability and the boundedness of the U-rank. (shrink)

We study model theory of actions of finite groups on substructures of a stable structure. We give an abstract description of existentially closed actions as above in terms of invariants and PAC structures. We show that if the corresponding PAC property is first order, then the theory of such actions has a model companion. Then, we analyze some particular theories of interest (mostly various theories of fields of positive characteristic) and show that in all the cases considered the PAC property (...) is first order. (shrink)

We exhibit a bridge between the theory of cellular categories, used in algebraic topology and homological algebra, and the model-theoretic notion of stable independence. Roughly speaking, we show that the combinatorial cellular categories (those where, in a precise sense, the cellular morphisms are generated by a set) are exactly those that give rise to stable independence notions. We give two applications: on the one hand, we show that the abstract elementary classes of roots of Ext studied by Baldwin–Eklof–Trlifaj are stable (...) and tame. On the other hand, we give a simpler proof (in a special case) that combinatorial categories are closed under 2-limits, a theorem of Makkai and Rosický. (shrink)

We prove that every type of finite Cantor-Bendixson rank over a model of a first-order theory without the strict order property is definable and has a unique nonforking extension to a global type. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We develop the theory of Kim-independence in the context of NSOP $_{1}$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP $_{1}$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP $_{1}$ theories.

We introduce and study (weakly) semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong honest definitions; demonstrate that certain trees are semi-equational, while algebraically closed valued fields are not weakly semi-equational; and obtain a general criterion for weak semi-equationality of an expansion of a distal structure by a new predicate.

A theory is stable up to Δ if any Δ-type over a model has a few extensions up to complete types. We prove that a theory has no the independence property iff it is stable up to some Δ, where each equation image has no the independence property.

In this paper, we study a stability transfer theorem in d-tame metric abstract elementary classes, in a similar way as in 2, but using superstability-like assumptions which involves a new independence notion instead of ℵ0-locality.

We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type p over a set B does not divide over $C\subseteq B$, then no extension of p to a complete type over $\operatorname {acl}(B)$ divides over C. Two of our examples are also the first known theories where all sets are extension bases for nonforking, but forking and dividing differ for complete types (answering a question of Adler). One example is an $\mathrm {NSOP}_1$ theory (...) with a complete type that forks, but does not divide, over a model (answering a question of d’Elbée). Moreover, dividing independence fails to imply M-independence in this example (which refutes another folklore claim). In addition to these counterexamples, we summarize various related properties of dividing that are still true. We also address consequences for previous literature, including an earlier unpublished result about forking and dividing in free amalgamation theories, and some claims about dividing in the theory of generic $K_{m,n}$ -free incidence structures. (shrink)

Motivated by the theory of domination for types, we introduce a notion of domination for Keisler measures called extension domination. We argue that this variant of domination behaves similarly to its typesetting counterpart. We prove that extension domination extends domination for types and that it forms a preorder on the space of global Keisler measures. We then explore some basic properties related to this notion (e.g., approximations by formulas, closure under localizations, convex combinations). We also prove a few preservation theorems (...) and provide some explicit examples. (shrink)

We study the $\kappa $ -Borel-reducibility of isomorphism relations of complete first-order theories by using coloured trees. Under some cardinality assumptions, we show the following: For all theories T and T’, if T is classifiable and T’ is unsuperstable, then the isomorphism of models of T’ is strictly above the isomorphism of models of T with respect to $\kappa $ -Borel-reducibility.

We generalise the properties $\mathsf {OP}$, $\mathsf {IP}$, k- $\mathsf {TP}$, $\mathsf {TP}_{1}$, k- $\mathsf {TP}_{2}$, $\mathsf {SOP}_{1}$, $\mathsf {SOP}_{2}$, and $\mathsf {SOP}_{3}$ to positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having $\mathsf {TP}$ and dividing (...) having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory T has $\mathsf {OP}$ iff it has $\mathsf {IP}$ or $\mathsf {SOP}_{1}$ and that T has $\mathsf {TP}$ iff it has $\mathsf {SOP}_{1}$ or $\mathsf {TP}_{2}$, analogous to the well-known results in full first-order logic where $\mathsf {SOP}_{1}$ is replaced by $\mathsf {SOP}$ in the former and by $\mathsf {TP}_{1}$ in the latter. Our proofs of these final two theorems are new and make use of Kim-independence. (shrink)

We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to Corollary 4.2 from that paper, and in the process discuss “theories with selectors.”.

We study tree-like decompositions of models of a theory and a related complexity measure called partition width. We prove a dichotomy concerning partition width and definable pairing functions: either the partition width of models is bounded, or the theory admits definable pairing functions. Our proof rests on structure results concerning indiscernible sequences and finitely satisfiable types for theories without definable pairing functions. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

We give two slight generalizations of results of Poizat about elementary theories of groups obtained by free constructions. The first-one concerns generic types and the non-superstability of such groups in many cases. The second-one concerns the connectedness of most free products of groups without amalgamation.

Assume$G\prec H$are groups and${\cal A}\subseteq {\cal P}(G),\ {\cal B}\subseteq {\cal P}(H)$are algebras of sets closed under left group translation. Under some additional assumptions we find algebraic connections between the Ellis [semi]groups of theG-flow$S({\cal A})$and theH-flow$S({\cal B})$. We apply these results in the model theoretic context. Namely, assumeGis a group definable in a modelMand$M\prec ^* N$. Using weak heirs and weak coheirs we point out some algebraic connections between the Ellis semigroups$S_{ext,G}(M)$and$S_{ext,G}(N)$. Assuming every minimal left ideal in$S_{ext,G}(N)$is a group we prove (...) that the Ellis groups of$S_{ext,G}(M)$are isomorphic to closed subgroups of the Ellis groups of$S_{ext,G}(N)$. (shrink)

In this paper, the connections between model theory and the theory of infinite permutation groups are used to study the n-existence and the n-uniqueness for n-amalgamation problems of stable theories. We show that, for any n ⩾ 2, there exists a stable theory having -existence and k-uniqueness, for every k ⩽ n, but has neither -existence nor -uniqueness. In particular, this generalizes the example, for n = 2, due to Hrushovski given in 3. © 2011 WILEY-VCH Verlag GmbH & Co. (...) KGaA, Weinheim. (shrink)

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal {B}$ on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type (...) and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures—in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types. (shrink)

The classes stable, simple, and NSOP $_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP $_1$ theories it must come from Kim-dividing. We generalise this work to the framework of Abstract Elementary Categories (AECats) (...) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple, and NSOP $_1$ -like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking, and long Kim-dividing. (shrink)

We solve two problems from a work of Haskel and Pillay concerning maximal stable quotients of groups ∧-definable in NIP theories. The first result says that if G is a ∧-definable group in a distal theory, then Gst=G00 (where Gst is the smallest ∧-definable subgroup with G∕Gst stable, and G00 is the smallest ∧-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion Theq. The second result (...) is an example of a group G definable in a nondistal NIP theory for which G=G00, but Gst is not an intersection of definable groups. Our example is a saturated extension of (R,+,[0,1]). Moreover, we make some observations on the question whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyperdefinable sets involving continuous logic. (shrink)

We show that a complete first-order theory T is distal provided it has a model M such that the theory of the Shelah expansion of M is distal.

We show that any formula with two free variables in a Vapnik–Chervonenkis (VC) minimal theory has VC-codensity at most 2. Modifying the argument slightly, we give a new proof of the fact that, in a VC-minimal theory where acleq= dcleq, the VC-codensity of a formula is at most the number of free variables (from the work of Aschenbrenner et al., the author, and Laskowski).

We give a new approach to the failure of the Canonical Base Property (CBP) in the so far only known counterexample, produced by Hrushovski, Palacín and Pillay. For this purpose, we will give an alternative presentation of the counterexample as an additive cover of an algebraically closed field. We isolate two fundamental weakenings of the CBP, which already appeared in work of Chatzidakis and Moosa-Pillay and show that they do not hold in the counterexample. In order to do so, a (...) study of imaginaries in additive covers is developed. As a by-product of the presentation, we observe that a pure binding-group-theoretic account of the CBP is unlikely. (shrink)

We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This improves previous work of Ben Yaacov and Chernikov. We propose a line of investigation of NTP2 theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.

Motivated by the Ax–Kochen/Ershov principle, a large number of questions about Henselian valued fields have been shown to reduce to analogous questions about the value group and residue field. In this article, we investigate the burden of Henselian valued fields in the three-sorted Denef–Pas language. If T is a theory of Henselian valued fields admitting relative quantifier elimination (in any characteristic), we show that the burden of T is equal to the sum of the burdens of its value group and (...) residue field. As a consequence, T is NTP2 if and only if its residue field and value group are; the same is true for the statements “T is strong” and “T has finite burden.”. (shrink)

In model theory, a branch of mathematical logic, we can classify mathematical structures based on their logical complexity. This yields the so-called stability hierarchy. Independence relations play an important role in this stability hierarchy. An independence relation tells us which subsets of a structure contain information about each other, for example, linear independence in vector spaces yields such a relation.Some important classes in the stability hierarchy are stable, simple, and NSOP $_1$, each being contained in the next. For each of (...) these classes there exists a so-called Kim-Pillay style theorem. Such a theorem describes the interaction between independence relations and the stability hierarchy. For example, simplicity is equivalent to admitting a certain independence relation, which must then be unique.All of the above classically takes place in full first-order logic. Parts of it have already been generalised to other frameworks, such as continuous logic, positive logic, and even a very general category-theoretic framework. In this thesis we continue this work.We introduce the framework of AECats, which are a specific kind of accessible category. We prove that there can be at most one stable, simple, or NSOP $_1$ -like independence relation in an AECat. We thus recover (part of) the original stability hierarchy. For this we introduce the notions of long dividing, isi-dividing, and long Kim-dividing, which are based on the classical notions of dividing and Kim-dividing but are such that they work well without compactness.Switching frameworks, we generalise Kim-dividing in NSOP $_1$ theories to positive logic. We prove that Kim-dividing over existentially closed models has all the nice properties that it is known to have in full first-order logic. We also provide a full Kim-Pillay style theorem: a positive theory is NSOP $_1$ if and only if there is a nice enough independence relation, which then must be given by Kim-dividing.Abstract prepared by Mark Kamsma.E-mail:[email protected]:https://markkamsma.nl/phd-thesis. (shrink)

We study the preservation of certain properties under products of classes of finite structures. In particular, we examine indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interacts with the lexicographic product, full product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products. In particular, we show that, under mild assumptions, the products considered in this article do not yield (...) new classes of theories. (shrink)

We study Martsinkovsky–Russell torsion modules with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pure-injective and the Σ-pure-injective modules relative to the class of torsion modules assuming that the torsion submodule is a pure submodule. Our characterization of relative Σ-pure-injective modules extends the classical characterization of Gruson and Jenson as well as Zimmermann. We study the limit models of the class and determine when the class is superstable assuming that the torsion submodule is a (...) pure submodule. As a corollary, we show that the class of torsion abelian groups with pure embeddings is strictly stable (i.e., stable not superstable). (shrink)

A first-order structure $\mathfrak {A}$ is called monadically stable iff every expansion of $\mathfrak {A}$ by unary predicates is stable. In this paper we give a classification of the class $\mathcal {M}$ of $\omega $ -categorical monadically stable structure in terms of their automorphism groups. We prove in turn that $\mathcal {M}$ is the smallest class of structures which contains the one-element pure set, is closed under isomorphisms, and is closed under taking finite disjoint unions, infinite copies, and finite index (...) first-order reducts. Using our classification we show that every structure in $\mathcal {M}$ is first-order interdefinable with a finitely bounded homogeneous structure. We also prove that every structure in $\mathcal {M}$ has finitely many reducts up to interdefinability, thereby confirming Thomas’ conjecture for the class $\mathcal {M}$. (shrink)

We refine results of Gannon [6, Theorem 4.7] and Simon [22, Lemma 2.8] on convergence of Morley sequences. We then introduce the notion of eventual $NIP$, as a property of a model, and prove a variant of [15, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of Pillay [17] on the model-theoretic meaning of Grothendieck’s double limit theorem.

We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

For a NIP theory T, a sufficiently saturated model ${\mathfrak C}$ of T, and an invariant (over some small subset of ${\mathfrak C}$ ) global type p, we prove that there exists a finest relatively type-definable over a small set of parameters from ${\mathfrak C}$ equivalence relation on the set of realizations of p which has stable quotient. This is a counterpart for equivalence relations of the main result of [2] on the existence of maximal stable quotients of type-definable groups (...) in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in [3]. (shrink)