We study cofinal systems of finite subsets of $\omega _1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: In an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.

The Lovász local lemma (LLL) is a technique from combinatorics for proving existential results. There are many different versions of the LLL. One of them, the lefthanded local lemma, is particularly well suited for applications to two player games. There are also constructive and computable versions of the LLL. The chief object of this thesis is to prove an effective version of the lefthanded local lemma and to apply it to effectivise constructions of non-repetitive sequences.

The second goal of this thesis is to categorize some classes of cohesive powers. We completely describe both the isomorphism types of cohesive powers of equivalence structures and injection structures, as well as clarify the relationship between these cohesive powers and their original structures. We also describe the finite condensation of cohesive powers of computable copies of the integers as a linear order by cohesive sets whose complement are computably enumerable.

Finally, we investigate the possibility of decomposing problems in the Weihrauch degrees into a product of first order part and second order part. We give a preliminary result in this direction.

Abstract prepared by Daniel Mourad.

E-mail: daniel.mourad@uconn.edu

URL: https://daniel-mourad.scholar.uconn.edu/wp-content/uploads/sites/2530/2023/06/ThesisFinalWithRevisions-2.pdf

We study ordered groups in the context of both algebra and computability. Ordered groups are groups that admit a linear order that is compatible with the group operation. We explore some properties of ordered groups and discuss some related topics. We prove results about the semidirect product in relation to orderability and computability. In particular, we give a criteria for when a semidirect product of orderable groups is orderable and for when a semidirect product is computably categorical. We also give an example of a semidirect product that has the halting set coded into its multiplication structure but it is possible to construct a computable presentation of this semidirect product.

We examine a family of orderable groups that admit exactly countably many orders and show that their space of orders has arbitrary finite Cantor–Bendixson rank. Furthermore, this family of groups is also shown to be computably categorical, which in particular will allow us to conclude that any computable presentation of the groups does not admit any noncomputable orders. Lastly, we construct an example of an orderable computable group with no computable Archimedean orders but at least one computable non-Archimedean order.

Abstract prepared by Waseet Kazmi.

E-mail: waseet.kazmi@uconn.edu

URL: http://hdl.handle.net/11134/20002:860745910

Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class $\Omega $ of uniform ultrafilters generates a $\Delta $-closed logic ${\mathcal {L}}_\Omega $. ${\mathcal {L}}_\Omega $ is $\omega $-relatively compact iff some $D\in \Omega $ fails to be $\omega _1$-complete iff ${\mathcal {L}}_\Omega $ does not contain the quantifier “there are uncountably many.” If $\Omega $ is a set, or if it contains a countably incomplete ultrafilter, then ${\mathcal {L}}_\Omega $ is not generated by Mostowski cardinality quantifiers. Assuming $\neg 0^\sharp $ or $\neg L^{\mu }$, if $D\in \Omega $ is a uniform ultrafilter over a regular cardinal $\nu $, then every family $\Psi $ of formulas in ${\mathcal {L}}_\Omega $ with $|\Phi |\leq \nu $ satisfies the compactness theorem. In particular, if $\Omega $ is a proper class of uniform ultrafilters over regular cardinals, ${\mathcal {L}}_\Omega $ is compact.

We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answering a question by Anscombe–Fehm in a strong way. Along the way, we construct an existentially decidable field of positive characteristic with an existentially undecidable finite extension, modifying a construction due to Kesavan Thanagopal.

We introduce and study the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with the trivial Bohr compactification, that is, the class of minimally almost periodic groups.

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.

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].

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.

We begin by proving that any Presburger-definable image of one or more sets of powers has zero natural density. Then, by adapting the proof of a dichotomy result on o-minimal structures by Friedman and Miller, we produce a similar dichotomy for expansions of Presburger arithmetic on the integers. Combining these two results, we obtain that the expansion of the ordered group of integers by any number of sets of powers does not define multiplication.

In this paper, together with the preceding Part I [10], we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to [10] the treatment of the mixed characteristic case. Hensel minimality is inspired by o-minimality and its role in real geometry and diophantine applications. We develop geometric results and applications for Hensel minimal structures that were previously known only under stronger or less axiomatic assumptions, and which often have counterparts in o-minimal structures. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under Hensel minimality – more precisely, $1$-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves.

For any subset $Z \subseteq {\mathbb {Q}}$, consider the set $S_Z$ of subfields $L\subseteq {\overline {\mathbb {Q}}}$ which contain a co-infinite subset $C \subseteq L$ that is universally definable in L such that $C \cap {\mathbb {Q}}=Z$. Placing a natural topology on the set ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$ of subfields of ${\overline {\mathbb {Q}}}$, we show that if Z is not thin in ${\mathbb {Q}}$, then $S_Z$ is meager in ${\operatorname {Sub}({\overline {\mathbb {Q}}})}$. Here, thin and meager both mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fields L have the property that the ring of algebraic integers $\mathcal {O}_L$ is universally definable in L. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every $\exists $-definable subset of an algebraic extension of ${\mathbb Q}$ is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials.

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 $\mathcal {M}$.

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.”

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 the G-flow $S({\cal A})$ and the H-flow $S({\cal B})$. We apply these results in the model theoretic context. Namely, assume G is a group definable in a model M and $M\prec ^* N$. 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 $S_{ext,G}(N)$.

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. 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. 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:

   1. (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 $M_1\cong _{M_0}M_2$;

   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.

Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. As an application, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures.

We aim at developing a systematic method of separating omniscience principles by constructing Kripke models for intuitionistic predicate logic $\mathbf {IQC}$ and first-order arithmetic $\mathbf {HA}$ from a Kripke model for intuitionistic propositional logic $\mathbf {IPC}$. To this end, we introduce the notion of an extended frame, and show that each IPC-Kripke model generates an extended frame. By using the extended frame generated by an IPC-Kripke model, we give a separation theorem of a schema from a set of schemata in $\mathbf {IQC}$ and a separation theorem of a sentence from a set of schemata in $\mathbf {HA}$. We see several examples which give us separations among omniscience principles.

We prove that the class of all the rings $\mathbb {Z}/m\mathbb {Z}$ for all $m>1$ is decidable. This gives a positive solution to a problem of Ax asked in his celebrated 1968 paper on the elementary theory of finite fields [1, Problem 5, p. 270]. In our proof, we reduce the problem to the decidability of the ring of adeles $\mathbb {A}_{\mathbb {Q}}$ of $\mathbb {Q}$.

Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $\mathbb {Q}_p$, we show that $G^0 = G^{00}$, and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb {Q}_p$.