A notion of interpretation between arbitrary logics is introduced, and the poset $\mathsf {Log}$ of all logics ordered under interpretability is studied. It is shown that in $\mathsf {Log}$ infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between $\mathsf {Log}$ and the lattice of interpretability types of varieties are investigated.

We study the first-order theories of some natural and important classes of coloured trees, including the four classes of trees whose paths have the order type respectively of the natural numbers, the integers, the rationals, and the reals. We develop a technique for approximating a tree as a suitably coloured linear order. We then present the first-order theories of certain classes of coloured linear orders and use them, along with the approximating technique, to establish complete axiomatisations of the four classes of trees mentioned above.

Fix an abelian group $\Gamma $ and an injective endomorphism $F\colon \Gamma \to \Gamma $ . Improving on the results of [2], new characterizations are here obtained for the existence of spanning sets, F-automaticity, and F-sparsity. The model theoretic status of these sets is also investigated, culminating with a combinatorial description of the F-sparse sets that are stable in $(\Gamma ,+)$ , and a proof that the expansion of $(\Gamma ,+)$ by any F-sparse set is NIP. These methods are also used to show for prime $p\ge 7$ that the expansion of $(\mathbb {F}_p[t],+)$ by multiplication restricted to $t^{\mathbb {N}}$ is NIP.

We show that if $X$ is an $m$-dimensional definable set in $\mathbb {R}_\text {an}^\text{pow}$, the structure of real subanalytic sets with real power maps added, then for any positive integer $r$ there exists a $C^{r}$-parameterization of $X$ consisting of $cr^{m^{3}}$ maps for some constant $c$. Moreover, these maps are real analytic and this bound is uniform for a definable family.

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm {ACFG}$ . This theory was introduced in [16] as a new example of $\mathrm {NSOP}_{1}$ nonsimple theory. In this paper we describe more features of $\mathrm {ACFG}$ , such as imaginaries. We also study various independence relations in $\mathrm {ACFG}$ , such as Kim-independence or forking independence, and describe interactions between them.

This thesis is concerned with the expansions of algebraic structures and their fit in Shelah’s classification landscape.

The first part deals with the expansion of a theory by a random predicate for a substructure model of a reduct of the theory. Let T be a theory in a language $\mathcal {L}$ . Let $T_0$ be a reduct of T. Let $\mathcal {L}_S = \mathcal {L}\cup \{S\}$ , for S a new unary predicate symbol, and $T_S$ be the $\mathcal {L}_S$ -theory that axiomatises the following structures: $(\mathscr {M},\mathscr {M}_0)$ consist of a model $\mathscr {M}$ of T and S is a predicate for a model $\mathscr {M}_0$ of $T_0$ which is a substructure of $\mathscr {M}$ . We present a setting for the existence of a model-companion $TS$ of $T_S$ . As a consequence, we obtain the existence of the model-companion of the following theories, for $p>0$ a prime number:

• • $\mathrm {ACF}_p$ , $\mathrm {SCF}_{e,p}$ , $\mathrm {Psf}_p$ , $\mathrm {ACFA}_p$ , $\mathrm {ACVF}_{p,p}$ in appropriate languages expanded by arbitrarily many predicates for additive subgroups;

• • $\mathrm {ACF}_p$ , $\mathrm {ACF}_0$ in the language of rings expanded by a single predicate for a multiplicative subgroup;

• • $\mathrm {PAC}_p$ -fields, in an appropriate language expanded by arbitrarily many predicates for additive subgroups.

$TS$

$TS$

$TS$

$\mathrm {NSOP}_{1}$

Configuration $T_0\subseteq T$	Generic expansion $TS$
$T_0 = T$	Preserves stability
$T_0\subseteq T$	Preserves $\mathrm {NSOP}_{1}$
$T_0 = \emptyset $	Preserves simplicity

In particular, this construction produces new examples of $\mathrm {NSOP}_{1}$ not simple theories, and we study in depth a particular example: the expansion of an algebraically closed field of positive characteristic by a generic additive subgroup. We give a full description of imaginaries, forking, and Kim-forking in this example.

The second part studies expansions of the group of integers by p-adic valuations. We prove quantifier elimination in a natural language and compute the dp-rank of these expansions: it equals the number of independent p-adic valuations considered. Thus, the expansion of the integers by one p-adic valuation is a new dp-minimal expansion of the group of integers. Finally, we prove that the latter expansion does not admit intermediate structures: any definable set in the expansion is either definable in the group structure or is able to “reconstruct” the valuation using only the group operation.

Abstract prepared by Christian d’Elbée.

E-mail: [email protected]

URL: https://choum.net/~chris/page_perso

The thesis pseudofinite structures and counting dimensions is about the model theory of pseudofinite structures with the focus on groups and fields. The aim is to deepen our understanding of how pseudofinite counting dimensions can interact with the algebraic properties of underlying structures and how we could classify certain classes of structures according to their counting dimensions. Our approach is by studying examples. We treat three classes of structures: The first one is the class of H-structures, which are generic expansions of existing structures. We give an explicit construction of pseudofinite H-structures as ultraproducts of finite structures. The second one is the class of finite difference fields. We study properties of coarse pseudofinite dimension in this class, show that it is definable and integer-valued and build a partial connection between this dimension and transformal transcendence degree. The third example is the class of pseudofinite primitive permutation groups. We generalise Hrushovski’s classical classification theorem for stable permutation groups acting on a strongly minimal set to the case where there exists an abstract notion of dimension, which includes both the classical model theoretic ranks and pseudofinite counting dimensions. In this thesis, we also generalise Schlichting’s theorem for groups to the case of approximate subgroups with a notion of commensurability.

Abstract prepared by Tingxiang Zou.

E-mail: [email protected]

URL: https://tel.archives-ouvertes.fr/tel-02283810/document

In this thesis, we study transfer principles in the context of certain Henselian valued fields, namely Henselian valued fields of equicharacteristic $0$ , algebraically closed valued fields, algebraically maximal Kaplansky valued fields, and unramified mixed characteristic Henselian valued fields with perfect residue field. First, we compute the burden of such a valued field in terms of the burden of its value group and its residue field. The burden is a cardinal related to the model theoretic complexity and a notion of dimension associated to $\text {NTP}_2$ theories. We show, for instance, that the Hahn field $\mathbb {F}_p^{\text {alg}}((\mathbb {Z}[1/p]))$ is inp-minimal (of burden 1), and that the ring of Witt vectors $W(\mathbb {F}_p^{\text {alg}})$ over $\mathbb {F}_p^{\text {alg}}$ is not strong (of burden $\omega $ ). This result extends previous work by Chernikov and Simon and realizes an important step toward the classification of Henselian valued fields of finite burden. Second, we show a transfer principle for the property that all types realized in a given elementary extension are definable. It can be written as follows: a valued field as above is stably embedded in an elementary extension if and only if its value group is stably embedded in the corresponding extension of value groups, its residue field is stably embedded in the corresponding extension of residue fields, and the extension of valued fields satisfies a certain algebraic condition. We show, for instance, that all types over the power series field $\mathbb {R}((t))$ are definable. Similarly, all types over the quotient field of $W(\mathbb {F}_p^{\text {alg}})$ are definable. This extends previous work of Cubides and Delon and of Cubides and Ye.

These distinct results use a common approach, which has been developed recently. It consists of establishing first a reduction to an intermediate structure called the leading term structure, or $\operatorname {\mathrm {RV}}$ -sort, and then of reducing to the value group and residue field. This leads us to develop similar reduction principles in the context of pure short exact sequences of abelian groups.

Abstract prepared by Pierre Touchard.

E-mail: [email protected]

URL: https://miami.uni-muenster.de/Record/a612cf73-0a2f-42c4-b1e4-7d28934138a9

The purpose of this thesis is to develop a paraconsistent Model Theory. The basis for such a theory was launched by Walter Carnielli, Marcelo Esteban Coniglio, Rodrigo Podiack, and Tarcísio Rodrigues in the article ‘On the Way to a Wider Model Theory: Completeness Theorems for First-Order Logics of Formal Inconsistency’ [The Review of Symbolic Logic, vol. 7 (2014)].

Naturally, a complete theory cannot be fully developed in a single work. Indeed, the goal of this work is to show that a paraconsistent Model Theory is a sound and worthy possibility. The pursuit of this goal is divided in three tasks: The first one is to give the theory a philosophical meaning. The second one is to transpose as many results from the classical theory to the new one as possible. The third one is to show an application of the theory to practical science.

The response to the first task is a Paraconsistent Reasoning System. The start point is that paraconsistency is an epistemological concept. The pursuit of a deeper understanding of the phenomenon of paraconsistency from this point of view leads to a reasoning system based on the Logics of Formal Inconsistency. Models are regarded as states of knowledge and the concept of isomorphism is reformulated so as to give raise to a new concept that preserves a portion of the whole knowledge of each state. Based on this, a notion of refinement is created which may occur from inside or from outside the state.

In order to respond to the second task, two important classical results, namely the Omitting Types Theorem and Craig’s Interpolation Theorem are shown to hold in the new system and it is also shown that, if classical results in general are to hold in a paraconsistent system, then such a system should be in essence how it was developed here.

Finally, the response to the third task is a proposal of what a Paraconsistent Logic Programming may be. For that, the basis for a paraconsistent PROLOG is settled in the light of the ideas developed so far.

Abstract prepared by Bruno Costa Coscarelli.

E-mail: [email protected]

URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/331697

In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G-compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types have a certain model theoretic property that we call divisible amalgamation. The main result of this paper is that if c is a finite tuple algebraic over a tuple a, the Lascar group of $\operatorname {stp}(ac)$ is abelian, and the underlying theory is G-compact, then the Lascar groups of $\operatorname {stp}(ac)$ and of $\operatorname {stp}(a)$ are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of $\operatorname {stp}(ac)$ is abelian, nor the assumption of c being finite can be removed.

We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety.

We introduce and study a natural class of fields in which certain first-order definable sets are existentially definable, and characterise this class by a number of equivalent conditions. We show that global fields belong to this class, and in particular obtain a number of new existential (or diophantine) predicates over global fields.

We use the model theoretic notion of coheir to give short proofs of old and new theorems in Ramsey Theory. As an illustration we start from Ramsey’s theorem itself. Then we prove Hindman’s theorem and the Hales–Jewett theorem. Finally, we prove two Ramsey theoretic principles that have among their consequences partition theorems due to Carlson and to Gowers.

We define a new type of “shatter function” for set systems that satisfies a Sauer–Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah’s two-rank instead of VC dimension. We identify the least exponent bounding the rate of growth of the shatter function, the quantity analogous to VC density, with Shelah’s $\omega $ -rank.

A complete list of one dimensional groups definable in the p-adic numbers is given, up to a finite index subgroup and a quotient by a finite subgroup.

We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation subclasses whose axiomatisations are recursively enumerable in our second-order fragment can also be recursively axiomatised in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. As applications of these results, by showing that certain classes can be straightforwardly defined as separation subclasses, we obtain first-order axiomatisability results for these classes. In particular we apply this technique to graph colourings and a class of partial algebras arising from separation logic.

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property.

We build a new spectrum of recursive models ( $ \operatorname {\mathrm {SRM}}(T)$ ) of a strongly minimal theory. This theory is non-disintegrated, flat, model complete, and in a language with a finite signature.

We prove that if an $\omega $ -categorical structure has an $\omega $ -categorical homogeneous Ramsey expansion, then so does its model-complete core.

We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.