The notion $J$ is independent in $(M,M_0,N)$ was established by Shelah, for an AEC (abstract elementary class) which is stable in some cardinal $\lambda$ and has a non-forking relation, satisfying the good $\lambda$-frame axioms and some additional hypotheses. Shelah uses independence to define dimension. Here, we show the connection between the continuity property and dimension: if a non-forking satisfies natural conditions and the continuity property, then the dimension is well-behaved. As a corollary, we weaken the stability hypothesis and two additional (...) hypotheses, that appear in Shelah's theorem. (shrink)

We observe simple links between equivalence relations, groups, fields and groupoids (and between preorders, semi-groups, rings and categories), which are type-definable in an arbitrary structure, and apply these observations to the particular context of small and simple structures. Recall that a structure is small if it has countably many n-types with no parameters for each natural number n. We show that a θ-type-definable group in a small structure is the conjunction of definable groups, and extend the result to semi-groups, fields, (...) rings, categories, groupoids and preorders which are θ-type-definable in a small structure. For an A-type-definable group G A (where the set A may be infinite) in a small and simple structure, we deduce that (1) if G A is included in some definable set X such that boundedly many translates of G A cover X, then G A is the conjunction of definable groups. (2) for any finite tuple ḡ in G A , there is a definable group containing ḡ. (shrink)

In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the forma/Ewhereais a possibly (...) infinite tuple andEa type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2].Throughout this paper we will work in a large, saturated modelMof a complete theoryT. All types, sets and sequences will have size smaller than the size ofM. We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9]. (shrink)

We prove that for every simple theory T there is a compact abstract theory T.

We show that equational independence in the sense of Srour equals local non-forking. We then examine so-called almost equational theories where equational independence is a symmetric relation.

We develop the theory of germs of generic functions in simple theories. Starting with an algebraic quadrangle (or other similar hypotheses), we obtain an "almost" generic group chunk, where the product is denned up to a bounded number of possible values. This is the first step towards the proof of the group configuration theorem for simple theories, which is completed in [3].

As a continuation of ideas initiated in [19], we study bi-colored (generic) expansions of geometric theories in the style of the Fraïssé–Hrushovski construction method. Here we examine that the properties $NTP_{2}$, strongness, $NSOP_{1}$, and simplicity can be transferred to the expansions. As a consequence, while the corresponding bi-colored expansion of a red non-principal ultraproduct of p-adic fields is $NTP_{2}$, the expansion of algebraically closed fields with generic automorphism is a simple theory. Furthermore, these theories are strong with $\operatorname {\mathrm {bdn}}(\text (...) {"}x=x\text {"})=(\aleph _0)_{-}$. (shrink)

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 study model theoretic tree properties and their associated cardinal invariants. In particular, we obtain a quantitative refinement of Shelah’s theorem for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable and that weak [Formula: see text] is equivalent to [Formula: see text]. Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are (...) indeed [Formula: see text]. (shrink)

Disjoint n-amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this article, we show that if a countably categorical theory T admits an expansion with disjoint n-amalgamation for all n, then T is pseudofinite. All theories which admit an expansion with disjoint n-amalgamation for all n are simple, but the method can be extended, using filtrations of Fraïssé classes, to show that certain nonsimple theories are pseudofinite. As (...) case studies, we examine two generic theories of equivalence relations, Tfeq∗ and TCPZ, and show that both are pseudofinite. The theories Tfeq∗ and TCPZ are not simple, but they have NSOP1. This is established here for TCPZ for the first time. (shrink)

For n≥3, define Tn to be the theory of the generic Kn-free graph, where Kn is the complete graph on n vertices. We prove a graph-theoretic characterization of dividing in Tn and use it to show that forking and dividing are the same for complete types. We then give an example of a forking and nondividing formula. Altogether, Tn provides a counterexample to a question of Chernikov and Kaplan.

We examine fields in which model theoretic algebraic closure coincides with relative field theoretic algebraic closure. These are perfect fields with nice model theoretic behaviour. For example, they are exactly the fields in which algebraic independence is an abstract independence relation in the sense of Kim and Pillay. Classes of examples are perfect PAC fields, model complete large fields and henselian valued fields of characteristic 0.

We study structures equipped with generic predicates and/or automorphisms, and show that in many cases we obtain simple theories. We also show that a bounded PAC field is simple. 1998 Published by Elsevier Science B.V. All rights reserved.

The notion of a strongly determined type over A extending p is introduced, where p .S. A strongly determined extension of p over A assigns, for any model M )- A, a type q S extending p such that, if realises q, then any elementary partial map M → M which fixes acleq pointwise is elementary over . This gives a crude notion of independence which arises very frequently. Examples are provided of many different kinds of theories with strongly determined (...) types, and some without. We investigate a notion of multiplicity for strongly determined types with applications to ‘involved’ finite simple groups, and an analogue of the Finite Equivalence Relation Theorem. Lifting of strongly determined types to covers of a structure is discussed, and an application to finite covers is given. (shrink)

We give an axiomatic framework for the non-modular simple 0-categorical structures constructed by Hrushovski. This allows us to verify some of their properties in a uniform way, and to show that these properties are preserved by iterations of the construction.

We study generalized amalgamation properties in simple theories. We formulate a notion of generalized amalgamation in such a way so that the properties are preserved when we pass from T to Teq or Theq; we provide several equivalent ways of formulating the notion of generalized amalgamation.We define two distinct hierarchies of simple theories characterized by their amalgamation properties; examples are given to show the difference between the hierarchies.

We prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence Theorem for Lascar strong types. We also characterize the relation of starting an infinite indiscernible sequence in terms of coheirs.

An elliptic curve over a supersimple field with exactly one extension of degree 2 has an s-generic point.

In this paper, we study the notions related to tree property 1 , or, equivalently, SOP2. Among others, we supply a type-counting criterion for TP1 and show the equivalence of TP1 and k- TP1. Then we introduce the notions of weak k- TP1 for k≥2, and also supply type-counting criteria for those. We do not know whether weak k- TP1 implies TP1, but at least we prove that each weak k- TP1 implies SOP1. Our generalization of the tree-indiscernibility results in (...) Džamonja and Shelah [5] is crucially used throughout the paper. (shrink)

We study the groups Gal L and Gal KP, and the associated equivalence relations EL and EKP, attached to a first order theory T. An example is given where EL≠ EKP. It is proved that EKP is the composition of EL and the closure of EL. Other examples are given showing this is best possible.

Under [Formula: see text]-amalgamation, we obtain the canonical hyperdefinable group from the group configuration.

We introduce and study the notions of a PAC-substructure of a stable structure, and a bounded substructure of an arbitrary substructure, generalizing [10]. We give precise definitions and equivalences, saying what it means for properties such as PAC to be first order, study some examples (such as differentially closed fields) in detail, relate the material to generic automorphisms, and generalize a "descent theorem" for pseudo-algebraically closed fields to the stable context. We also point out that the elementary invariants of pseudo-algebraically (...) closed fields from [6] are also valid for pseudo-differentially closed fields. (shrink)

We investigate the geometry of forking for SU-rank 2 elements in supersimple ω-categorical theories and prove stable forking and some structural properties for such elements. We extend this analysis to the case of SU-rank 3 elements.

In a simple theory with elimination of finitary hyperimaginaries if tp(a) is real and analysable over a definable set Q, then there exists a finite sequence ( $a_{i}|i \leq n^{*}$ ) $\subseteq dcl^{eq}$ (a) with $a_{n}*$ = a such that for every $i \leq n*$ , if $p_{i} = tp(a_{i}/{a_{i}|j < i}$ ) then $Aut(p_{i}/Q)$ is type-definable with its action on $p_{i}^{c}$ . A unidimensional simple theory eliminates the quantifier $\exists^{\infty}$ and either interprets (in $C^{eq}$ ) an infinite type-definable group (...) or has the property that ACL(Q) = C for every infinite definable set Q. (shrink)

The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory T in which all partially recursive functions are representable, yet T does not interpret Robinson’s theory R. To this end, we borrow tools from model theory — specifically, we investigate model-theoretic properties of the model completion of the empty theory in a language with function symbols. We obtain a certain characterization of ∃∀ theories interpretable in (...) existential theories in the process. (shrink)

This paper investigates a connection between the semantic notion provided by the ordering * among theories in model theory and the syntactic SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1. It is shown here that SOP3 implies SOP2 implies SOP1. In Shelah's article 229) it was shown that SOP3 implies *-maximality and we prove here that *-maximality in a model of GCH implies a property called SOP2″. It has been (...) subsequently shown by Shelah and Usvyatsov that SOP2″ and SOP2 are equivalent, so obtaining an implication between *-maximality and SOP2. It is not known if SOP2 and SOP3 are equivalent. Together with the known results about the connection between the SOPn hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classification of unstable theories without the strict order property. (shrink)

We continue work of Shelah on the cardinality of families of pairwise incompatible types in simple theories obtaining characterizations of simple and supersimple theories. We develop a local analysis of the number of types in simple theories and we find a new example of a simple unstable theory.

We introduce Lascar strong types in excellent classes and prove that they coincide with the orbits of the group generated by automorphisms fixing a model. We define a new independence relation using Lascar strong types and show that it is well-behaved over models, as well as over finite sets. We then develop simplicity and show that, under simplicity, the independence relation satisfies all the properties of nonforking in a stable first order theory. Further, simplicity for an excellent class, as well (...) as the independence relation itself, is uniquely determined. Finally, we show that an excellent class is simple if and only if it has extensible U-rank . We deduce that any excellent class of finite U-rank is simple, and that any uncountably categorical excellent class has an expansion with countably many constants which is simple. (shrink)

Let T be a countable, small simple theory. In this paper, we prove that for such T, the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

This communication deals with positive model theory, a non first order model theoretic setting which preserves compactness at the cost of giving up negation. Positive model theory deals transparently with hyperimaginaries, and accommodates various analytic structures which defy direct first order treatment. We describe the development of simplicity theory in this setting, and an application to the lovely pairs of models of simple theories without the weak non finite cover property.

We explore Shelah's model‐theoretic dividing line methodology. In particular, we discuss how problems in model theory motivated new techniques in model theory, for example classifying theories by their potential (consistently with Zermelo–Fraenkel set theory with the axiom of choice (ZFC)) spectrum of cardinals in which there is a universal model. Two other examples are the study (with Malliaris) of the Keisler order leading to a new ZFC result on cardinal invariants and attempts to clarify the “main gap” by reducing the (...) dependence of certain versions on (highly independent) cardinal arithmetic. (shrink)

We introduce the framework of AECats, generalizing both the category of models of some first-order theory and the category of subsets of models. Any AEC and any compact abstract theory forms an AECat. In particular, we find applications in positive logic and continuous logic: the category of models of a positive or continuous theory is an AECat. The Kim–Pillay theorem for first-order logic characterizes simple theories by the properties dividing independence has. We prove a version of the Kim–Pillay theorem for (...) AECats with the amalgamation property, generalizing the first-order version and existing versions for positive logic. (shrink)

A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be (...) closely related to modular pairs in the lattice of algebraically closed sets. (shrink)

We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP1 theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.

We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are (...) made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's Algebraic Regularity Lemma is noted. (shrink)

We introduce the notion of a lovely pair of models of a simple theory T, generalizing Poizat's “belles paires” of models of a stable theory and the third author's “generic pairs” of models of an SU-rank 1 theory. We characterize when a saturated model of the theory TP of lovely pairs is a lovely pair , finding an analog of the nonfinite cover property for simple theories. We show that, under these hypotheses, TP is also simple, and we study forking (...) and canonical bases in TP. We also prove that assuming only that T is low, the existentially universal models of the universal part of a natural expansion TP+ of TP, are lovely pairs, and “simple Robinson universal domains”. (shrink)

We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic types and stabilizers for groups definable in simple theories. The general ideology is that the role of formulas (or definability) in stable theories is replaced by partial types (or ∞-definability) in simple theories.

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.

A study of smooth classes whose generic structures have simple theory is carried out in a spirit similar to Hrushovski 147; Simplicity and the Lascar group, preprint, 1997) and Baldwin–Shi 1). We attach to a smooth class K0, of finite -structures a canonical inductive theory TNat, in an extension-by-definition of the language . Here TNat and the class of existentially closed models of =T+,EX, play an important role in description of the theory of the K0,-generic. We show that if M (...) is the K0,-generic then MEX. Furthermore, if this class is an elementary class then Th=Th). The investigations by Hrushovski and Pillay , provide a general theory for forking and simplicity for the nonelementary classes, and using these ideas, we show that if K0,, where {,*}, has the joint embedding property and is closed under the Independence Theorem Diagram then EX is simple. Moreover, we study cases where EX is an elementary class. We introduce the notion of semigenericity and show that if a K0,-semigeneric structure exists then EX is an elementary class and therefore the -theory of K0,-generic is near model complete. By this result we are able to give a new proof for a theorem of Baldwin and Shelah 1359). We conclude this paper by giving an example of a generic structure whose first-order theory is simple. (shrink)

Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T, canonical base of an amalgamation class P is the union of names of ψ-definitions of P, ψ ranging over stationary L-formulas in P. Also, we prove that the same is true with stable formulas for an 1-based theory having (...) elimination of hyperimaginaries. For such a theory, the stable forking property holds, too. (shrink)

The non‐forking‐instances topology (NFI topology) is a topology on the Stone space of a theory T that depends on a reduct of T. This topology has been used in [6] to describe the set of universal transducers for (invariants sets that translate forking‐open sets in to forking‐open sets in T). In this paper we show that in contrast to the stable case, the NFI topology need not be invariant over parameters in but a weak version of this holds for any (...) simple T. We also note that for the lovely pair expansions, of theories with the weak non‐finite cover property (wnfcp), the topology is invariant over in. (shrink)