We continue the study of the connection between the “geometric” properties of SU -rank 1 structures and the properties of “generic” pairs of such structures, started in [8]. In particular, we show that the SU-rank of the theory of generic pairs of models of an SU -rank 1 theory T can only take values 1 , 2 or ω, generalizing the corresponding results for a strongly minimal T in [3]. We also use pairs to derive the implication from pseudolinearity to (...) linearity for ω -categorical SU -rank 1 structures, established in [7], from the conjecture that an ω -categorical supersimple theory has finite SU -rank, and find a condition on generic pairs, equivalent to pseudolinearity in the general case. (shrink)

We extend the dichotomy between 1-basedness and supersimplicity proved in Shami :309–332, 2011). The generalization we get is to arbitrary language, with no restrictions on the topology [we do not demand type-definabilty of the open set in the definition of essential 1-basedness from Shami :309–332, 2011)]. We conclude that every hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the (...) language is countable. (shrink)

Let T be a simple L-theory and let \ be a reduct of T to a sublanguage \ of L. For variables x, we call an \-invariant set \\) in \ a universal transducer if for every formula \\in L^-\) and every a, $$\begin{aligned} \phi ^-\ L^-\text{-forks } \text{ over }\ \emptyset \ \text{ iff } \Gamma \wedge \phi ^-\ L\text{-forks } \text{ over }\ \emptyset. \end{aligned}$$We show that there is a greatest universal transducer \ and it is type-definable. In (...) particular, the forking topology on \\) refines the forking topology on \\) for all y. Moreover, we describe the set of universal transducers in terms of certain topology on the Stone space and show that \ is the unique universal transducer that is \-type-definable with parameters. If \ is a theory with the wnfcp and T is the theory of its lovely pairs of models we show that \\) and give a more precise description of the set of universal transducers for the special case where \ has the nfcp. (shrink)

We show that in a simple theory T in which the τf-topologies are closed under projections every type analyzable in a supersimple τf-open set has ordinal SU-rank. In particular, if in addition T is unidimensional, the existence of a supersimple unbounded τf-open set implies T is supersimple. We also introduce the notion of a standard τ-metric and show that for simple theories its completeness is equivalent to the compactness of the τ-topology.

In a supersimple unidimensional theory, SU-rank is continuous and D-rank is definable.

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)

We prove that any countable simple unidimensional theory T is supersimple, under the additional assumptions that T eliminates hyperimaginaries and that the $D_\phi-ranks$ are finite and definable.

We consider the theory P of pairs F (...) is generically free. (shrink)

In the present paper we shall prove that countable ω-categorical simple CM-trivial theories and countable ω-categorical simple theories with strong stable forking are low. In addition, we observe that simple theories of bounded finite weight are low.

Bounded PAC substructures of models of stable theory T are generalizations of bounded PAC fields and bounded PAC beautiful pairs generalize Poizat's beautiful pairs. Both notions were introduced in the authors Ph.D. thesis. In this paper, we prove that under the assumption that the PAC property is first order for T, the theory of any bounded PAC structure is simple. Moreover, if the PAC property is first order for T and T does not have the finite cover property, then the (...) theory of any bounded PAC beautiful pair is simple. We, also, give a characterization of dividing in both cases. (shrink)

A first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of the theory of proper extensions of algebraically closed fields and of the theory of separably closed fields of arbitrary imperfection degree.

Dividing chains have been used as conditions to isolate adequate subclasses of simple theories. In the first part of this paper we present an introduction to the area. We give an overview on fundamental notions and present proofs of some of the basic and well-known facts related to dividing chains in simple theories. In the second part we discuss various characterizations of the subclass of low theories. Our main theorem generalizes and slightly extends a well-known fact about the connection between (...) dividing chains and Morley sequences (in our case: independent sequences). Moreover, we are able to give a proof that is shorter than the original one. This result motivates us to introduce a special property of formulas concerning independent dividing chains: For any dividing chain there exists an independent dividing chain of the same length. We study this property in the context of low, short and ω -categorical simple theories, outline some examples and define subclasses of low and short theories, which imply this property. The results give rise to further studies of the relationships between some subclasses of simple theories. (shrink)

We prove that the definition of supersimplicity in metric structures from [7] is equivalent to an a priori stronger variant. This stronger variant is then used to prove that if T is a supersimple Hausdorff cat then so is its theory of lovely pairs.

We study groups definable in tame expansions of ω-stable theories. Assuming several tameness conditions, we obtain structural theorems for groups definable and interpretable in these expansions. As our main example, by characterizing independence in the pair, where K is an algebraically closed field and G is a multiplicative subgroup of K× with the Mann property, we show that the pair satisfies the assumptions. In particular, this provides a characterization of definable and interpretable groups in in terms of algebraic groups in (...) K and interpretable groups in G. Furthermore, we compute the Morley rank and the U-rank in and both ranks agree. (shrink)

In this paper, we study an algebraically closed field \ expanded by two unary predicates denoting an algebraically closed proper subfield k and a multiplicative subgroup \. This will be a proper expansion of algebraically closed field with a group satisfying the Mann property, and also pairs of algebraically closed fields. We first characterize the independence in the triple \\). This enables us to characterize the interpretable groups when \ is divisible. Every interpretable group H in \\) is, up to (...) isogeny, an extension of a direct sum of k-rational points of an algebraic group defined over k and an interpretable abelian group in \ by an interpretable group N, which is the quotient of an algebraic group by a subgroup \, which in turn is isogenous to a cartesian product of k-rational points of an algebraic group defined over k and an interpretable abelian group in \. (shrink)

A structure M is pregeometric if the algebraic closure is a pregeometry in all structures elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of Lascar U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding an integral domain, while not pregeometric in general, do have a unique existential matroid. Generalising previous results by van den Dries, we define dense elementary pairs of structures expanding an (...) integral domain and with an existential matroid, and we show that the corresponding theories have natural completions, whose models also have a unique existential matroid. We also extend the above result to dense tuples of structures. (shrink)

The weak non-finite cover property (wnfcp) was introduced in [1] in connection with "axiomatizability" of lovely pairs of models of a simple theory. We find a combinatorial condition on a simple theory equivalent to the wnfcp, yielding a direct proof that the non-finite cover property implies the wnfcp, and that the wnfcp is preserved under reducts. We also study the question whether the wnfcp is preserved when passing from a simple theory T to the theory TP of lovely pairs of (...) models of T (true in the stable case). While the question remains open, we show, among other things, that if (for a T with the wnfcp) TP is low, then TP has the wnfcp. To study this question, we describe "double lovely pairs", and, along the way, we develop the notion of a "lovely n-tuple" of models of a simple theory, which is an analogue of the notion of a beautiful tuple of models of stable theories [2]. (shrink)

We prove the NTP1 property of a geometric theory T is inherited by theories of lovely pairs and H‐structures associated to T. We also provide a class of examples of nonsimple geometric NTP1 theories.

Let T be a complete geometric theory and let $$T_P$$ T P be the theory of dense pairs of models of T. We show that if T is superrosy with "Equation missing"-rank 1 then $$T_P$$ T P is superrosy with "Equation missing"-rank at most $$\omega $$ ω.

We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: "weak one-basedness", absence of type definable "almost quasidesigns", and "generic linearity". Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over (...) a finite field. (shrink)

We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.

Based on the work done in [][] in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking‐independent elements that is dense inside a partial type, which we call H‐structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again H‐structures. We prove that under these assumptions the expansion is supersimple and (...) characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one‐basedness and CM‐triviality. In the one‐based case, when T has SU‐rank and the SU‐rank is continuous, we take to be the type of elements of SU‐rank and we describe a natural “geometry of generics modulo H” associated with such expansions and show it is modular. (shrink)

We study the theory of lovely pairs of geometric structures, in particular o-minimal structures. We use the pairs to isolate a class of geometric structures called weakly locally modular which generalizes the class of linear structures in the settings of SU-rank one theories and o-minimal theories. For o-minimal theories, we use the Peterzil–Starchenko trichotomy theorem to characterize for a sufficiently general point, the local geometry around it in terms of the thorn U-rank of its type inside a lovely pair.

We study the theory of the structure induced by parameter free formulas on a “dense” algebraically independent subset of a model of a geometric theory T. We show that while being a trivial geometric theory, inherits most of the model theoretic complexity of T related to stability, simplicity, rosiness, the NIP and the NTP2. In particular, we show that T is strongly minimal, supersimple of SU‐rank 1, has the NIP or the NTP2 exactly when has these properties. We show that (...) if T is superrosy of thorn rank 1, then so is, and that the converse holds if T satisfies acl = dcl. (shrink)

We show that if T is any geometric theory having the NTP2 then the corresponding theories of lovely pairs of models of T and of H‐structures associated to T also have the NTP2. We also prove that if T is strong then the same two expansions of T are also strong.

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.

Given a lovely pair P ≺ M of models of a simple theory T, we study the structure whose universe is P and whose relations are the traces on P of definable (in ℒ with parameters from M) sets in M. We give a necessary and sufficient condition on T (which we call weak lowness) for this structure to have quantifier-elimination. We give an example of a non-weakly-low simple theory.

The weak non-finite cover property was introduced in [1] in connection with “axiomatizability” of lovely pairs of models of a simple theory. We find a combinatorial condition on a simple theory equivalent to the wnfcp, yielding a direct proof that the non-finite cover property implies the wnfcp, and that the wnfcp is preserved under reducts. We also study the question whether the wnfcp is preserved when passing from a simple theory T to the theory TP of lovely pairs of models (...) of T. While the question remains open, we show, among other things, that if TP is low, then TP has the wnfcp. To study this question, we describe “double lovely pairs”, and, along the way, we develop the notion of a “lovely n-tuple” of models of a simple theory, which is an analogue of the notion of a beautiful tuple of models of stable theories [2]. (shrink)