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.

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)

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.

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 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 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 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 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 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.

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)

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 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.

We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterizes linearity in the setting of geometric þ-rank 1structures and that lovely pairs of weakly one-based geometric þ-rank 1 structures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite-dimensional vector spaces over division rings.

We characterize þ-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.