In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable.

We introduce the notion of normal hyperimaginary and we develop its basic theory. We present a new proof of the Lascar-Pillay theorem on bounded hyperimaginaries based on properties of normal hyperimaginaries. However, the use of the Peter–Weyl theorem on the structure of compact Hausdorff groups is not completely eliminated from the proof. In the second part, we show that all closed sets in Kim-Pillay spaces are equivalent to hyperimaginaries and we use this to introduce an approximation of φ-types for (...) bounded hyperimaginaries. (shrink)

This is an expository note on the Lascar group. We also study the Lascar group over hyperimaginaries and make some new observations on the strong types over those. In particular, we show that in a simple theory $\operatorname{Ltp}\equiv\operatorname{stp}$ in real context implies that for hyperimaginary context.

We solve two problems from a work of Haskel and Pillay concerning maximal stable quotients of groups ∧-definable in NIP theories. The first result says that if G is a ∧-definable group in a distal theory, then Gst=G00 (where Gst is the smallest ∧-definable subgroup with G∕Gst stable, and G00 is the smallest ∧-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion Theq. The second (...) result is an example of a group G definable in a nondistal NIP theory for which G=G00, but Gst is not an intersection of definable groups. Our example is a saturated extension of (R,+,[0,1]). Moreover, we make some observations on the question whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyperdefinable sets involving continuous logic. (shrink)

We continue [2], developing simplicity in the framework of compact abstract theories. Due to the generality of the context we need to introduce definitions which differ somewhat from the ones use in first order theories. With these modified tools we obtain more or less classical behaviour: simplicity is characterized by the existence of a certain notion of independence, stability is characterized by simplicity and bounded multiplicity, and hyperimaginary canonical bases exist.

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]. (shrink)

A Steiner triple system (STS) is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite STS has a Fraïssé limit M_F. Here, we show that the theory T of M_F is the model completion of the theory of STSs. We also prove that T is not small and it has quantifier elimination, (...) TP2, NSOP1, elimination of hyperimaginaries and weak elimination of imaginaries. (shrink)

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)

Journal of Mathematical Logic, Ahead of Print. Given a structure [math] and a stably embedded [math]-definable set [math], we prove tameness preservation results when enriching the induced structure on [math] by some further structure [math]. In particular, we show that if [math] and [math] are stable (respectively, superstable, [math]-stable), then so is the theory [math] of the enrichment of [math] by [math]. Assuming simplicity of [math], elimination of hyperimaginaries and a further condition on [math] related to the behavior of algebraic (...) closure, we also show that simplicity and NSOP1 pass from [math] to [math]. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of [math]. More generally, we show that any stable (respectively, superstable, simple, NIP, NTP2, NSOP1) countable graph can be defined in a stable (respectively, superstable, simple, NIP, NTP2, NSOP1) expansion of [math] by some unary predicate [math]. (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 study the notion of weak canonical bases in an NSOP $_{1}$ theory T with existence. Given $p=\operatorname {tp}$ where $B=\operatorname {acl}$ in ${\mathcal M}^{\operatorname {eq}}\models T^{\operatorname {eq}}$, the weak canonical base of p is the smallest algebraically closed subset of B over which p does not Kim-fork. With this aim we firstly show that the transitive closure $\approx $ of collinearity of an indiscernible sequence is type-definable. Secondly, we prove that given a total $\mathop {\smile \hskip -0.9em ^| \ (...) }^K$ -Morley sequence I in p, the weak canonical base of $\operatorname {tp}$ is $\operatorname {acl}$, if the hyperimaginary $I/\approx $ is eliminable to e, a sequence of imaginaries. We also supply a couple of criteria for when the weak canonical base of p exists. In particular the weak canonical base of p is the intersection of the weak canonical bases of all total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequences in p over B. However, while we investigate some examples, we point out that given two weak canonical bases of total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequences in p need not be interalgebraic, contrary to the case of simple theories. Lastly we suggest an independence relation relying on weak canonical bases, when T has those. The relation, satisfying transitivity and base monotonicity, might be useful in further studies on NSOP $_1$ theories. (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)

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.

1. We show that if p is a real type which is internal in a set $\sigma$ of partial types in a simple theory, then there is a type p' interbounded with p, which is finitely generated over $\sigma$ , and possesses a fundamental system of solutions relative to $\sigma$ . 2. If p is a possibly hyperimaginary Lascar strong type, almost \sigma-internal$ , but almost orthogonal to $\sigma^{\omega}$ , then there is a canonical non-trivial almost hyperdefinable polygroup which (...) multi-acts on p while fixing $\sigma$ generically. In case p is $\sigma-internal$ and T is stable, this is the binding group of p over \sigma$. (shrink)

We show that any (atomic) excellent class K can be expanded with hyperimaginaries to form an (atomic) excellent class Keq which has canonical bases. When K is, in addition, of finite U-rank, then Keq is also simple and has a full canonical bases theorem. This positive situation contrasts starkly with homogeneous model theory for example, where the eq-expansion may fail to be homogeneous. However, this paper shows that expanding an ω-stable, homogeneous class K gives rise to an excellent class, which (...) is simple if K is of finite U-rank. (shrink)

In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or in the $\omega$-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity. The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, and (...) the study of polygroups. (shrink)

Journal of Mathematical Logic, Volume 23, Issue 03, December 2023. The aim of this paper is to generalize and improve two of the main model-theoretic results of “Stable group theory and approximate subgroups” by Hrushovski to the context of piecewise hyperdefinable sets. The first one is the existence of Lie models. The second one is the Stabilizer Theorem. In the process, a systematic study of the structure of piecewise hyperdefinable sets is developed. In particular, we show the most significant properties (...) of their logic topologies. (shrink)