We define the interpolative fusion T∪∗ of a family i∈I of first-order theories over a common reduct T∩, a notion that generalizes many examples of random or generic structures in the model-theo...

We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type p over a set B does not divide over $C\subseteq B$, then no extension of p to a complete type over $\operatorname {acl}(B)$ divides over C. Two of our examples are also the first known theories where all sets are extension bases for nonforking, but forking and dividing differ for complete types (answering a question of Adler). One example is an $\mathrm {NSOP}_1$ theory (...) with a complete type that forks, but does not divide, over a model (answering a question of d’Elbée). Moreover, dividing independence fails to imply M-independence in this example (which refutes another folklore claim). In addition to these counterexamples, we summarize various related properties of dividing that are still true. We also address consequences for previous literature, including an earlier unpublished result about forking and dividing in free amalgamation theories, and some claims about dividing in the theory of generic $K_{m,n}$ -free incidence structures. (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)

We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups (...) and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups. (shrink)

We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum‐sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit.

An action is a pair of sets, C and S, and a function \. Rothschild and Yalcin gave a simple axiomatic characterization of those actions arising from set intersection, i.e. for which the elements of C and S can be identified with sets in such a way that elements of S act on elements of C by intersection. We introduce and axiomatically characterize two natural classes of actions which arise from set intersection and union. In the first class, the \-actions, (...) each element of S is identified with a pair of sets \\), which act on a set c by intersection with \ and union with \. In the second class, the \-biactions, each element of S is labeled as an intersection or a union, and acts accordingly on C. We give intuitive examples of these actions, one involving conversations and another a university’s changing student body. The examples give some motivation for considering these actions, and also help give intuitive readings of the axioms. The class of \-actions is closely related to a class of single-sorted algebras, which was previously treated by Margolis et al., albeit in another guise, and we note this connection. Along the way, we make some useful, though very general, observations about axiomatization and representation problems for classes of algebras. (shrink)

This paper explores relational syllogistic logics, a family of logical systems related to reasoning about relations in extensions of the classical syllogistic. These are all decidable logical systems. We prove completeness theorems and complexity results for a natural subfamily of relational syllogistic logics, parametrized by constructors for terms and for sentences.