Many notions in model theory base their intuition on phenomenas in algebraically closed fields. So does the notion of equationality which can be understood as an analog of noetherianity for instances of first order formulas: In algebraically closed fields, instances of first order formulas are boolean combinations of varieties, i.e. Zarriski closed sets. These are noetherian, meaning that an infinite intersection of varieties is already given by a finite subintersection. Transferring this principle to model theory, we say that a first order formula is equational if any infinite intersection of its instances is equivalent to a finite subintersection. We then call a theory equational, if any formula is the boolean combination of equational formulas. An easy proof shows that any equational theory is necessarily stable. The converse question is more complex. Until recently, the only known natural example of a stable, non-equational theory was given by the non-abelian free group. The proof of Zlil Sela herefor relies on deep geometric tools and was not accessible to the community of model theorists. We will present a new criterion for the non-equationality of a theory, which yields a short elementary proof of the non-equationality of the free group and generalizes to the larger class of free products of stable groups. That indicates that the difference between equational and stable theories is much larger then previously assumed.
This is joint work with Rizos Sklinos.