We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to Corollary 4.2 from that paper, and in the process discuss “theories with selectors.”.

We introduce and study (weakly) semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong honest definitions; demonstrate that certain trees are semi-equational, while algebraically closed valued fields are not weakly semi-equational; and obtain a general criterion for weak semi-equationality of an expansion of a distal structure by a new predicate.

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.

We present a structure theorem for superstable quasi-varieties without DOP. We show that every algebra in such a quasi-variety weakly decomposes as the product of an affine algebra and a combinational algebra, that is, it is bi-interpretable with a two sorted structure where one sort is an affine algebra, the other sort is a combinatorial algebra and the only non-trivial polynomials between the two sorts are certain actions of the affine sort on the combinatorial sort.

Several attempts have been done to distinguish “positive” information in an arbitrary first order theory, i.e., to find a well behaved class of closed sets among the definable sets. In many cases, a definable set is said to be closed if its conjugates are sufficiently distinct from each other. Each such definition yields a class of theories, namely those where all definable sets are constructible, i.e., boolean combinations of closed sets. Here are some examples, ordered by strength:Weak normality describes a (...) rather small class of theories which are well understood by now (see, for example, [P]). On the other hand, normalization is so weak that all theories, in a suitable context, are normalizable (see [HH]). Thus equational theories form an interesting intermediate class of theories. Little work has been done so far. The original work of Srour [S1, S2, S3] adopts a point of view that is closer to universal algebra than to stability theory. The fundamental definitions and model theoretic properties can be found in [PS], though some easy observations are missing there. Hrushovski's example of a stable non-equational theory, the first and only one so far, is described in the unfortunately unpublished manuscript [HS]. In fact, it is an expansion of the free pseudospace constructed independently by Baudisch and Pillay in [BP] as an example of a strictly 2-ample theory. Strong equationality, defined in [Hr], is also investigated in [HS]. (shrink)

Independently and pursuing different aims, Hrushovski and Srour (On stable non-equational theories. Unpublished manuscript, 1989) and Baudisch and Pillay (J Symb Log 65(1):443–460, 2000) have introduced two free pseudospaces that generalize the well know concept of Lachlan’s free pseudoplane. In this paper we investigate the relationship between these free pseudospaces, proving in particular, that the pseudospace of Baudisch and Pillay is a reduct of the pseudospace of Hrushovski and Srour.