Semi-equational theories

Abstract

We introduce and study semi-equational and 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.

Version 2: 28 pages. This version of the article was significantly shortened for the journal publication, resulting in many details, examples and general results about semi-equationality being omitted. For the full version of the article see arXiv:2204.13790v1. Accepted to the Journal of Symbolic Logic