Semi-equational theories

TL;DR

This paper introduces semi-equational and weakly semi-equational theories, extending equationality concepts to NIP theories, and explores their properties, connections to distality, and criteria for expansions.

Contribution

It generalizes equationality to NIP theories, links semi-equationality to distality, and provides criteria for expansions of distal structures.

Findings

Certain trees are semi-equational.

Algebraically closed valued fields are not weakly semi-equational.

A criterion for weak semi-equationality of expansions of distal structures.

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.