Coordinate recognition: General theory, Groups, and other surprises

TL;DR

This paper develops a general theory of coordinate recognition in structures, revealing its implications for reduced products, set-theoretic assumptions, and quantifier elimination, with applications to groups and other classes.

Contribution

It introduces equivalent characterizations of coordinate recognition, linking model-theoretic and set-theoretic properties, and applies these to various classes including groups.

Findings

Reduced products exhibit rigidity phenomena under coordinate recognition.

Isomorphisms between reduced products lift to original structures under set-theoretic assumptions.

Coordinate recognition implies strong quantifier elimination under mild conditions.

Abstract

A class of structures recognizes coordinates if any reduced product of structures from said class witnesses a certain kind of rigidity phenomena. We provide several equivalent characterizations of this property. This property has (at least) two remarkable consequences, one set-theoretic and one model-theoretic, for reduced products of structures of the said class. First, under appropriate set-theoretic assumptions every isomorphism between such reduced products associated with the Fr\' echet ideal lifts (modulo a finite change) to an isomorphism between products of the original structures. Second, with an additional mild assumption, it implies a strong quantifier elimination result. Of note, we show that a class recognizes coordinates if and only if an individual formula witnesses a certain syntactic property. We also consider many concrete classes of structures and determine whether or not they recognize coordinates. We place heavy emphasis on well known classes of groups, but we also discuss other classes of structures.