---
tag: MEC3ZMC4F1
kind: concept
name: Elementary class
msc: ["03C52", "03C07"]
sources:
  - url: https://en.wikipedia.org/wiki/Elementary_class
same_as:
  wikidata: Q1327762
  wikipedia: Elementary class
---

## Summary

An elementary class is the class of all [[models|MVCH6N9SNW]] of a first-order [[theory|92SWXST1ZM]] in a fixed signature.
Groups, rings, fields of characteristic $0$ and dense linear orders are elementary classes.
An elementary class is closed under [[elementary equivalence|P3B1V182N4]] and under ultraproducts.
By the [[compactness theorem|1NR3HAWG9D]], many natural classes, such as torsion groups, are not elementary.
Universal classes, universal Horn classes, quasivarieties and varieties are elementary classes with axioms of special shapes.

## Explanations

### L1 human

Pick a list of first-order axioms, such as the group axioms.
The elementary class of the list is everything that satisfies all of them.
First-order axioms may quantify over elements but not over sets of elements or over natural numbers.
So "every element has finite order" cannot be written as one first-order axiom, and torsion groups are not an elementary class.