---
tag: F1M51B4MDG
kind: concept
name: Equational theory
msc: [08B05, 03C05]
sources:
  - url: https://en.wikipedia.org/wiki/Equational_logic
---

## Summary

The equational theory of a class of algebras is the set of all [[identities|PKD440G2XW]] that hold in every member of the class.
A set of identities is called an equational theory when it arises in this way.
Equational theories correspond one to one to varieties of algebras, with larger theories giving smaller varieties.

## Explanations

### L1 human

Collect every law that holds in all groups: associativity, $x \cdot e \approx x$, $(x y)^{-1} \approx y^{-1} x^{-1}$, and infinitely many more.
This collection is the equational theory of groups.
It does not contain $x y \approx y x$, because some groups are not commutative.
The equational theory of abelian groups is bigger: it contains $x y \approx y x$ and everything that follows from it.