---
tag: 9SNTXYEEBQ
kind: concept
name: Equivalence of varieties
msc: [08B05]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Summary

Two [[varieties|AJNZJD8442]] are equivalent, or term equivalent, when each can be interpreted in the other and the two translations undo each other.
Equivalent varieties have the same algebras up to a change of the basic operations, so they have the same subalgebras, congruences and term operations.
The standard example is the equivalence between Boolean algebras and Boolean rings.

## Explanations

### L1 human

The same structure can often be described with different basic operations.
A Boolean algebra uses "or", "and" and "not", while a Boolean ring uses "exclusive or" and "and".
Each set of operations can be written in terms of the other, and translating there and back gives the original operations.
So the two descriptions are equivalent: they are two names for one kind of structure.