---
tag: PZMGQMDN25
kind: concept
name: Isomorphism
msc: [08A05, 03C07]
specializes: [KBXSZC6BV1]
sources:
  - url: https://en.wikipedia.org/wiki/Isomorphism
---

## Summary

An isomorphism between two [[structures|983GZBQYT4]] of the same signature is a bijection that preserves and reflects all operations, constants and relations.
Two structures are isomorphic when an isomorphism between them exists; then they have the same properties expressible in their signature.
[[Group isomorphisms|08MG7H8JW5]], [[ring isomorphisms|0KN2S81K72]], [[linear isomorphisms|0B8T10CE7N]] and [[order isomorphisms|0D3WA36K2Q]] are special cases.
An isomorphism of a structure with itself is an [[automorphism|DSQ17TNW1C]].

## Explanations

### L1 human

Two structures are isomorphic when they are the same up to renaming of their elements.
The even integers with addition and the integers with addition are isomorphic: the map $x \mapsto 2x$ renames each integer as an even integer and turns sums into sums.
The integers and the rational numbers with addition are not isomorphic, although both are infinite: in $\Q$ every element is twice another, and in $\Z$ the element $1$ is not.