---
tag: SCRB74SWQN
kind: concept
name: Embedding of structures
msc: [08A05, 03C07]
specializes: [KBXSZC6BV1]
sources:
  - url: https://en.wikipedia.org/wiki/Embedding
---

## Summary

An embedding of one [[structure|983GZBQYT4]] into another is an injective [[homomorphism|KBXSZC6BV1]] that also reflects the relations.
It identifies the first structure with a [[substructure|676H7EGTKJ]] of the second, up to [[isomorphism|PZMGQMDN25]].
Embeddings let one say that a structure "sits inside" another, as $\Z$ sits inside $\Q$.
[[Elementary embeddings|CGH8C8EA9N]] are the embeddings that preserve all first-order formulas.

## Explanations

### L1 human

An embedding places a copy of one structure inside another without losing any information.
The map that sends an integer $n$ to the rational number $n/1$ is an embedding of the ring of integers into the field of rational numbers.
Different elements go to different elements, sums go to sums and products go to products.
For relational structures, such as graphs, an embedding must also keep non-edges as non-edges.