---
tag: 2TQD8NMG9P
kind: concept
name: Generated subalgebra
msc: [08A30, 03C07]
specializes: [676H7EGTKJ]
sources:
  - url: https://en.wikipedia.org/wiki/Substructure_(mathematics)
---

## Summary

The subalgebra generated by a subset $X$ of an [[algebra|QJGPT2W0FT]] is the smallest [[subalgebra|676H7EGTKJ]] that contains $X$.
Its elements are the values of all terms in the operations applied to elements of $X$ and to the constants.
The [[subgroup generated by a subset|0RP7VWQ0YN]] and the [[ideal generated by a subset|0GSW76Q14N]] are special cases of this idea.
The map that sends $X$ to the subalgebra it generates is a [[closure operator|2HJGRABSPZ]], and an algebra generated by a finite set is [[finitely generated|WR4WF1K2KW]].

## Explanations

### L1 human

Start with some elements, then apply the operations to them again and again, and collect everything you get.
The result is the generated subalgebra: it contains the starting elements and is closed under the operations, and nothing smaller has both properties.
In the integers with addition, negation and $0$, starting from $4$ and $6$ you get $10$, $2 = 6 - 4$, and then all multiples of $2$.
So $\set{4, 6}$ generates the even integers.