---
tag: W251JWP6PS
kind: definition
name: Finitely generated algebra
msc: [08A30, 08A05]
defines: WR4WF1K2KW
uses: [QJGPT2W0FT, 2TQD8NMG9P, 070R982G4B]
sources:
  - url: https://en.wikipedia.org/wiki/Substructure_(mathematics)
---

## Definition

Let $\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.
For $X \subseteq A$, let $\operatorname{Sg}(X)$ denote the [[subuniverse generated by|2TQD8NMG9P]] $X$.
It is the smallest subset of $A$ that contains $X$ and is closed under all fundamental operations of $\mathbf{A}$.
The algebra $\mathbf{A}$ is *finitely generated* when there is a [[finite set|070R982G4B]] $X \subseteq A$ with $\operatorname{Sg}(X) = A$.

## Remarks

Example: the group $(\Z, +, -, 0)$ is finitely generated.
Indeed $\operatorname{Sg}(\set{1})$ contains $0$, all sums $1 + \cdots + 1$, and their negatives, so it is all of $\Z$.

Non-example: the group $(\Q, +, -, 0)$ is not finitely generated.
If $X = \set{p_1/q, \dots, p_k/q}$ with a common denominator $q \ge 1$, then every element of $\operatorname{Sg}(X)$ is an integer multiple of $1/q$, so $1/(2q) \notin \operatorname{Sg}(X)$.

Every finite algebra is finitely generated, with $X = A$.
In model theory the same notion is used for structures in a signature with relation symbols, where the relations play no role in generation.