---
tag: 2JFAZSGB6J
kind: concept
name: Countably generated structure
msc: [08A30, 03C07]
sources:
  - url: https://en.wikipedia.org/wiki/Substructure_(mathematics)
---

## Summary

A structure is countably generated when some [[countable|01TQB88H4C]] subset of it generates the whole structure under its operations.
It is the countable analogue of a [[finitely generated algebra|WR4WF1K2KW]].
In a countable signature, a structure is countably generated exactly when its universe is countable.

## Explanations

### L1 human

Start with a list of elements, one for each natural number, and close it up under the operations.
If this reaches every element, the structure is countably generated.
The rational numbers with addition, negation and $0$ are countably generated, by the list of all fractions $1/n$.
They are not finitely generated, so the countable notion is really weaker.
The real numbers with the same operations are not countably generated, since closing a countable set under these operations gives only a countable set.