---
tag: T0TNG7AGXT
kind: concept
name: Subalgebra lattice
msc: [08A30, 06B23]
sources:
  - url: https://en.wikipedia.org/wiki/Compact_element
---

## Summary

The subalgebra lattice of an [[algebra|QJGPT2W0FT]] is the set of all its subuniverses, ordered by inclusion.
It is a [[complete lattice|D7A0ZGZT5S]], and in fact an [[algebraic lattice|3J0BTXSCVN]] whose [[compact elements|9F7TYES4KE]] are the finitely generated subuniverses.
It generalizes the [[lattice of subgroups|0B8BW4YWTF]] of a group.
By the [[Birkhoff–Frink theorem|YFT50501A0]], every algebraic lattice arises in this way.

## Explanations

### L1 human

Collect all the subsets of an algebra that are closed under its operations, and order them by inclusion.
Two of them always have a largest common lower bound, their intersection, and a smallest common upper bound, the subalgebra generated by their union.
So they form a lattice, which records how the algebra is built from its pieces.
For the group $\Z/4\Z$ the lattice is a chain of three subgroups: $\set{0}$, $\set{0, 2}$ and the whole group.