---
tag: 3J0BTXSCVN
kind: concept
name: Algebraic lattice
msc: [06B23, 08A30]
specializes: [D7A0ZGZT5S]
sources:
  - url: https://en.wikipedia.org/wiki/Compact_element
---

## Summary

An algebraic lattice is a complete lattice in which every element is the supremum of the [[compact elements|9F7TYES4KE]] below it.
Lattices of subgroups, of ideals, of subspaces and of congruences of algebras are algebraic, which explains the name.
The [[Birkhoff–Frink theorem|DZGNHKE3X1]] shows that the algebraic lattices are exactly the lattices of subalgebras of algebras.
Algebraic lattices are also called compactly generated lattices.

## Explanations

### L1 human

In the lattice of subgroups of a group, every subgroup is the union of its finitely generated subgroups.
An algebraic lattice is a complete lattice with the same feature: every element is built from "finite" elements below it.
The power set of a set is algebraic, since every set is the union of its finite subsets.
The lattice of closed subsets of the interval $[0, 1]$ is not algebraic.
Its only compact element is the empty set, so no nonempty closed set is a join of compact elements.