Algebraic lattice

Summary

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