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  "body": "\n## Summary\n\nAn algebraic lattice is a complete lattice in which every element is the supremum of the [[compact elements|9F7TYES4KE]] below it.\nLattices of subgroups, of ideals, of subspaces and of congruences of algebras are algebraic, which explains the name.\nThe [[Birkhoff–Frink theorem|DZGNHKE3X1]] shows that the algebraic lattices are exactly the lattices of subalgebras of algebras.\nAlgebraic lattices are also called compactly generated lattices.\n\n## Explanations\n\n### L1 human\n\nIn the lattice of subgroups of a group, every subgroup is the union of its finitely generated subgroups.\nAn algebraic lattice is a complete lattice with the same feature: every element is built from \"finite\" elements below it.\nThe power set of a set is algebraic, since every set is the union of its finite subsets.\nThe lattice of closed subsets of the interval $[0, 1]$ is not algebraic.\nIts only compact element is the empty set, so no nonempty closed set is a join of compact elements.\n",
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