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  "body": "\n## Statement\n\nLet $L$ be a [[complete lattice|D7A0ZGZT5S]].\nThe following are equivalent.\n\n1. $L$ is an [[algebraic lattice|3J0BTXSCVN]].\n2. $L$ is isomorphic to the lattice of subalgebras of some algebra $A$, a set with a family of finitary operations, ordered by inclusion.\n3. $L$ is isomorphic to the lattice of closed sets of an algebraic [[closure operator|2HJGRABSPZ]] $c$ on some set $X$.\n   Algebraic means that $c(S) = \\bigcup \\set{c(F) : F \\subseteq S \\text{ finite}}$ for every $S \\subseteq X$.\n\n## Remarks\n\nFrom 2 to 3: the subalgebra generated by a set is an algebraic closure operator, since each operation has finitely many arguments.\nFrom 3 to 1: in the lattice of closed sets, the [[compact elements|9F7TYES4KE]] are the closures of finite sets, and every closed set is the union of these.\nFrom 1 to 2: one equips the set $K$ of compact elements of $L$ with operations whose subalgebras are exactly the ideals of the join-semilattice $K$.\nThe map $a \\mapsto \\set{k \\in K : k \\le a}$ identifies $L$ with these ideals.\nFor example, the lattice of subgroups of a group and the lattice of ideals of a ring are algebraic.\nThe lattice of closed subsets of $[0, 1]$ is complete but not algebraic, so it is not a subalgebra lattice.\n\n## Explanations\n\n### L1 human\n\nThe subgroups of a group, the subspaces of a vector space and the ideals of a ring all form complete lattices of a special kind, called algebraic.\nThe theorem says that the lattices of \"sub-things\" of algebraic structures are exactly the algebraic lattices.\nEvery algebraic lattice is the lattice of substructures of some suitable algebra.\n\n### L2 human\n\nThe theorem explains why compactness is the right abstraction of \"finitely generated\": the compact elements of a subalgebra lattice are the finitely generated subalgebras.\nIt is the first of two representation results: the [[Grätzer–Schmidt theorem|2BZ2ZXH03B]] shows that the same lattices are also exactly the congruence lattices of algebras.\nTogether with the description of algebraic lattices as ideal lattices of join-semilattices with $0$, it gives several equivalent pictures of the same class.\n",
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