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  "body": "\n## Statement\n\nLet $L$ be an [[algebraic lattice|3J0BTXSCVN]].\nThen there is an algebra $A$, a set with a family of finitary operations, with the following property.\nThe lattice $\\operatorname{Con} A$ of congruences of $A$, ordered by inclusion, is isomorphic to $L$.\n\n## Remarks\n\nA congruence of an algebra $A$ is an equivalence relation on $A$ that is compatible with every operation of $A$.\nFor every algebra $A$, the lattice $\\operatorname{Con} A$ is algebraic: its [[compact elements|9F7TYES4KE]] are the finitely generated congruences, and every congruence is the join of the congruences generated by single pairs.\nSo the theorem says that the algebraic lattices are exactly the congruence lattices of algebras, up to isomorphism.\nThe proof is an explicit construction: the operations of $A$ are chosen so that the finitely generated congruences of $A$ behave like the compact elements of $L$.\nThe algebra produced may have infinitely many operations; the representation of finite lattices as congruence lattices of finite algebras is a different and harder question.\n\n## Explanations\n\n### L1 human\n\nEvery algebra, such as a group, a ring or a lattice, has congruences, which are the ways to glue its elements together compatibly with the operations.\nThe congruences of one algebra form a lattice with a special property called algebraic.\nThe theorem says that this is the only restriction: every algebraic lattice arises in this way from some algebra.\n\n### L2 human\n\nIn a course on universal algebra this theorem closes the description of congruence lattices.\nIt sits next to the [[Birkhoff–Frink theorem|DZGNHKE3X1]], which says the same thing for lattices of subalgebras.\nFor special classes of algebras the question changes: the congruence lattice of a lattice is always distributive, and which distributive algebraic lattices occur there is the congruence lattice problem.\n",
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