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  "body": "\n## Summary\n\nThe congruence lattice of an [[algebra|QJGPT2W0FT]] is the set of all its [[congruences|DCSXXT4SER]], ordered by inclusion.\nIt is an [[algebraic lattice|3J0BTXSCVN]], and it records all the [[quotient algebras|N5KE7FH6CX]] of the algebra and how they relate.\nMany properties of algebras and varieties, such as simplicity, subdirect irreducibility and congruence modularity, are properties of this lattice.\nBy a theorem of Grätzer and Schmidt, every algebraic lattice is isomorphic to the congruence lattice of some algebra.\n\n## Explanations\n\n### L1 human\n\nEach congruence is one way of gluing elements of an algebra together.\nSome gluings are finer than others, and comparing them by inclusion gives a lattice.\nAt the bottom is \"glue nothing\", and at the top is \"glue everything\".\nFor a group, this lattice is the same as the lattice of normal subgroups.\nFor the group $\\Z/6\\Z$ it has four elements, matching the subgroups $\\set{0}$, $\\set{0, 3}$, $\\set{0, 2, 4}$ and the whole group.\n",
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