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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.\nLet $\\operatorname{Con} \\mathbf{A}$ be the set of all [[congruences|DCSXXT4SER]] on $\\mathbf{A}$.\nThe *congruence lattice* of $\\mathbf{A}$ is the partially ordered set $(\\operatorname{Con} \\mathbf{A}, \\subseteq)$.\nIts meet and join are\n$$\\theta \\wedge \\psi = \\theta \\cap \\psi, \\qquad \\theta \\vee \\psi = \\operatorname{Cg}(\\theta \\cup \\psi),$$\nwhere $\\operatorname{Cg}$ is the [[congruence generated by a relation|QP06QH2Q7C]].\n\n## Remarks\n\nThe join $\\theta \\vee \\psi$ is also the [[transitive closure|05J9NWX07X]] of $\\theta \\cup \\psi$, that is, the union of the relations $\\theta \\circ \\psi$, $\\theta \\circ \\psi \\circ \\theta$, and so on.\n$\\operatorname{Con} \\mathbf{A}$ is a [[complete lattice|D7A0ZGZT5S]] with least element $\\Delta_A$ and greatest element $\\nabla_A = A \\times A$.\nIt is an [[algebraic lattice|3J0BTXSCVN]], and its [[compact elements|9F7TYES4KE]] are the finitely generated congruences $\\operatorname{Cg}(R)$ with $R$ finite.\n\nExample: for the group $\\Z/6\\Z$, the congruences correspond to the subgroups $\\set{0}$, $\\set{0, 3}$, $\\set{0, 2, 4}$ and $\\Z/6\\Z$.\nSo $\\operatorname{Con}(\\Z/6\\Z)$ has four elements, with two incomparable elements in the middle.\n\nNon-example: the union of two congruences need not be their join.\nOn the group $\\Z$, the union of congruence modulo $2$ and modulo $3$ contains $(0, 2)$ and $(2, 5)$ but not $(0, 5)$, so it is not transitive.\n",
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    "name": "Congruence lattice",
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