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  "body": "\n## Statement\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$ and let $\\theta$ be a [[congruence|DCSXXT4SER]] on $\\mathbf{A}$.\nFor a congruence $\\psi \\supseteq \\theta$ on $\\mathbf{A}$, put\n$$\\psi/\\theta = \\set{(a/\\theta, b/\\theta) : (a, b) \\in \\psi}.$$\nThen $\\psi \\mapsto \\psi/\\theta$ is an isomorphism from the interval $\\set{\\psi \\in \\operatorname{Con} \\mathbf{A} : \\theta \\subseteq \\psi}$ of the [[congruence lattice|REE07CSXHR]] onto $\\operatorname{Con}(\\mathbf{A}/\\theta)$.\n\n## Remarks\n\nProof idea.\nLet $\\nu \\colon \\mathbf{A} \\to \\mathbf{A}/\\theta$ be the natural map of the [[quotient algebra|N5KE7FH6CX]].\nFor $\\psi \\supseteq \\theta$, the relation $\\psi/\\theta$ is a congruence, and its preimage under $\\nu \\times \\nu$ is $\\psi$ again, because $\\psi$ contains $\\theta$.\nConversely, the preimage of any congruence on $\\mathbf{A}/\\theta$ is a congruence on $\\mathbf{A}$ containing $\\theta$.\nThese two maps are inverse to each other and preserve inclusion, so they are lattice isomorphisms.\n\nExample: in the group $\\Z$, take $\\theta$ to be congruence modulo $12$.\nThe congruences above $\\theta$ are congruence modulo $d$ for the divisors $d$ of $12$, and they match the six subgroups of $\\Z/12\\Z$.\n\nThe hypothesis $\\psi \\supseteq \\theta$ is needed.\nFor $\\psi$ not containing $\\theta$, the relation $\\psi/\\theta$ can fail to be transitive, and different $\\psi$ can give the same relation.\n\n## Explanations\n\n### L1 human\n\nGluing elements of an algebra in two steps is the same as gluing them in one step.\nSo the ways to glue the quotient $\\mathbf{A}/\\theta$ further are exactly the gluings of $\\mathbf{A}$ that are coarser than $\\theta$.\nThe theorem says this matching keeps the order, so the two lattices have the same shape.\n\n### L2 human\n\nThis is the general form of the correspondence theorems for groups, rings and modules, where congruences are replaced by normal subgroups, ideals and submodules.\nTogether with the second isomorphism theorem $(\\mathbf{A}/\\theta)/(\\psi/\\theta) \\cong \\mathbf{A}/\\psi$, it shows that a quotient of a quotient is again a quotient.\nIt is used to prove that $\\mathbf{A}/\\theta$ is [[simple|9CJCXNYNK8]] exactly when $\\theta$ is a maximal proper congruence, and in the proof of [[Birkhoff's subdirect representation theorem|CZF5696TPY]].\n",
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  "name": "Correspondence theorem for congruences",
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