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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$ having at least two elements, and let $\\Delta_A = \\set{(a, a) : a \\in A}$.\nThe algebra $\\mathbf{A}$ is *subdirectly irreducible* when the set of [[congruences|DCSXXT4SER]] on $\\mathbf{A}$ different from $\\Delta_A$ has a least element.\nThis least element is called the *monolith* of $\\mathbf{A}$.\n\n## Remarks\n\nEquivalently, $\\mathbf{A}$ is subdirectly irreducible when, for every subdirect embedding $\\alpha \\colon \\mathbf{A} \\to \\prod_{i \\in I} \\mathbf{A}_i$, some composite $\\pi_i \\circ \\alpha$ is an isomorphism.\nThis is the form that gives the name, through [[subdirect products|88WN3AD2VQ]].\nIn terms of the [[congruence lattice|REE07CSXHR]]: the intersection of all congruences other than $\\Delta_A$ is not $\\Delta_A$.\n\nExample: the group $\\Z/4\\Z$ is subdirectly irreducible.\nIts congruences come from the subgroups $\\set{0} \\subset \\set{0, 2} \\subset \\Z/4\\Z$, which form a chain, so the congruence of $\\set{0, 2}$ is the monolith.\n\nNon-example: the group $\\Z/6\\Z$ is not subdirectly irreducible.\nThe congruences modulo $2$ and modulo $3$ are both different from $\\Delta$, and their intersection is $\\Delta$, so there is no least one.\n\nThe ring $\\Z$ is not subdirectly irreducible either, since the congruences modulo the primes intersect in $\\Delta_\\Z$.\n",
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