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  "body": "\n## Summary\n\nA subdirectly irreducible algebra is a nontrivial [[algebra|QJGPT2W0FT]] that cannot be split as a [[subdirect product|88WN3AD2VQ]] in a nontrivial way.\nIn every subdirect representation of it, one of the projections is already an isomorphism.\nEquivalently, among its [[congruences|DCSXXT4SER]] other than equality there is a smallest one, called the monolith.\nBy [[Birkhoff's subdirect representation theorem|CZF5696TPY]], these algebras are the building blocks of all algebras, and a variety is determined by its subdirectly irreducible members.\n\n## Explanations\n\n### L1 human\n\nAny nontrivial gluing of a subdirectly irreducible algebra, however small, must glue a certain fixed pair of elements.\nSo the algebra cannot be recovered from several smaller pictures of it, since each picture loses that pair.\nEvery [[simple algebra|9CJCXNYNK8]] is subdirectly irreducible.\nThe group $\\Z/4\\Z$ is subdirectly irreducible but not simple: every nontrivial gluing identifies $0$ and $2$.\nThe group $\\Z/6\\Z$ is not, because it is the product of $\\Z/2\\Z$ and $\\Z/3\\Z$.\n",
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