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  "body": "\n## Summary\n\nAn [[algebra|QJGPT2W0FT]] is finitely generated when some finite subset of it generates the whole algebra.\nEvery element can then be built from finitely many fixed elements by repeated use of the operations.\nFinitely generated groups, rings and lattices are all special cases.\nIn the [[subalgebra lattice|T0TNG7AGXT]], the finitely generated subalgebras are exactly the compact elements.\n\n## Explanations\n\n### L1 human\n\nPick a few elements of an algebra and apply the operations to them again and again.\nIf every element of the algebra shows up sooner or later, the algebra is finitely generated.\nFor example, the integers with addition, negation and $0$ are generated by the single element $1$.\nEvery integer is $1 + 1 + \\cdots + 1$, its negative, or $0$.\nThe rational numbers with the same operations are not finitely generated, because finitely many fractions only reach the multiples of one fixed fraction.\n",
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