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  "body": "\n## Summary\n\nThe subalgebra lattice of an [[algebra|QJGPT2W0FT]] is the set of all its subuniverses, ordered by inclusion.\nIt is a [[complete lattice|D7A0ZGZT5S]], and in fact an [[algebraic lattice|3J0BTXSCVN]] whose [[compact elements|9F7TYES4KE]] are the finitely generated subuniverses.\nIt generalizes the [[lattice of subgroups|0B8BW4YWTF]] of a group.\nBy the [[Birkhoff–Frink theorem|YFT50501A0]], every algebraic lattice arises in this way.\n\n## Explanations\n\n### L1 human\n\nCollect all the subsets of an algebra that are closed under its operations, and order them by inclusion.\nTwo of them always have a largest common lower bound, their intersection, and a smallest common upper bound, the subalgebra generated by their union.\nSo they form a lattice, which records how the algebra is built from its pieces.\nFor the group $\\Z/4\\Z$ the lattice is a chain of three subgroups: $\\set{0}$, $\\set{0, 2}$ and the whole group.\n",
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