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  "body": "\n## Definition\n\nLet $\\mathbf{A}$ be an [[algebra|QJGPT2W0FT]] with universe $A$.\nA *subuniverse* of $\\mathbf{A}$ is a subset of $A$ closed under all fundamental operations, as for a [[subalgebra|7CHG6SD365]].\nLet $\\operatorname{Sub} \\mathbf{A}$ be the set of all subuniverses of $\\mathbf{A}$.\nThe *subalgebra lattice* of $\\mathbf{A}$ is the partially ordered set $(\\operatorname{Sub} \\mathbf{A}, \\subseteq)$.\nIts meet and join are\n$$B \\wedge C = B \\cap C, \\qquad B \\vee C = \\operatorname{Sg}(B \\cup C),$$\nwhere $\\operatorname{Sg}$ is the [[generated subuniverse|2TQD8NMG9P]].\n\n## Remarks\n\nThe intersection of any family of subuniverses is a subuniverse, so $\\operatorname{Sub} \\mathbf{A}$ is a [[complete lattice|D7A0ZGZT5S]].\nIt is an [[algebraic lattice|3J0BTXSCVN]].\nIts [[compact elements|9F7TYES4KE]] are the subuniverses $\\operatorname{Sg}(X)$ with $X$ finite, that is, the [[finitely generated|WR4WF1K2KW]] ones.\n\nExample: for the group $(\\Z, +, -, 0)$ the subuniverses are the sets $n\\Z$ with $n \\ge 0$.\nHere $m\\Z \\subseteq n\\Z$ if and only if $n$ divides $m$, and $m\\Z \\vee n\\Z = d\\Z$ with $d = \\gcd(m, n)$.\n\nNon-example: the lattice of all subsets of $\\Z$ is not the subalgebra lattice of this group.\nThe subset $\\set{1}$ is not closed under addition, so it is not a subuniverse.\n\nWhen the signature has no constant symbols, the empty set is a subuniverse and is the least element of $\\operatorname{Sub} \\mathbf{A}$.\nSome authors exclude it and call only nonempty subuniverses subalgebras.\n",
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