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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.\nThe *lattice of subvarieties* of $\\mathcal{V}$ is the set $L(\\mathcal{V})$ of all [[subvarieties|4YSCK1DF26]] of $\\mathcal{V}$, ordered by inclusion.\nIn it, the meet of a family $(\\mathcal{W}_i)_{i \\in I}$ is the intersection $\\bigcap_i \\mathcal{W}_i$.\nThe join is the smallest subvariety that contains $\\bigcup_i \\mathcal{W}_i$, which is $HSP\\left(\\bigcup_i \\mathcal{W}_i\\right)$.\n\n## Remarks\n\nAn intersection of varieties is a variety, since it is defined by the union of their identities.\nSo $L(\\mathcal{V})$ is a [[complete lattice|D7A0ZGZT5S]], with least element the trivial variety and greatest element $\\mathcal{V}$.\nThe maps $\\mathcal{W} \\mapsto \\operatorname{Id}(\\mathcal{W})$ and $T \\mapsto \\operatorname{Mod}(T)$ are inverse dual isomorphisms between $L(\\mathcal{V})$ and the [[lattice of equational theories|2R8QQTP1CE]] that contain $\\operatorname{Id}(\\mathcal{V})$.\nExample: for the variety of Boolean algebras, $L(\\mathcal{V})$ has exactly two elements, the trivial variety and $\\mathcal{V}$ itself.\nExample: for abelian groups, the subvarieties are $\\mathcal{A}_n = \\operatorname{Mod}(nx \\approx 0)$ for $n \\ge 1$ and the whole variety, and $\\mathcal{A}_n \\subseteq \\mathcal{A}_m$ if and only if $n \\mid m$.\nNon-example: the join is not the union in general, since the union of $\\mathcal{A}_2$ and $\\mathcal{A}_3$ does not contain $\\Z / 2\\Z \\times \\Z / 3\\Z \\in \\mathcal{A}_6$.\n",
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