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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of algebras.\nThe variety $\\mathcal{V}$ is *minimal* if both of these hold:\n\n1. $\\mathcal{V}$ contains an algebra with more than one element.\n2. Every [[subvariety|4YSCK1DF26]] of $\\mathcal{V}$ is either $\\mathcal{V}$ itself or the trivial variety of one-element algebras.\n\n## Remarks\n\nExample: the variety of Boolean algebras is minimal.\nEvery nontrivial Boolean algebra contains the two-element Boolean algebra $\\mathbf{2}$ as a subalgebra, and $\\mathbf{2}$ generates the variety of all Boolean algebras.\nSo a subvariety with a nontrivial member contains $\\mathbf{2}$ and hence every Boolean algebra.\nNon-example: the variety of all groups is not minimal.\nIts subvariety of abelian groups contains $\\Z/2\\Z$, so it is not trivial, and it is a proper subvariety.\nIn the [[lattice of subvarieties|GJAYCVYZYF]] of a variety, the minimal varieties are exactly the atoms, the elements that cover the trivial variety.\n\n## Explanations\n\n### L1 human\n\nA minimal variety cannot be cut down further by identities without becoming trivial.\n",
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