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  "body": "\n## Example\n\nTake the signature with two binary operations $\\vee$ and $\\wedge$.\nThe *variety of lattices* is the class of algebras $(L, \\vee, \\wedge)$ with the [[identities|PKD440G2XW]]\n$$x \\vee y = y \\vee x, \\qquad (x \\vee y) \\vee z = x \\vee (y \\vee z), \\qquad x \\vee (x \\wedge y) = x,$$\ntogether with the same three identities with $\\vee$ and $\\wedge$ exchanged.\nThese algebras are exactly the [[lattices|WH4SFAACAS]].\nThe order is recovered by $x \\le y$ if and only if $x \\vee y = y$.\nIn this order, $x \\vee y$ and $x \\wedge y$ are the join and the meet.\nThe class is a [[variety|AJNZJD8442]], since it is defined by identities.\n\n## Remarks\n\nThe idempotent laws $x \\vee x = x$ and $x \\wedge x = x$ follow from the absorption laws: $x \\vee x = x \\vee (x \\wedge (x \\vee x)) = x$.\nThe [[distributive lattices|0HM0HQCK6N]] and the [[modular lattices|QX7WZ92QAK]] form subvarieties, each defined by one more identity.\nThe class of complete lattices is not a variety in this signature.\nFor example, the rational numbers in $[0, 1]$ form a sublattice of the complete lattice $[0, 1]$, but this sublattice is not complete.\n\n## Explanations\n\n### L1 human\n\nA lattice is usually defined as an ordered set in which any two elements have a least upper bound and a greatest lower bound.\nThe same structures can be described by a short list of equations for the two operations join and meet.\nSo lattices form a variety, and the tools of universal algebra apply to them.\n",
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