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  "body": "\n## Example\n\nTake the signature with a binary operation $\\cdot$, a unary operation ${}^{-1}$ and a constant $e$.\nThe *variety of groups* is the class of algebras $(G, \\cdot, {}^{-1}, e)$ of this signature that satisfy the [[identities|PKD440G2XW]]\n$$(x \\cdot y) \\cdot z = x \\cdot (y \\cdot z), \\qquad x \\cdot e = e \\cdot x = x, \\qquad x \\cdot x^{-1} = x^{-1} \\cdot x = e.$$\nThese algebras are exactly the [[groups|G5S1M9]], with the inverse map and the identity element listed as operations.\nThe class is a [[variety|AJNZJD8442]], since it is defined by identities.\nIn this signature, a subalgebra of a group is a subgroup, a homomorphism is a group homomorphism, and a congruence corresponds to a normal subgroup.\n\n## Remarks\n\nThe choice of signature matters.\nWith the binary operation $\\cdot$ alone, the class of groups is not closed under subalgebras.\nFor example, $(\\N, +)$ is closed under $+$ inside the group $(\\Z, +)$, but it is not a group.\nSo the class of groups is not a variety in that signature.\nSubvarieties of the variety of groups include the abelian groups, defined by $x \\cdot y = y \\cdot x$, and the groups of exponent $n$, defined by $x^n = e$.\n\n## Explanations\n\n### L1 human\n\nGroups can be described entirely by equations, once the inverse and the identity element are treated as operations.\nThis makes the groups a variety, so all the general theorems about varieties apply to them.\nFor instance, free groups are the free algebras of this variety.\n",
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