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  "body": "\n## Definition\n\nLet $\\sigma$ be a signature with only function symbols and let $\\mathcal{K}$ be a class of algebras of signature $\\sigma$, with [[equational theory|F1M51B4MDG]] $\\operatorname{Id}(\\mathcal{K})$.\nA set $\\Sigma$ of [[identities|PKD440G2XW]] over $\\sigma$ is an *equational basis* of $\\mathcal{K}$ if the following hold:\n\n1. $\\Sigma \\subseteq \\operatorname{Id}(\\mathcal{K})$;\n2. every identity in $\\operatorname{Id}(\\mathcal{K})$ is satisfied by every algebra that satisfies all identities of $\\Sigma$.\n\nAn equational basis of a single algebra $A$ is an equational basis of $\\set{A}$.\n\n## Remarks\n\nEquivalently, the class of all algebras satisfying $\\Sigma$ is the [[variety|AJNZJD8442]] generated by $\\mathcal{K}$.\nBy Birkhoff's completeness theorem, clause 2 can be read syntactically: every identity of $\\mathcal{K}$ is derivable from $\\Sigma$ in equational logic.\nThe whole theory $\\operatorname{Id}(\\mathcal{K})$ is always an equational basis of $\\mathcal{K}$, so the interest lies in small bases.\n\nExample: in the signature $(\\cdot, {}^{-1}, e)$, the set $\\set{x(yz) \\approx (xy)z, \\ x e \\approx x, \\ x x^{-1} \\approx e}$ is an equational basis of the class of groups.\nClause 1 holds because every group satisfies these laws, and clause 2 holds because every algebra satisfying them is a group.\n\nNon-example: in the same signature, $\\set{x(yz) \\approx (xy)z}$ is not an equational basis of the class of groups, since clause 2 fails.\nLet $A = \\set{0, 1}$ with $a \\cdot b = 0$ for all $a, b$, with $a^{-1} = a$, and with $e = 1$.\nThen $A$ satisfies associativity, since all products are $0$, but $1 \\cdot e = 0 \\ne 1$, so $A$ does not satisfy $x e \\approx x$.\n",
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