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  "body": "\n## Definition\n\nLet $A$ be an algebra of a signature with only function symbols.\nThe algebra $A$ is *finitely based* if it has a finite [[equational basis|9Q008GMZSW]].\nThat is, there is a finite set $\\Sigma$ of [[identities|PKD440G2XW]] satisfied by $A$ such that every identity of $A$ holds in all algebras satisfying $\\Sigma$.\nA [[variety|AJNZJD8442]] is *finitely based* if it is the class of all algebras satisfying some finite set of identities.\n\n## Remarks\n\nAn algebra is finitely based exactly when the variety it generates is finitely based.\nAn algebra is finitely based exactly when its equational theory is the set of consequences of finitely many identities.\nEvery two-element algebra of finite signature is finitely based, by a theorem of Lyndon, while Lyndon also found a finite algebra that is not finitely based.\n\nExample: the group $(\\Z, +, -, 0)$ is finitely based.\nIt generates the variety of abelian groups, so the identities $x + (y + z) \\approx (x + y) + z$, $x + 0 \\approx x$, $x + (-x) \\approx 0$ and $x + y \\approx y + x$ form an equational basis.\n\nNon-example: the six-element Brandt monoid $B_2^1$ is not finitely based as a semigroup, by a theorem of Perkins.\nIt consists of the four $2 \\times 2$ matrix units, the zero matrix and the identity matrix, under multiplication.\n",
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