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  "body": "\n## Definition\n\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]], let $X$ be a set, and let $F = F_{\\mathcal{V}}(X)$ be the [[free algebra|GTS4DRRMZR]] of $\\mathcal{V}$ on $X$.\nLet $R$ be a set of pairs $(s, t)$ of elements of $F$, called *relations* and written $s = t$.\nLet $\\theta_R$ be the [[congruence generated|QP06QH2Q7C]] by $R$ on $F$.\nThe algebra *presented* by $X$ and $R$ in $\\mathcal{V}$ is the [[quotient algebra|N5KE7FH6CX]]\n$$\\langle X \\mid R \\rangle_{\\mathcal{V}} = F / \\theta_R.$$\nA *presentation* of an algebra $A \\in \\mathcal{V}$ is a pair $(X, R)$ with an [[isomorphism|PZMGQMDN25]] $\\langle X \\mid R \\rangle_{\\mathcal{V}} \\to A$.\nThe algebra $A$ is *finitely presented* in $\\mathcal{V}$ if it has a presentation with $X$ and $R$ finite.\n\n## Remarks\n\nThe presented algebra has this universal property.\nLet $B \\in \\mathcal{V}$ and let $f \\colon X \\to B$ be a map under which each relation $s = t$ holds in $B$.\nThen there is exactly one homomorphism $\\langle X \\mid R \\rangle_{\\mathcal{V}} \\to B$ that extends $f$.\nEvery $A \\in \\mathcal{V}$ has a presentation: take $X = A$ and $R$ the kernel of the homomorphism $F_{\\mathcal{V}}(A) \\to A$ that extends the identity map.\nExample: in the variety of commutative monoids, $\\langle a \\mid a^2 = a^3 \\rangle$ is the monoid $\\set{1, a, a^2}$ with $a^3 = a^2$.\nThe relation holds there, and the universal property is checked by sending $a$ to any element $b$ with $b^2 = b^3$.\nNon-example: the same pair $(\\set{a}, \\set{a^2 = a^3})$ does not present $\\Z / 2\\Z$ as a commutative monoid, since in $\\Z / 2\\Z = \\set{1, a}$ one has $a^2 = 1$ and $a^3 = a$, so the relation $a^2 = a^3$ fails.\nThe presented algebra depends on the variety: $\\langle a, b \\mid \\ \\rangle$ is $\\N^2$ in commutative monoids and the free monoid on two letters in monoids.\n",
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