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  "body": "\n## Definition\n\nLet $(M, \\cdot)$ be a [[magma|9689SFPVE9]].\nIt is a *unital magma* if there is an element $e \\in M$ with $e \\cdot x = x$ and $x \\cdot e = x$ for all $x \\in M$.\nSuch an $e$ is an [[identity element|07WYQPB8E3]] of $M$.\n\n## Remarks\n\nThe identity is unique: if $e$ and $e'$ are identities, then $e = e \\cdot e' = e'$.\n\nExample: the set $\\set{e, a, b}$ with $e \\cdot x = x \\cdot e = x$ for all $x$ and $u \\cdot v = e$ for $u, v \\in \\set{a, b}$.\nBoth identity equations hold by construction, so this is a unital magma.\nIt is not associative: $(a \\cdot a) \\cdot b = e \\cdot b = b$, but $a \\cdot (a \\cdot b) = a \\cdot e = a$.\n\nNon-example: $(\\Z, -)$ has no identity.\nIf $e$ were an identity, then $e - x = x$ for all $x \\in \\Z$, so $e = 2x$ for every $x$, which is impossible.\n\nIn Mathlib the class `MulOneClass` is a unital magma written multiplicatively.\n",
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