= Inverse element
{title2=$a^{-1}a=aa^{-1}=e$}
{wiki}
An <inverse element> of $a$ in a <monoid> is an element $b$ satisfying $a\ast b=b\ast a=e$, where $e$ is the <identity element>. Its notation is $a^{-1}$. Some elements of a <monoid> can lack an <inverse element>; a <group> requires one for every element. <Associativity> makes a two-sided <inverse element> unique: if $b,c$ both invert $a$, then $b=b\ast(a\ast c)=(b\ast a)\ast c=c$.
For the <full transformation monoid>, a <self-map> has an <inverse element> exactly when it is a <bijection>; that element is its <inverse function>. On $\{0,1\}$, the swap $s(0)=1,s(1)=0$ is its own <inverse element>, while a <constant function> has none because it merges the two <inputs>.
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