= Identity element
{title2=$e\ast a=a\ast e=a$}
{wiki}
An <identity element> $e$ for a <binary operation> satisfies $e\ast a=a\ast e=a$ for every element $a$ of the underlying <set>. It is a two-sided identity: merely satisfying one of the two equations is insufficient. It is unique when it exists, since two candidates satisfy $e=e\ast e'=e'$. This uniqueness does not require the <associative property>.
Zero is the <identity element> for <addition> of <integers>, one for <multiplication> of <integers>, and the <identity function> for <function composition> of <self-maps>. The <empty word> is the <identity element> for <concatenation>. An <associative operation> with an <identity element> makes a <monoid>; <inverse elements> are an additional requirement for a <group>.
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