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Revision difference : identity element |
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| Let $G$ be a groupoid, that is a set with a binary operation $G \times G \to G$, written muliplicatively so that $(x, y) \mapsto xy$. |
An \emph{identity element} is an element, traditionally represented by the letter $e$, of a group or monoid $G$ such that $ge=eg=g$ for any $g\in G$. The identity element is unique for any given group, and its existence is guaranteed by the definition of a group. |
| An \emph{identity element} for $G$ is an element $e$ such that $ge = eg = g$ for all $g \in G$. |
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| The symbol $e$ is most commonly used for identity elements. Another common symbol for an identity element is $1$, particularly in semigroup theory (and ring theory, considering the multiplicative structure as a semigroup). |
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| Groups, monoids, and loops are classes of groupoids that, by definition, always have an identity element. |
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