Category:Definitions/Identity Elements

This category contains definitions related to Identity Elements.
Related results can be found in Category:Identity Elements.

An element $e \in S$ is called an identity (element) if and only if it is both a left identity and a right identity:

$\forall x \in S: x \circ e = x = e \circ x$

In Identity is Unique it is established that an identity element, if it exists, is unique within $\struct {S, \circ}$.

Thus it is justified to refer to it as the identity (of a given algebraic structure).

This identity is often denoted $e_S$, or $e$ if it is clearly understood what structure is being discussed.