Definition:Invertible Element

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Let $\struct {S, \circ}$ be an algebraic structure which has an identity $e_S$.

If $x \in S$ has an inverse, then $x$ is said to be invertible for $\circ$.

That is, $x$ is invertible if and only if:

$\exists y \in S: x \circ y = e_S = y \circ x$

Invertible Operation

The operation $\circ$ is invertible if and only if:

$\forall a, b \in S: \exists r, s \in S: a \circ r = b = s \circ a$

Also known as

Some sources refer to an invertible element as a unit, consistent with the definition of unit of ring.

Also see

  • Results about inverse elements can be found here.