Definition:Invertible Element

Definition
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 :


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

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

Also see

 * Identity is Unique
 * Inverse in Monoid is Unique


 * Definition:Unit of Ring: In the context of a ring $\struct {R, +, \circ}$, an element that is invertible in the semigroup $\struct {R, \circ}$ is called a unit of $\struct {R, +, \circ}$.