Inverse Elements

Why

Is the inverse of an element function the element function of a different element?

Definition

The inverse of an element of an algebra (also called the inverse element) is the element (if it exists) whose corresponding element function under the operation is the inverse of the first element's function.

Notation

Let $(A, +)$ be an algebra. Let $a \in A$. If the inverse element for $a$ exists and is unique we denote it by $\inv{a}$. In other words $+^{\inv{a}} \circ +^{a} = \idfn{A}$