# Definition:Invertible Element

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## 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** 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

- 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 $\struct {R, +, \circ}$**.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): Chapter $4$. Groups: Exercise $13$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 4$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $2$: Examples of Groups and Homomorphisms: $2.3$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids