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 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

• 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 $\struct {R, +, \circ}$.

• Results about inverse elements can be found here.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $4$. Groups: Exercise $13$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses
• 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