Definition:Inverse (Abstract Algebra)
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Definition
Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.
Left Inverse
An element $x_L \in S$ is called a left inverse of $x$ if and only if:
- $x_L \circ x = e_S$
Right Inverse
An element $x_R \in S$ is called a right inverse of $x$ if and only if:
- $x \circ x_R = e_S$
Inverse
The element $y$ is an inverse of $x$ if and only if:
- $y \circ x = e_S = x \circ y$
that is, if and only if $y$ is both:
- a left inverse of $x$
and:
- a right inverse of $x$.