Definition:Inverse (Abstract Algebra)/Left Inverse

This page is about Left Inverse Element in the context of Abstract Algebra. For other uses, see Left Inverse.

Definition

Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$.

An element $x_L \in S$ is called a left inverse of $x$ if and only if:

$x_L \circ x = e_S$

Also see

• Definition:Right Inverse Element
• Definition:Inverse Element

• Left Inverse for All is Right Inverse

• Results about inverse elements can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Exercise $4.9$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inverse: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inverse: 2.