Inverse of Inverse/Monoid

Theorem

Let $\struct {S, \circ}$ be a monoid.

Let $x \in S$ be invertible, and let its inverse be $x^{-1}$.


Then $x^{-1}$ is also invertible, and:

$\paren {x^{-1} }^{-1} = x$


Proof

By Inverse in Monoid is Unique, any inverse of $x$ is unique, and can be denoted $x^{-1}$.

From Inverse of Inverse in General Algebraic Structure:

$x^{-1}$ is invertible and its inverse is $x$.


That is:

$\paren {x^{-1} }^{-1} = x$

$\blacksquare$


Sources

  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses: Theorem $4.3$
  • 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids
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