Inverse of Inverse/Monoid

From ProofWiki
Jump to navigation Jump to search


Let $\left({S, \circ}\right)$ be a monoid.

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

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

$\left({x^{-1}}\right)^{-1} = x$


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:

$\left({x^{-1}}\right)^{-1} = x$