Inverse of Inverse/Monoid
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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