Invertible Element of Associative Structure is Cancellable

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Theorem

Let $\struct {S, \circ}$ be an algebraic structure where $\circ$ is associative.

Let $\struct {S, \circ}$ have an identity element $e_S$.

An element of $\struct {S, \circ}$ which is invertible is also cancellable.


Corollary

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

An element of $\struct {S, \circ}$ which is invertible is also cancellable.


Proof

Let $a \in S$ be invertible.

Suppose $a \circ x = a \circ y$.


Then:

\(\ds x\) \(=\) \(\ds e_S \circ x\) Definition of Identity Element
\(\ds \) \(=\) \(\ds \paren {a^{-1} \circ a} \circ x\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds a^{-1} \circ \paren {a \circ x}\) Associativity of $\circ$
\(\ds \) \(=\) \(\ds a^{-1} \circ \paren {a \circ y}\) by hypothesis
\(\ds \) \(=\) \(\ds \paren {a^{-1} \circ a} \circ y\) Associativity of $\circ$
\(\ds \) \(=\) \(\ds e_S \circ y\) Definition of Inverse Element
\(\ds \) \(=\) \(\ds y\) Definition of Identity Element


A similar argument shows that $x \circ a = y \circ a \implies x = y$.

$\blacksquare$


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