Invertible Element of Associative Structure is Cancellable
Jump to navigation
Jump to search
 It has been suggested that this page or section be merged into Invertible Element of Monoid is Cancellable. (Discuss) |
Theorem
Let $\left({S, \circ}\right)$ be an algebraic structure where $\circ$ is associative.
Let $\left({S, \circ}\right)$ have an identity element $e_S$.
An element of $\left({S, \circ}\right)$ which is invertible is also cancellable.
Corollary
Let $\left({S, \circ}\right)$ be a monoid whose identity is $e_S$.
An element of $\left({S, \circ}\right)$ which is invertible is also cancellable.
Proof
Let $a \in S$ be invertible.
Suppose $a \circ x = a \circ y$.
Then:
| \(\ds x\) | \(=\) | \(\ds e_S \circ x\) | Behaviour of Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left({a^{-1} \circ a}\right) \circ x\) | Behaviour of Inverse | |||||||||||
| \(\ds \) | \(=\) | \(\ds a^{-1} \circ \left({a \circ x}\right)\) | Associativity of $\circ$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds a^{-1} \circ \left({a \circ y}\right)\) | By Hypothesis | |||||||||||
| \(\ds \) | \(=\) | \(\ds \left({a^{-1} \circ a}\right) \circ y\) | [Definition:Associative Operation | |||||||||||
| \(\ds \) | \(=\) | \(\ds e_S \circ y\) | Behaviour of Inverse | |||||||||||
| \(\ds \) | \(=\) | \(\ds y\) | Behaviour of Identity |
A similar argument shows that $x \circ a = y \circ a \implies x = y$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 7$: Theorem $7.1$