Isomorphism Preserves Cancellability
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Theorem
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures.
Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an isomorphism.
Then:
- $a \in S$ is cancellable in $\struct {S, \circ}$ if and only if $\map \phi a \in T$ is cancellable in $\struct {T, *}$.
Proof
Let $\struct {S, \circ}$ be an algebraic structure in which $a$ is cancellable.
From Isomorphism Preserves Left Cancellability and Isomorphism Preserves Right Cancellability:
- $a \in S$ is left cancellable in $\struct {S, \circ}$ if and only if $\map \phi a \in T$ is left cancellable in $\struct {T, *}$
and
- $a \in S$ is right cancellable in $\struct {S, \circ}$ if and only if $\map \phi a \in T$ is right cancellable in $\struct {T, *}$.
It follows that:
- $a \in S$ is cancellable in $\struct {S, \circ}$
- $\map \phi a \in T$ is cancellable in $\struct {T, *}$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.6$