Homomorphism with Cancellable Codomain Preserves Identity

Theorem

Let $\struct{S, \circ}$ and $\struct{T, *}$ be algebraic structures.

Let $\phi: \struct{S, \circ} \to \struct{T, *}$ be a homomorphism.

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

Let $\struct{T, *}$ have an identity $e_T$.

Let every element of $\struct{T, *}$ be cancellable.

Then $\map \phi {e_S}$ is the identity $e_T$.

Proof

Let $\struct{S, \circ}$ be an algebraic structure in which $\circ$ has an identity $e_S$.

Let $\struct{T, *}$ be an algebraic structure in which $*$ has an identity $e_T$.

Let $\struct{T, *}$ be such that every element is cancellable.

Let $\phi: \struct{S, \circ} \to \struct{T, *}$ be a homomorphism.

Every element of $\struct{T, *}$ is cancellable.

Suppose there is an idempotent element in $\struct{T, *}$

So from Identity is only Idempotent Cancellable Element, it must be the identity $e_T$.

Thus:

\(\displaystyle \map \phi {e_S} * \map \phi {e_S}\)

\(=\)

\(\displaystyle \map \phi {e_S \circ e_S}\)

by the morphism property of $\circ$ under $\phi$

\(\displaystyle \)

\(=\)

\(\displaystyle \map \phi {e_S}\)

as $e_S$ is the identity of $\left({S, \circ}\right)$

So $\map \phi {e_S}$ is idempotent in $\struct{T, *}$ and the result follows.

$\blacksquare$

Also see

• Epimorphism Preserves Identity

• Homomorphism with Identity Preserves Inverses

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 12$: Theorem $12.3: \ 1^\circ$