Canonical Injection is Monomorphism

Theorem
Let $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ be algebraic structures with identities $e_1, e_2$ respectively.

The canonical injections:


 * $\inj_1: \struct {S_1, \circ_1} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_1: \map {\inj_1} x = \tuple {x, e_2}$


 * $\inj_2: \struct {S_2, \circ_2} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_2: \map {\inj_2} x = \tuple {e_1, x}$

are monomorphisms.

Proof
From Canonical Injection is Injection we have that the canonical injections are in fact injective.

It remains to prove the morphism property.

Let $x, y \in \struct {S_1, \circ_1}$.

Then:

and the morphism property has been demonstrated to hold for $\inj_1$.

Thus $\inj_1: \struct {S_1, \circ_1} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}$ has been shown to be an injective homomorphism and therefore a monomorphism.

The same argument applies to $\inj_2$.