Canonical Injection is Monomorphism/General Result

Theorem
Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \dotsc, \struct {S_j, \circ_j}, \dotsc, \struct {S_n, \circ_n}$ be algebraic structures with identities $e_1, e_2, \dotsc, e_j, \dotsc, e_n$ respectively.

Then the canonical injection:
 * $\ds \inj_j: \struct {S_j, \circ_j} \to \prod_{i \mathop = 1}^n \struct {S_i, \circ_i}$

defined as:


 * $\map {\inj_j} x = \tuple {e_1, e_2, \dotsc, e_{j - 1}, x, e_{j + 1}, \dotsc, e_n}$

is a monomorphism.

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

It remains to prove the morphism property.

Let $x, y \in \struct {S_j, \circ_j}$.

Then:

and the morphism property has been demonstrated to hold.

Thus $\ds \inj_j: \struct {S_j, \circ_j} \to \prod_{i \mathop = 1}^n \struct {S_i, \circ_i}$ has been shown to be an injective homomorphism and therefore a monomorphism.