Canonical Injection is Monomorphism

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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.

General Result

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.


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

It remains to prove the morphism property.

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


\(\ds \map {\inj_1} {x \circ_1 y}\) \(=\) \(\ds \tuple {x \circ_1 y, e_2}\)
\(\ds \) \(=\) \(\ds \tuple {x \circ_1 y, e_2 \circ_2 e_2}\)
\(\ds \) \(=\) \(\ds \tuple {x, e_2} \circ \tuple {y, e_2}\)
\(\ds \) \(=\) \(\ds \map {\inj_1} x \circ \map {\inj_1} y\)

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 $\inj_2$.