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.
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.
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:
\(\ds \map {\inj_1} {x \circ_1 y}\) | \(=\) | \(\ds \tuple {x \circ_1 y, e_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x \circ_1 y, e_2 \circ_2 e_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x, e_2} \circ \tuple {y, e_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\inj_1} x \circ \map {\inj_1} y\) |
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 $\inj_2$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Theorem $13.3$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Exercise $13.3$