Composite of Monomorphisms is Monomorphism
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Theorem
Let:
- $\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$
- $\struct {S_2, *_1, *_2, \ldots, *_n}$
- $\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
Let:
- $\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
- $\psi: \struct {S_2, *_1, *_2, \ldots, *_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
be monomorphisms.
Then the composite of $\phi$ and $\psi$ is also a monomorphism.
Proof
From Composite of Homomorphisms on Algebraic Structure is Homomorphism, $\psi \circ \phi$ is a homomorphism.
From Composite of Injections is Injection, $\psi \circ \phi$ is an injection.
A monomorphism is an injective homorphism.
Hence $\psi \circ \phi$ is a monomorphism.
$\blacksquare$
Sources
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$