Composite of Isomorphisms is Isomorphism
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Theorem
Algebraic Structures
Let:
- $\struct {S_1, \odot_1, \odot_2, \ldots, \odot_n}$
- $\struct {S_2, *_1, *_2, \ldots, *_n}$
- $\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
Let:
- $\phi: \struct {S_1, \odot_1, \odot_2, \ldots, \odot_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
- $\psi: \struct {S_2, *_1, *_2, \ldots, *_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$
be isomorphisms.
Then the composite of $\phi$ and $\psi$ is also an isomorphism.
R-Algebraic Structures
Let:
- $\struct {S_1, \ast_1}_R$
- $\struct {S_2, \ast_2}_R$
- $\struct {S_3, \ast_3}_R$
be $R$-algebraic structures with the same number of operations.
Let:
- $\phi: \struct {S_1, \ast_1}_R \to \struct {S_2, \ast_2}_R$
- $\psi: \struct {S_2, \ast_2}_R \to \struct {S_3, \ast_3}_R$
be isomorphisms.
Then the composite of $\phi$ and $\psi$ is also an isomorphism.