Composite of Homomorphisms is Homomorphism/R-Algebraic Structure

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Theorem

Let:

$\left({S_1, *_1}\right)_R$
$\left({S_2, *_2}\right)_R$
$\left({S_3, *_3}\right)_R$

be $R$-algebraic structures with the same number of operations.

Let:

$\phi: \left({S_1, *_1}\right)_R \to \left({S_2, *_2}\right)_R$
$\psi: \left({S_2, *_2}\right)_R \to \left({S_3, *_3}\right)_R$

be homomorphisms.


Then the composite of $\phi$ and $\psi$ is also a homomorphism.


Proof


Sources