# 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

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 28$: Theorem $28.1$