# Composite of Homomorphisms is Homomorphism/Algebraic Structure

## Theorem

Let:

- $\struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n}$
- $\struct {S_2, *_1, *_2, \ldots, *_n}$
- $\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$

Let:

- $\phi: \struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
- $\psi: \struct {S_2, *_1, *_2, \ldots, *_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$

be homomorphisms.

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

## Proof

Let $\psi \circ \phi$ denote the composite of $\phi$ and $\psi$.

Then what we are trying to prove is denoted:

- $\paren {\psi \circ \phi}: \struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$ is a homomorphism.

To prove the above is the case, we need to demonstrate that the morphism property is held by each of the operations $\otimes_1, \otimes_2, \ldots, \otimes_n$ under $\psi \circ \phi$.

Let $\otimes_k$ be one of these operations.

We take two elements $x, y \in S_1$, and put them through the following wringer:

\(\ds \map {\paren {\psi \circ \phi} } {x \otimes_k y}\) | \(=\) | \(\ds \map \psi {\map \phi {x \otimes_k y} }\) | Definition of Composition of Mappings | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \psi {\map \phi x *_k \map \phi y}\) | Definition of Morphism Property | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \psi {\map \phi x} \oplus_k \map \psi {\map \phi y}\) | Definition of Morphism Property | |||||||||||

\(\ds \) | \(=\) | \(\ds \map {\paren {\psi \circ \phi} } x \oplus_k \map {\paren {\psi \circ \phi} } y\) | Definition of Composition of Mappings |

Disentangling the confusing and tortuous expressions above, we (eventually) see that this shows that the morphism property is indeed held by $\otimes_k$ under $\psi \circ \phi$.

As this holds for any arbitrary operation $\otimes_k$ in $\struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n}$, it follows that it holds for all of them.

Thus $\paren {\psi \circ \phi}: \struct {S_1, \otimes_1, \otimes_2, \ldots, \otimes_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$ is indeed a homomorphism.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Theorem $12.4$ - 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\S 1.2$