Composition of Homomorphisms

Theorem

Algebraic Structures

Let:

• $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$
• $\left({S_2, *_1, *_2, \ldots, *_n}\right)$
• $\left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$

be algebraic structures.

Let:

• $\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S_2, *_1, *_2, \ldots, *_n}\right)$
• $\psi: \left({S_2, *_1, *_2, \ldots, *_n}\right) \to \left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$

be homomorphisms.

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

R-Algebraic Structures

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.

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

be homomorphisms.

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