-Morphisms

Homomorphism: A structure preserving map between 2 objects.

• e.g., $f: \mathbb{Z} \rightarrow \mathbb{Z}_4, f(x) = x \mod 4$

• homomorphism: \begin{align*} f(m + n) &= (m + n) \mod 4\\ &=(m \mod 4) + (n \mod 4) \\ &= f(m) + f(n) \end{align*}

Endomorphism: A homomorphism where domain and codomain are the same object.

• e.g., $f: \mathbb{Z} \rightarrow \mathbb{Z}, f(x) = 2\cdot x$

• homomorphism: \begin{align*} f(m + n) &= 2(m + n) \\ &= 2m + 2n \\ &= f(m) + f(n) \end{align*}

Isomorphism: A bijective homomorphism. (i.e., it has an inverse that is an homomorphism).

• e.g., $f: \mathbb{Z}_4 \rightarrow \mathbb{Z}_4, f(x) = 3 \cdot x \mod 4$

• homomorphism: \begin{align*} f(m + n) &= 3(m + n) \mod 4 \\ &= (3m + 3n) \mod 4 \\ &= (3m \mod 4) + (3n \mod 4) \\ &= f(m) + f(n) \end{align*}

• bijection: Since $\gcd(3, 4) = 1$, multiplication by $3 \mod 4$ is invertible. Its inverse is multiplication by $3$ itself, because $3 \cdot 3 = 9 \equiv 1\mod 4$.

Automorphism: An isomorphism from an object to itself. Equivalently, it is a bijective endomorphism.

• e.g., the example above is also an example of automorphism. TODO: Find a better example.

Written by ryan Kim from A41