-Morphisms

Homomorphism: A structure preserving map between 2 objects.

  • e.g., f:ZZ4,f(x)=xmod4f: \mathbb{Z} \rightarrow \mathbb{Z}_4, f(x) = x \mod 4

  • homomorphism:

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

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

  • homomorphism:

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

  • e.g., f:Z4Z4,f(x)=3xmod4f: \mathbb{Z}_4 \rightarrow \mathbb{Z}_4, f(x) = 3 \cdot x \mod 4

  • homomorphism:

  • bijection: Since gcd(3,4)=1\gcd(3, 4) = 1, multiplication by 3mod43 \mod 4 is invertible. Its inverse is multiplication by 33 itself, because 33=91mod43 \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

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