Homomorphisms
# Group Homomorphisms
They are obvious.
# Ring Homomorphisms
A ring homomorphism $ \phi $ must satisfy
- $\phi (g*h) = \phi (g) * \phi (h)$
- $\phi (g+h) = \phi (g) + \phi (h)$
- $\phi (1) = 1$
The third condition make the homomorphism more strict.
examples (Non trivial)
- $ \mathbb{C} $ homomorphism ( $ z \mapsto z^n $ ) are ring homomorphisms.
# Rng Homomorphisms
A rng homomorphism $ \phi $ is a ring homomorphism without `i` (identity law)
- $\phi (g*h) = \phi (g) * \phi (h)$
- $\phi (g+h) = \phi (g) + \phi (h)$
# Rig Homomorphism
A rig is a ring without `n` (additive negation `-`)
e.g. Natural Numbers $\mathbb{N}$ is a rig.
A rig homomorphism is the same with ring homomorphism except they do not have negation.
- $\phi (g*h) = \phi (g) * \phi (h)$
- $\phi (g+h) = \phi (g) + \phi (h)$
- $\phi (1) = 1$
# Magma Homomorphisms
similar
# SemiGroup Homomorphisms
similar
# Monoid Homomorphisms
similar