Definition:Monoid Homomorphism
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Definition
Let $\struct {S, \circ}$ and $\struct {T, *}$ be monoids.
Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$.
That is, $\forall a, b \in S$:
- $\map \phi {a \circ b} = \map \phi a * \map \phi b$
Suppose further that $\phi$ preserves identities, that is:
- $\map \phi {e_S} = e_T$
Then $\phi: \struct {S, \circ} \to \struct {T, *}$ is a monoid homomorphism.
Also see
- Definition:Homomorphism (Abstract Algebra)
- Definition:Group Homomorphism
- Definition:Ring Homomorphism
- Definition:Monoid Epimorphism: a surjective monoid homomorphism
- Definition:Monoid Monomorphism: an injective monoid homomorphism
- Definition:Monoid Isomorphism: a bijective monoid homomorphism
- Definition:Monoid Endomorphism: a monoid homomorphism from a monoid to itself
- Definition:Monoid Automorphism: a monoid isomorphism from a monoid to itself
- Results about monoid homomorphisms can be found here.
Linguistic Note
The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.
Thus homomorphism means similar structure.
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.13$