Definition:Semigroup Homomorphism
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Definition
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be semigroups.
Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$.
That is, $\forall a, b \in S$:
- $\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$
Then $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ is a semigroup homomorphism.
Also see
- Definition:Homomorphism (Abstract Algebra)
- Definition:Group Homomorphism
- Definition:Ring Homomorphism
- Definition:Semigroup Endomorphism: a semigroup homomorphism from a semigroup to itself
- Definition:Semigroup Automorphism: a semigroup isomorphism from a semigroup to itself
- Results about semigroup 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
- 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$: Definition $2.1$