Definition:Homomorphism (Abstract Algebra)/Cartesian Product

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Definition

Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a mapping from one algebraic structure $\struct {S, \circ}$ to another $\struct {T, *}$.

We define the cartesian product $\phi \times \phi: S \times S \to T \times T$ as:

$\forall \tuple {x, y} \in S \times S: \map {\paren {\phi \times \phi} } {x, y} = \tuple {\map \phi x, \map \phi y}$


Hence we can state that $\phi$ is a homomorphism if and only if:

$\map \ast {\map {\paren {\phi \times \phi} } {x, y} } = \map \phi {\map \circ {x, y} }$

using the notation $\map \circ {x, y}$ to denote the operation $x \circ y$.


The point of doing this is so we can illustrate what is going on in a commutative diagram:

$\begin{xy} \[email protected]+2mu@+1em{ S \times S \ar[r]^*{\circ} \ar[d]_*{\phi \times \phi} & S \ar[d]^*{\phi} \\ T \times T \ar[r]_*{\ast} & T }\end{xy}$

Thus we see that $\phi$ is a homomorphism if and only if both of the composite mappings from $S \times S$ to $T$ have the same effect on all elements of $S \times S$.


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

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