φ(a ∗ b) = φ(a) ∘ φ(b)
A map between algebraic structures becomes a homomorphism the moment this single equality is required to hold. C
ompute the product in the source and then send the result across, or send the factors first and multiply them in the target: both routes must arrive at the same element.
The demand is independent of the particular operations; it is the same idea whether one is working with groups, rings or algebras.
When the map is also bijective the two structures are algebraically indistinguishable and the homomorphism is an isomorphism.
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