homomorphism
A homomorphism is a map between two algebraic structures that respects their operations — it carries the “grammar” of one over to the other without scrambling it. The slogan is: do the operation first and then map, or map first and then do the operation, and you get the same answer. It is a translation that preserves meaning, even if it loses some detail.
Formally, for groups, a function f from G to H is a homomorphism if f(a * b) = f(a) * f(b) for all a, b — the left side uses G's operation, the right side uses H's. For rings, the map must respect both addition and multiplication. A homomorphism automatically sends the identity to the identity and inverses to inverses, so structure is genuinely carried across.
Homomorphisms need not be one-to-one or onto; they may collapse information. What survives the collapse is captured by the kernel (everything sent to the identity) and the image. The kernel of a group homomorphism is always a normal subgroup, and of a ring homomorphism always an ideal — which is how these maps connect directly to quotient structures.
The map f from the integers (ℤ, +) to itself given by f(n) = 2n is a homomorphism: f(a + b) = 2(a + b) = 2a + 2b = f(a) + f(b). The exponential map exp(x + y) = exp(x)·exp(y) is another, turning addition into multiplication.
n ↦ 2n preserves addition; exp turns + into ×.