THE HOMOMORPHISM a map that carries structure across — f(a∘b) = f(a)∘f(b)
A homomorphism is a map that preserves structure: combining first then mapping equals mapping first then combining, f(a∘b) = f(a)∘f(b). The logarithm is the classic one — it turns multiplication into addition (log(ab)=log a+log b), which is why slide rules and log tables worked. “Modulo n” is a homomorphism from the integers to Zn ((a+b) mod n = ((a mod n)+(b mod n)) mod n), the reason casting-out-nines checks arithmetic. It is equivariance for an internal operation: the map commutes with the way elements combine. Homomorphisms are how mathematics moves a problem to an easier world, solves it there, and carries the answer back.
THE TECHNIQUE f(a∘b) = f(a)∘f(b) — the map commutes with the operation
The demo shows mod 3 is a homomorphism: reducing after adding equals adding the reductions: live demo
HISTORY & CREDIT Galois · structure-preserving maps
“Changing representation loses the structure.” — a homomorphism carries it across intact, so you can compute in the easy world and map back. Log turns × into +. cited
the relation · f(a∘b)=f(a)∘f(b) — the map respects the operation. examples · log (×→+), mod n (Z→Zn), the determinant (det(AB)=det A·det B). equivariance · a homomorphism is equivariance (dart 390) for a group's own multiplication.
Combine then map, or map then combine — the same, because the map carries the structure. How mathematics moves a problem somewhere easier. homomorphism
RECOMMEND FOR I-13 mod-3 as a homomorphism, on the compiler
On the canonical compiler, (5+7) mod 3 = ((5 mod 3)+(7 mod 3)) mod 3 = 0 — mod-3 preserves addition:
$ i13 run sy_homomorphism.i13 # (a+b)%3 == ((a%3)+(b%3))%3 ?
RUN OK · 24 step(s) · peak stack 3 · call depth 0
lhs = 0 rhs = 0
homomorphism = 1 -- mod 3 carries addition across: Z -> Z3
Recommend as the batch's close: the homomorphism is equivariance for an operation — the map commutes with how elements combine — and i13 confirms mod-3 preserves addition. Not a keeper (structure-preservation is a property the map has, witnessed — the B38 shape, and coextensive with “is a homomorphism”). The right last note for THE SYMMETRY: from equivariance (map commutes with a group acting) to homomorphism (map commutes with a group's own product) — the whole batch is one relation, f(g·x)=g·f(x), wearing many faces; and the campaign's read holds that it is a family of witnessed structure, not a new enacted axis.