English
Let f: R → S be surjective and suppose R has a NonUnitalCommSemiring structure. If f preserves addition and multiplication, then S inherits a NonUnitalCommSemiring structure transported along f.
Русский
Пусть f: R → S — суръекция и R имеет структуру NonUnitalCommSemiring. Если f сохраняет сложение и умножение, то S наследует структуру NonUnitalCommSemiring, перенесённую по f.
LaTeX
$$$\text{Pushforward}_{f}({\text{NonUnitalCommSemiring}}\,R) = {\text{NonUnitalCommSemiring}}\,S.$$$
Lean4
/-- Pushforward a `NonUnitalCommSemiring` instance along a surjective function. -/
-- See note [reducible non-instances]
protected abbrev nonUnitalCommSemiring [NonUnitalCommSemiring R] (zero : f 0 = 0) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) (nsmul : ∀ (n : ℕ) (x), f (n • x) = n • f x) : NonUnitalCommSemiring S
where
toNonUnitalSemiring := hf.nonUnitalSemiring f zero add mul nsmul
__ := hf.commSemigroup f mul
