Theorems · Theorem · group theory

MonoidHom.exists_mrange_eq_mgraph

∀ {G : Type u_1} {H : Type u_2} {I : Type u_3} [inst : Monoid G] [inst_1 : Monoid H] [inst_2 : Monoid I]
  {f : G →* H × I},
  Function.Surjective (Prod.fst ∘ ⇑f) →
    (∀ (g₁ g₂ : G), (f g₁).1 = (f g₂).1 → (f g₁).2 = (f g₂).2) → ∃ f', MonoidHom.mrange f = f'.mgraph

Vertical line test for monoid homomorphisms. Let f : G → H × I be a homomorphism to a product of monoids. Assume that f is surjective on the first factor and that the image of f intersects every "vertical line" {(h, i) | i : I} at most once. Then the image of f is the graph of some monoid homomorphism f' : H → I.

Defined in

Mathlib.Algebra.Group.Graph

Cited by

3 results in Mathlib

Foundations

Depth 21 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MonoidMonoidMonoid