Theorems · Definition · group theory

AddSubmonoid.LocalizationMap.map

{M : Type u_1} →
  [inst : AddCommMonoid M] →
    {S : AddSubmonoid M} →
      {N : Type u_2} →
        [inst_1 : AddCommMonoid N] →
          {P : Type u_3} →
            [inst_2 : AddCommMonoid P] →
              S.LocalizationMap N →
                {g : M →+ P} →
                  {T : AddSubmonoid P} →
                    (∀ (y : ↥S), g ↑y ∈ T) → {Q : Type u_4} → [inst : AddCommMonoid Q] → T.LocalizationMap Q → N →+ Q

Given an AddCommMonoid homomorphism g : M →+ P where for AddSubmonoids S ⊆ M, T ⊆ P we have g(S) ⊆ T, the induced AddMonoid homomorphism from the Localization of M at S to the Localization of P at T: if f : M →+ N and k : P →+ Q are Localization maps for S and T respectively, we send z : N to k (g x) - k (g y), where (x, y) : M × S are such that z = f x - f y.

Defined in

Mathlib.GroupTheory.MonoidLocalization.Maps

Cited by

15 results in Mathlib

Foundations

Depth 48 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommMonoidAddCommMonoidAddCommMonoidAddCommMonoid