Theorems · Definition · group theory

Submonoid.LocalizationMap.map

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

Given a CommMonoid homomorphism g : M →* P where for Submonoids S ⊆ M, T ⊆ P we have g(S) ⊆ T, the induced Monoid 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))⁻¹, where (x, y) : M × S are such that z = f x * (f y)⁻¹.

Defined in

Mathlib.GroupTheory.MonoidLocalization.Maps

Cited by

16 results in Mathlib

Foundations

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

Assumes

CommMonoidCommMonoidCommMonoidCommMonoid