Theorems · Inductive type · group theory

DirectSum.Decomposition

{ι : Type u_1} →
  {M : Type u_3} →
    {σ : Type u_4} →
      [DecidableEq ι] →
        [inst : AddCommMonoid M] → [inst_1 : SetLike σ M] → [AddSubmonoidClass σ M] → (ι → σ) → Type (max u_1 u_3)

A decomposition is an equivalence between an additive monoid M and a direct sum of additive submonoids ℳ i of that M, such that the "recomposition" is canonical. This definition also works for additive groups and modules. This is a version of DirectSum.IsInternal which comes with a constructive inverse to the canonical "recomposition" rather than just a proof that the "recomposition" is bijective. Often it is easier to construct a term of this type via Decomposition.ofAddHom or Decomposition.ofLinearMap.

Defined in

Mathlib.Algebra.DirectSum.Decomposition

Cited by

57 results in Mathlib

Foundations

Depth 6 from the axioms · uses no axioms

Assumes

DecidableEqAddCommMonoidSetLikeAddSubmonoidClass