Theorems · Theorem · group theory

AddSubmonoid.closure_image_isAddIndecomposable_baseOf

∀ {ι : Type u_1} {M : Type u_2} {S : Type u_4} [inst : AddMonoid M] [inst_1 : LinearOrder S] [Finite ι]
  [inst_3 : AddCommMonoid S] [IsOrderedCancelAddMonoid S] (v : ι → M) (f : M →+ S),
  AddSubmonoid.closure (v '' IsAddIndecomposable.baseOf v f) = AddSubmonoid.closure (v '' {i | 0 < f (v i)})

Given a finite family of points v in an additive monoid M, together with a morphism into a linearly-ordered additive monoid f : M →+ S, the submonoid generated by those points of v which lie in the half space where f > 0 is generated by the subset of such points which are indecomposable with respect to points in this half space. If v is the set of roots of a crystallographic root system and S = ℚ, then this is [serre1965](Ch. V, §9, Lemma 2) and it may be used to prove that the root system has a base.

Defined in

Mathlib.Algebra.Group.Irreducible.Indecomposable

Cited by

2 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddMonoidLinearOrderFiniteAddCommMonoidIsOrderedCancelAddMonoid