Theorems · Theorem · group theory

Submonoid.closure_image_isMulIndecomposable_baseOf

∀ {ι : Type u_1} {M : Type u_2} {S : Type u_4} [inst : Monoid M] [inst_1 : LinearOrder S] [Finite ι]
  [inst_3 : CommMonoid S] [IsOrderedCancelMonoid S] (v : ι → M) (f : M →* S),
  Submonoid.closure (v '' IsMulIndecomposable.baseOf v f) = Submonoid.closure (v '' {i | 1 < f (v i)})

Given a finite family of points v in a monoid M, together with a morphism into a linearly-ordered monoid f : M →* S, the submonoid generated by those points of v which lie in the "half space" where f > 1 is generated by the subset of such points which are indecomposable with respect to points in this half space.

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

MonoidLinearOrderFiniteCommMonoidIsOrderedCancelMonoid