Theorems · Theorem · group theory
AddSubmonoid.exists_minimal_closure_eq_top
∀ (M : Type u_1) [inst : AddMonoid M] [AddMonoid.FG M], ∃ S, Minimal (fun S => AddSubmonoid.closure ↑S = ⊤) S
A finitely generated monoid has a minimal generating set.
- Defined in
- Mathlib.GroupTheory.Finiteness
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddMonoidAddMonoid.FG
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement · cited by 13,712
- Top.topstatement · cited by 9,680
- SetLike.coestatement · cited by 8,199
- AddMonoidstatement and proof · cited by 2,864
- AddSubmonoidstatement · cited by 1,178
- AddSubmonoid.closurestatement · cited by 224
- Minimalstatement · cited by 150
- AddMonoid.FGstatement and proof · cited by 31
- AddMonoid.FG.fg_topproof · cited by 10
- AddSubmonoid.FG.exists_minimal_closure_eqproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- AddSubmonoid.closure_addIrreducibleproof · cited by 0