Section Generated_monoid_def.
Variable M : MONOID.
Variable A : part_set M.
Definition generated_monoid : submonoid M :=
image_monoid_hom (FM_lift (inj_part A)).
End Generated_monoid_def.

Lemma generated_monoid_minimal :
forall (M : MONOID) (A : part_set M) (H : submonoid M),
included A H -> included (generated_monoid A) H.