(** The opposite of identity is maximal or full *)
Definition full {A} {JA: Join A}(sigma : A) : Prop :=
forall sigma', joins sigma sigma' -> identity sigma'.
Definition maximal {A} {JA: Join A} (sigma : A) : Prop :=
forall sigma', join_sub sigma sigma' -> sigma = sigma'.

Lemma full_maximal {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A} :
forall a, full a <-> maximal a.