Induced and non-induced poset saturation problems

A subfamily G⊆F⊆2[n] of sets is a non-induced (weak) copy of a poset P in F if there exists a bijection i:P→G such that p≤Pq implies i(p)⊆i(q). In the case where in addition p≤Pq holds if and only if i(p)⊆i(q), then G is an induced (strong) copy of P in F. We consider the minimum number sat(n,P) [resp. sat⁎(n,P)] of sets that a family F⊆2[n] can have without containing a non-induced [induced] copy of P and being maximal with respect to this property, i.e., the addition of any G∈2[n]∖F creates a non-induced [induced] copy of P.