Induced Saturation of the Poset 2C₂

Given a set X, the power set P (X), and a finite poset P, a family F P (X) is said to be induced-P-free if there is no injection: P P (X) such that (p) (q) if and only if p₏ q. The family F is induced-P-saturated if it is maximal with respect to being induced-P-free. If n=|X|, then the size of the smallest induced-P-saturated family in P (X) is denoted sat (n, P). The poset 2C₂ is two disjoint 2-chains (the Hasse diagram is two vertex-disjoint edges) and Keszegh, Lemons, Martin, P\'alv\"olgyi, and Patk\'os proved that n+2 sat (n, 2C₂) 2n and gave one isomorphism class of an induced-2C₂-saturated family that achieves the upper bound. We show that the lower bound can be improved to 3n/2 + 1/2 by examining the necessary structure of a saturated family. In addition, we provide many examples of induced-2C₂-saturated families of size 2n in P (X) where |X|=n.