Induced Saturation of the Poset 2C_2

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 φ: P→ P(X) such that φ(p)⊆φ(q) if and only if p≤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 $|X|=n$.