Induced Saturation of the Poset 2C_2

Given a set $X$, the power set $\mathbb{P}(X)$, and a finite poset $P$, a family $F\subset \mathbb{P}(X)$ is said to be induced-$P$-free if there is no injection $\phi :P\to \mathbb{F}$ such that $\phi (p)\subseteq \phi (q)$ if and only if $p{\le }_{P}q$, for all $p,q\in P$. The family $F$ is induced-$P$-saturated if it is maximal with respect to being induced-$P$-free. If $n=\mathrm{\mid }X\mathrm{\mid }$, then the size of the smallest induced-$P$-saturated family in $\mathbb{P}(X)$ is denoted $sat(n,P)$. The poset $2C_2$ is two incomparable 2-chains (the Hasse diagram is two vertex-disjoint edges) and Keszegh, Lemons, Martin, Pálvölgyi, and Patkós proved that $n+2\le sat(n,2C_2)\le 2n$ and gave one isomorphism class of an induced-$2C_2$-saturated family that achieves the upper bound. We show that the lower bound can be improved to $3n\mathrm{/}2+1\mathrm{/}2$ by examining the necessary structure of a saturated family. In addition, we provide many examples of induced-$2C_2$-saturated families of size $2n$ in $\mathbb{P}(X)$ where $\mathrm{\mid }X\mathrm{\mid }=n$.