  Given a finite poset $\mathcal P$, we call a family $\mathcal F$ of subsets
of $[n]$ $\mathcal P$-saturated if $\mathcal F$ does not contain an induced
copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an
induced copy of $\mathcal P$. The induced saturated number of $\mathcal P$,
denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal
P$-saturated family with ground set $[n]$. In this paper we are mainly
interested in the four-point poset called the butterfly. Ferrara, Kay, Kramer,
Martin, Reiniger, Smith and Sullivan showed that the saturation number for the
butterfly lies between $\log_2{n}$ and $n^2$. We give a linear lower bound of
$n+1$. We also prove some other results about the butterfly and the poset
$\mathcal N$.
