Given a finite poset $P$, we call a family $F$ of subsets of $[n]$ $P$-saturated if $F$ does not contain an induced copy of $P$, but adding any other set to $F$ creates an induced copy of $P$. The induced saturated number of $P$, denoted by $sat\ast (n,P)$, is the size of the smallest $P$-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 $\log 2n$ and $n2$. We give a linear lower bound of $n+1$. We also prove some other results about the butterfly and the poset $N$.