  Given a finite poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the
Boolean lattice is induced-$\mathcal{P}$-saturated if $\mathcal{F}$ contains no
copy of $\mathcal{P}$ as an induced subposet but every proper superset of
$\mathcal{F}$ contains a copy of $\mathcal{P}$ as an induced subposet. The
minimum size of an induced-$\mathcal{P}$-saturated family in the
$n$-dimensional Boolean lattice, denoted $\operatorname{sat}^*(n,\mathcal{P})$,
was first studied by Ferrara et al. (2017).
  Our work focuses on strengthening lower bounds. For the 4-point poset known
as the diamond, we prove $\operatorname{sat}^*(n,\mathcal{D}_2)\geq\sqrt{n}$,
improving upon a logarithmic lower bound. For the antichain with $k+1$
elements, we prove $\operatorname{sat}^*(n,\mathcal{A}_{k+1})\geq
(1-o_k(1))\frac{kn}{\log_2 k}$, improving upon a lower bound of $3n-1$ for
$k\geq 3$.
