  We study feebly compact topologies $\tau$ on the semilattice
$\left(\exp_n\lambda,\cap\right)$ such that $\left(\exp_n\lambda,\tau\right)$
is a semitopological semilattice. All compact semilattice $T_1$-topologies on
$\exp_n\lambda$ are described. Also we prove that for an arbitrary positive
integer $n$ and an arbitrary infinite cardinal $\lambda$ for a $T_1$-topology
$\tau$ on $\exp_n\lambda$ the following conditions are equivalent: $(i)$
$\left(\exp_n\lambda,\tau\right)$ is a compact topological semilattice; $(ii)$
$\left(\exp_n\lambda,\tau\right)$ is a countably compact topological
semilattice; $(iii)$ $\left(\exp_n\lambda,\tau\right)$ is a feebly compact
topological semilattice; $(iv)$ $\left(\exp_n\lambda,\tau\right)$ is a compact
semitopological semilattice; $(v)$ $\left(\exp_n\lambda,\tau\right)$ is a
countably compact semitopological semilattice. We construct a countably
pracompact $H$-closed quasiregular non-semiregular topology
$\tau_{\operatorname{\textsf{fc}}}^2$ such that
$\left(\exp_2\lambda,\tau_{\operatorname{\textsf{fc}}}^2\right)$ is a
semitopological semilattice with discontinuous semilattice operation and prove
that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal
$\lambda$ every $T_1$-semiregular feebly compact semitopological semilattice
$\exp_n\lambda$ is a compact topological semilattice.
