  We show that the solid hull of every weakly precompact set of a Banach
lattice $E$ is weakly precompact if and only if every order interval in $E$ is
weakly precompact, or equivalently, if and only if every disjoint weakly
compact set is weakly precompact. Some results on the domination property for
weakly precompact positive operators are obtained. Among other things, we show
that, for a pair of Banach lattices $E$ and $F$ with $E$ $\sigma$-Dedekind
complete, every positive operator from $E$ to $F$ dominated by a weakly
precompact operator is weakly precompact if and only if either the norm of
$E^{\prime}$ is order continuous or else every order interval in $F$ is weakly
precompact.
