  We study the problem of $P$-interpolation, where $P$ is a set of binary
predicate symbols, for certain classes of local extensions of a base theory.
For computing the $P$-interpolating terms, we use a hierarchic approach: This
allows us to compute the interpolating terms using a method for computing
interpolating terms in the base theory. We use these results for proving
$\leq$-interpolation in classes of semilattices with monotone operators; we
show, by giving a counterexample, that $\leq$-interpolation does not hold if by
"shared" symbols we mean just the common symbols. We use these results for the
study of $\sqsubseteq$-interpolation in the description logics ${\cal EL}$ and
${\cal EL}^+$.
