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 ${\cal EL}^+$.

翻译：我们研究了$P$-插值问题（其中$P$为一组二元谓词符号）在基础理论的某些局部扩展类中的性质。为了计算$P$-插值项，我们采用分层方法：这使得我们能够利用基础理论中计算插值项的方法来导出插值项。我们将这些结果用于证明带有单调算子的半格类中的$\leq$-插值性质；通过构造反例表明，若将"共享"符号仅理解为公共符号，则$\leq$-插值不成立。进一步地，我们将这些结果应用于描述逻辑${\cal EL}$和${\cal EL}^+$中$\sqsubseteq$-插值性质的研究。