  In this paper we show that subsumption problems in lightweight description
logics (such as $\mathcal{EL}$ and $\mathcal{EL}^+$) can be expressed as
uniform word problems in classes of semilattices with monotone operators. We
use possibilities of efficient local reasoning in such classes of algebras, to
obtain uniform PTIME decision procedures for CBox subsumption in
$\mathcal{EL}$, $\mathcal{EL}^+$ and extensions thereof. These locality
considerations allow us to present a new family of (possibly many-sorted)
logics which extend $\mathcal{EL}$ and $\mathcal{EL}^+$ with $n$-ary roles
and/or numerical domains. As a by-product, this allows us to show that the
algebraic models of ${\cal EL}$ and ${\cal EL}^+$ have ground interpolation and
thus that ${\cal EL}$, ${\cal EL}^+$, and their extensions studied in this
paper have interpolation. We also show how these ideas can be used for the
description logic $\mathcal{EL}^{++}$.
