Elliptic equations on configurations $W = W_{1} \cup \dots
\cup W_{N}$ with edge $Y$ and components $W_{j}$ of different
dimension can be treated in the frame of pseudo-differential
analysis on manifolds with geometric singularities, here edges.
Starting from edge-degenerate operators on $W_{j}$, $j=1,\dots,N$,
we construct an algebra with extra `transmission' conditions
on $Y$ that satisfy an analogue of the Shapiro-Lopatinskij
condition. Ellipticity refers to a two-component symbolic
hierarchy with an interior and an edge part; the latter one
is operator-valued, operating on the union of different dimensional
model cones. We construct parametrices within our calculus,
where exchange of information between the various components
is encoded in Green and Mellin operators that are smoothing
on $W \setminus Y$. Moreover, we obtain regularity of solutions
in weighted edge spaces with asymptotics.