  We investigate monomial labellings on cell complexes, giving a minimal
cellular resolution of the ideal generated by these monomials, and such that
the associated quotient ring is Cohen-Macaulay. We introduce a notion of such a
labelling being maximal. There is only a finite number of maximal labellings
for each cell complex, and we classify these for trees, partly for subdivisions
of polygons, and for some classes of selfdual polytopes.
