  We investigate the intersection problem for finite monoids, which asks for a
given set of regular languages, represented by recognizing morphisms to finite
monoids from a variety V, whether there exists a word contained in their
intersection. Our main result is that the problem is PSPACE-complete if V is
contained in DS and NP-complete if V is non-trivial and contained in DO. Our
NP-algorithm for the case that V is contained in DO uses novel methods, based
on compression techniques and combinatorial properties of DO. We also show that
the problem is log-space reducible to the intersection problem for
deterministic finite automata (DFA) and that a variant of the problem is
log-space reducible to the membership problem for transformation monoids. In
light of these reductions, our hardness results can be seen as a generalization
of both a classical result by Kozen and a theorem by Beaudry, McKenzie and
Therien.
