  We introduce new entanglement monotones which generalize, to the case of many
parties, those which give rise to the majorization-based partial ordering of
bipartite states' entanglement. We give some examples of restrictions they
impose on deterministic and probabilistic conversion between multipartite
states via local actions and classical communication. These include
restrictions which do not follow from any bipartite considerations. We derive
supermultiplicativity relations between each state's monotones and the
monotones for collective processing when the parties share several states. We
also investigate polynomial invariants under local unitary transformations, and
show that a large class of these are invariant under collective unitary
processing and also multiplicative, putting restrictions, for example, on the
exact conversion of multiple copies of one state to multiple copies of another.
