The problem definition of indexing relations is as follows.
We have chosen a system of
composite indices.
Since the fields have all been compressed to an integer
representation, the set of fields can be treated
as an orthogonal basis for a vector space, with each
tuple corresponding to a point in this space.
Given an n-ary relation we can construct one or more
m-ary indices, 
, each of which constitutes a
projection of the relation space onto an m dimensional sub space.
The specification of values for p fields, for 
,
then defines an m-p dimensional submanifold of the m
dimensional index space space within which the tuples that
constitute valid responsed to the query must lie.
Clearly for the special case of m=1 we have a conventional
single field index. The indices are implemented using
multi-dimensional hash tables.