  The traditional Pi-theorem tells us that for any dimensionally invariant
relation there exists a full set of independent dimensionless "Pi groups" which
can be used to nondimensionalise the relation. In this paper, we seek to
understand better the structure of dimensionally invariant relations and sets,
by giving a complete characterisation of them in terms of independent
dimensionless Pi groups. The traditional Pi-theorem only goes part of the way
towards achieving such a characterisation. Our characterisation presented here
can be viewed as the "complete Pi-theorem".
