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

A more "complete" version of the Pi-theorem: DRAFT