Some Properties of Transforms in Culture Theory

It is shown that, in certain circumstances, systems of cultural rules may be represented by doubly stochastic matrices denoted called possibility transforms, and by certain real valued possibility densities with inner product. Using such objects we may characterize a certain problem of ethnographic and ethological description as a problem of prediction, in which observations are predicted by properties of fixed points of transforms of pure systems, or by properties of convex combinations of such pure systems. That is, ethnographic description is an application of the Birkhoff theorem regarding doubly stochastic matrices on a space whose vertices are permutations.