A composition structure is a sequence of consistent probability distributions for compositions (ordered partitions) of n = 1, 2,. Any composition structure can be associated with an exchangeable random composition of the set of natural numbers. Following Donnelly and Joyce, we study the problem of characterizing a generic composition structure as a convex mixture of the "extreme" ones. We topologize the family U of open subsets of [0, 1] so that U becomes compact and show that U is homeomorphic to the set of extreme composition structures. The general composition struc-ture is related to a random element of U via a construction introduced by J. Pitman.
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Alexander Gnedin (1997) studied this question.
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