  Recently there has been significant interest in constructing ordered
analogues of Petrov's two-parameter extension of Ethier and Kurtz's
infinitely-many-neutral-alleles diffusion model. One method for constructing
these processes goes through taking an appropriate diffusive limit of Markov
chains on integer compositions called ordered Chinese Restaurant Process
up-down chains. The resulting processes are diffusions whose state space is the
set of open subsets of the open unit interval. In this paper we begin to study
nontrivial aspects of the order structure of these diffusions. In particular,
for a certain choice of parameters, we take the diffusive limit of the size of
the first component of ordered Chinese Restaurant Process up-down chains and
describe the generator of the limiting process. We then relate this to the size
of the leftmost maximal open subset of the open-set valued diffusions. This is
challenging because the function taking an open set to the size of its leftmost
maximal open subset is discontinuous. Our methods are based on establishing
intertwining relations between the processes we study.
