  We construct an infinitely exchangeable process on the set $\cate$ of subsets
of the power set of the natural numbers $\mathbb{N}$ via a Poisson point
process with mean measure $\Lambda$ on the power set of $\mathbb{N}$. Each
$E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone
subsets of $\cate$, and every monotone subset maps to an undirected graph
$G\in\catg$, the space of undirected graphs with vertex set $\mathbb{N}$. We
show a natural mapping $\cate\rightarrow\catf\rightarrow\catg$ which induces an
infinitely exchangeable measure on the projective system $\catg^{\rest}$ of
graphs $\catg$ under permutation and restriction mappings given an infinitely
exchangeable family of measures on the projective system $\cate^{\rest}$ of
subsets with permutation and restriction maps. We show potential connections of
this process to applications in cluster analysis, machine learning,
classification and Bayesian inference.
