  We present an objective function for similarity based hierarchical clustering
of partially ordered data that preserves the partial order. That is, if $x \le
y$, and if $[x]$ and $[y]$ are the respective clusters of $x$ and $y$, then
there is an order relation $\le'$ on the clusters for which $[x] \le' |y]$. The
theory distinguishes itself from existing theories for clustering of ordered
data in that the order relation and the similarity are combined into a
bi-objective optimisation problem to obtain a hierarchical clustering seeking
to satisfy both. In particular, the order relation is weighted in the range
$[0,1]$, and if the similarity and the order relation are not aligned, then
order preservation may have to yield in favor of clustering. Finding an optimal
solution is NP-hard, so we provide a polynomial time approximation algorithm,
with a relative performance guarantee of $O\!\left(\log^{3/2} \!\!\, n
\right)$, based on successive applications of directed sparsest cut. We provide
a demonstration on a benchmark dataset, showing that our method outperforms
existing methods for order preserving hierarchical clustering with significant
margin. The theory is an extension of the Dasgupta cost function for divisive
hierarchical clustering.
