  We consider databases in which each attribute takes values from a partially
ordered set (poset). This allows one to model a number of interesting scenarios
arising in different applications, including quantitative databases,
taxonomies, and databases in which each attribute is an interval representing
the duration of a certain event occurring over time. A natural problem that
arises in such circumstances is the following: given a database $\mathcal{D}$
and a threshold value $t$, find all collections of "generalizations" of
attributes which are "supported" by less than $t$ transactions from
$\mathcal{D}$. We call such collections infrequent elements. Due to
monotonicity, we can reduce the output size by considering only \emph{minimal}
infrequent elements. We study the complexity of finding all minimal infrequent
elements for some interesting classes of posets. We show how this problem can
be applied to mining association rules in different types of databases, and to
finding "sparse regions" or "holes" in quantitative data or in databases
recording the time intervals during which a re-occurring event appears over
time. Our main focus will be on these applications rather than on the
correctness or analysis of the given algorithms.
