  We consider the issue of answering unions of conjunctive queries (UCQs) with
disjunctive existential rules and mappings. While this issue has already been
well studied from a chase perspective, query rewriting within UCQs has hardly
been addressed yet. We first propose a sound and complete query rewriting
operator, which has the advantage of establishing a tight relationship between
a chase step and a rewriting step. The associated breadth-first query rewriting
algorithm outputs a minimal UCQ-rewriting when one exists. Second, we show that
for any ``truly disjunctive'' nonrecursive rule, there exists a conjunctive
query that has no UCQ-rewriting. It follows that the notion of finite
unification sets (fus), which denotes sets of existential rules such that any
UCQ admits a UCQ-rewriting, seems to have little relevance in this setting.
Finally, turning our attention to mappings, we show that the problem of
determining whether a UCQ admits a UCQ-rewriting through a disjunctive mapping
is undecidable. We conclude with a number of open problems.
