  We introduce a notion of relative primeness for equivalence relations,
strengthening the notion of non-reducibility, and show for many standard
benchmark equivalence relations that non-reducibility may be strengthened to
relative primeness. We introduce several analogues of cardinal properties for
Borel equivalence relations, including the notion of a prime equivalence
relation and Borel partition properties on quotient spaces. In particular, we
introduce a notion of Borel weak compactness, and characterize partition
properties for the equivalence relations ${\mathbb F}_2$ and ${\mathbb E}_1$.
We also discuss dichotomies related to primeness, and see that many natural
questions related to Borel reducibility of equivalence relations may be viewed
in the framework of relative primeness and Borel partition properties.
