  An $n$-valued map is a set-valued continuous function $f$ such that $f(x)$
has cardinality $n$ for every $x$. Some $n$-valued maps will "split" into a
union of $n$ single-valued maps. Characterizations of splittings has been a
major theme in the topological theory of $n$-valued maps.
  In this paper we consider the more general notion of "partitions" of an
$n$-valued map, in which a given map is decomposed into a union of other maps
which may not be single-valued. We generalize several splitting
characterizations which will describe partitions in terms of mixed
configuration spaces and mixed braid groups, and connected components of the
graph of $f$. We demonstrate the ideas with some examples on tori.
  We also discuss the fixed point theory of $n$-valued maps and their
partitions, and make some connections to the theory of finite-valued maps due
to Crabb.
