Partitions of $n$-valued maps

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.