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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 aunion of $n$ single-valued maps. Characterizations of splittings has been amajor 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 mapswhich may not be single-valued. We generalize several splittingcharacterizations which will describe partitions in terms of mixedconfiguration spaces and mixed braid groups, and connected components of thegraph of $f$. We demonstrate the ideas with some examples on tori.  We also discuss the fixed point theory of $n$-valued maps and theirpartitions, and make some connections to the theory of finite-valued maps dueto Crabb.

Partitions of $n$-valued maps