Decompositions of functions based on arity gap

We study the arity gap of functions of several variables defined on an arbitrary set AA and valued in another set BB. The arity gap of such a function is the minimum decrease in the number of essential variables when variables are identified. We establish a complete classification of functions according to their arity gap, extending existing results for finite functions. This classification is refined when the codomain BB has a group structure, by providing unique decompositions into sums of functions of a prescribed form. As an application of the unique decompositions, in the case of finite sets we count, for each nn and pp, the number of nn-ary functions that depend on all of their variables and have arity gap pp.

► We classify functions of several variables according to their arity gap.
► We decompose functions uniquely into sums of functions of a certain prescribed type.
► We use these results to count the number of functions with a given arity gap.