Decompositions of functions based on arity gap

Miguel Couceiro (1) , Erkko Lehtonen (2) , Tamas Waldhauser (1)

1 Mathematics Research Unit
2 Uni.lu - Université du Luxembourg

We study the arity gap of functions of several variables defined on an arbitrary set A and valued in another set B. 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 B 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 n and p, the number of n-ary functions that depend on all of their variables and have arity gap p.

Domaines: Mathématiques [math]

Miguel Couceiro, Erkko Lehtonen, Tamas Waldhauser. Decompositions of functions based on arity gap. Discrete Mathematics, 2012, 312 (2), pp.238-247. ⟨10.1016/j.disc.2011.08.028⟩. ⟨hal-01090606⟩