Decompositions of functions based on arity gap

Waldhauser, Tamás; Lehtonen, Erkko; Couceiro, Miguel

Waldhauser, Tamás; Lehtonen, Erkko; Couceiro, Miguel (2012), Decompositions of functions based on arity gap, Discrete Mathematics, 312, 2, p. 238-247. http://dx.doi.org/10.1016/j.disc.2011.08.028

Type

Article accepté pour publication ou publié

External document link

http://arxiv.org/abs/1003.1294

Date

2012

Journal name

Discrete Mathematics

Volume

312

Number

Publisher

Elsevier

Pages

238-247

Abstract (EN)

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.

Subjects / Keywords

Arity gap; variable identification minor; Boolean group