Decompositions of functions based on arity gap

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 accordi...

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Bibliographic Details
Published inarXiv.org
Main Authors Couceiro, Miguel, Lehtonen, Erkko, Waldhauser, Tamás
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 05.03.2010
Subjects
Classification
Decomposition
Mathematical analysis
Mathematics - Combinatorics
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1003.1294