Abstract
We propose a parametrized version of arity gap. The parametrized arity gap gap (f, ℓ) of a function \(f \colon A^n \to B\) measures the minimum decrease in the number of essential variables of f when ℓ consecutive identifications of pairs of essential variables are performed. We determine gap (f, ℓ) for an arbitrary function f and a nonnegative integer ℓ. We also propose other variants of arity gap and discuss further problems pertaining to the effect of identification of variables on the number of essential variables of functions.
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The first named author is supported by the internal research project F1R-MTH-PUL-09MRDO of the University of Luxembourg. The third named author acknowledges that the present project is supported by the TÁMOP-4.2.1/B-09/1/KONV-2010-0005 program of the National Development Agency of Hungary, by the Hungarian National Foundation for Scientific Research under grants no. K77409 and K83219, by the National Research Fund of Luxembourg, and cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND).
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Couceiro, M., Lehtonen, E. & Waldhauser, T. Parametrized Arity Gap. Order 30, 557–572 (2013). https://doi.org/10.1007/s11083-012-9261-5
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DOI: https://doi.org/10.1007/s11083-012-9261-5