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arXiv · 0712.1753

Generalizations of Swierczkowski's lemma and the arity gap of finite functions

Miguel Couceiro·Erkko Lehtonen
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Abstract

Swierczkowski's Lemma - as it is usually formulated - asserts that if f is an at least quaternary operation on a finite set A and every operation obtained from f by identifying a pair of variables is a projection, then f is a semiprojection. We generalize this lemma in various ways. First, it is extended to B-valued functions on A instead of operations on A and to essentially at most unary functions instead of projections. Then we characterize the arity gap of functions of small arities in terms of quasi-arity, which in turn provides a further generalization of Swierczkowski's Lemma. Moreover, we explicitly classify all pseudo-Boolean functions according to their arity gap. Finally, we present a general characterization of the arity gaps of B-valued functions on arbitrary finite sets A.

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BibTeXRIS

Miguel Couceiro, Erkko Lehtonen. 2009-03-17. Generalizations of Swierczkowski's lemma and the arity gap of finite functions. https://doi.org/10.1016/j.disc.2009.04.009

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