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

TL;DR

This paper generalizes Swierczkowski's Lemma to broader classes of functions, characterizes arity gaps in finite functions, and classifies pseudo-Boolean functions based on their arity gaps.

Contribution

It extends the lemma to B-valued functions and essentially unary functions, characterizes arity gaps via quasi-arity, and classifies pseudo-Boolean functions.

Findings

Generalized Swierczkowski's Lemma to B-valued functions.

Characterized arity gaps of small arity functions using quasi-arity.

Explicit classification of pseudo-Boolean functions by arity gap.

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.