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

Discrete Mathematics, 309(20) (2009), 5905-5912
http://dx.doi.org/10.1016/j.disc.2009.04.009

Swierczkowski's lemma - as it is usually formulated–asserts that if f: An -> A is an operation on a finite set A, n >= 4, 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.