On the generation of clones containing near-unanimity operations

Abstract

It is a well-known consequence of the Baker-Pixley-Theorem that any clone containing a near-unanimity operation is finitely generated, leading to the question what arity the generating functions must have. In this paper, we show that, for arbitrary d ≥ 2 and large enough n, (n − 1)^d − 1 is the smallest integer k such that, for every clone C on an n-element set that contains a (d + 1)-ary near-unanimity operation, C ^(k) generates C.