Generating clones with conservative near-unanimity operation

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Due to the Baker-Pixley theorem we know that every clone over a finite domain A containing a near-unanimity operation g is finitely generated. Therefore there exists an integer k such that the clone is generated by its k-ary part. In this paper we are interested in the size of k for a fixed A and fixed arity of a conservative g. We obtain lower bounds for all arities and they turn out to be sharp for arity three.