Maximal and Minimal Partial Clones

The following two problems are addressed in this paper. Let k ≥ 2, k be a k-element set and M be a family of maximal partial clones with trivial intersection over ks. What is the smallest possible cardinality of M? Dually, if F is a family of minimal partial clones whose join is the set of all partial functions on k, then what is the smallest possible cardinality of F? We show that the answer to these two problems is three.

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