On the intersection of maximal partial clones and the join of minimal partial clones

Let A be a nonsingleton finite set and M be a family of maximal partial clones with trivial intersection over A. 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 A, then what is the smallest possible cardinality of F? The purpose of this note is to present results related to these two problems.