Clones above the unary clone
  Let c be the cardinality of the continuum.   We give a family of pairwise incomparable clones (on a countable base set) 2^c members, all with the same unary fragment, namely the set of all unary operations.   We also give, for each n, a family of 2^c clones all with the same n-ary fragment, and all containing the set of all unary operations. 
Martin Goldstern and G\'abor S\'agi and Saharon Shelah
math.RA math.LO