A closed algebra with a non-Borel clone and an ideal with a Borel clone
  Algebras on the natural numbers and their clones of term operations can be classified according to their descriptive complexity. We give an example of a closed algebra which has only unary operations and whose clone of term operations is not Borel. Moreover, we provide an example of a coatom in the clone lattice whose obvious definition via an ideal of subsets of natural numbers would suggest that it is complete coanalytic, but which turns out to be a rather simple Borel set. 
Martin Goldstern, Michael Pinsker, Saharon Shelah
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