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A closed algebra with a non-Borel clone and an ideal with a Borel clone

2011/12/04 by Goldstern, Martin, Pinsker, Michael, Shelah, Saharon
#FOS: Mathematics #Logic (math.LO) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1112.0774

Abstract

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.

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