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

TL;DR

This paper constructs examples of algebraic structures on natural numbers with complex descriptive properties, including a non-Borel clone and a seemingly complex ideal that is actually Borel.

Contribution

It provides the first known example of a closed algebra with only unary operations having a non-Borel clone, and an ideal whose clone appears complex but is Borel.

Findings

A closed algebra with unary operations has a non-Borel clone.

An ideal's clone can be complex in definition but Borel in actuality.

Examples highlight the nuanced relationship between algebraic structure and descriptive complexity.

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.