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. 1. Two problems about clones on N 1.1. Descriptive set theory of algebras and clones on N. Let X be a set, and denote for all n ≥ 1 the set X n of all functions on X in n variables by O(n). Then O := ⋃ n≥1 O (n) is the set of all finitary functions on X. A clone is a subset C of O which contains all projections (i.e., all functions satisfying an equation of the form f(x1, . . . , xn) = xk) and which is closed under composition, i.e., for all n,m ≥ 1, all n-ary f ∈ C , and all m-ary g1, . . . , gn ∈ C , the m-ary function f(g1(x1, . . . , xm), . . . , gn(x1, . . . , xm)) is also an element of C . In other words, C is required to be closed under building of terms from its functions. The latter perspective shows that clones arise naturally as sets of term functions of algebras with domain X; in fact, the clones on X are precisely the sets of term functions of such algebras. Since many properties of an algebra (e.g., subalgebras, congruences) depend only on the clone of the algebra, clones are in that sense canonical representatives of algebras, and have been studied intensively in the literature; for a monograph on clones, see [Sze86]. While clones arise in this way on base sets X of arbitrary (finite or infinite) cardinality, there is an additional perspective on clones from the viewpoint of descriptive set theory that can only be enjoyed on a countably infinite base set, as we will outline in the following. For notational and conceptual convenience, let us identify X with the set of natural numbers N. Then, for every fixed n ≥ 1, we can view O(n) = N n as a topological space whose topology is naturally given by equipping N with the discrete topology and viewing N n as a product space. This space is homeomorphic to the Baire space (the metric space on N in which two functions are closer the later they start to differ), and a function f ∈ O(n) is in the closure of a set F ⊆ O(n) iff for every finite subset of N there exists g ∈ F which agrees with f on this set. The set O then becomes the sum space of the spaces O(n), i.e., the open subsets of O are those sets F for which the n-ary fragment F (n) := F ∩O(n) is open in O(n) for every Date: February 5, 2013. 2010 Mathematics Subject Classification. Primary 08A40; secondary 54H15; 22A30; 03E15.