Monotone reducibility over the Cantor space

Abstract

Define the partial ordering ⩽ \leqslant on the Cantor space ω 2 {}^\omega 2 by x ⩽ y x \leqslant y iff ∀ n x ( n ) ⩽ y ( n ) \forall n\,x(n) \leqslant y(n) (this corresponds to the subset relation on the power set of ω \omega ). A set A ⊆ ω 2 A \subseteq {}^\omega 2 is monotone reducible to a set B ⊆ ω 2 B \subseteq {}^\omega 2 iff there is a monotone (i.e., x ⩽ y ⇒ f ( x ) ⩽ f ( y ) x \leqslant y \Rightarrow f(x) \leqslant f(y) ) continuous function f : ω 2 → ω 2 f:{}^\omega 2 \to {}^\omega 2 such that x ∈ A x \in A iff f ( x ) ∈ B f(x) \in B . In this paper, we study the relation of monotone reducibility, with emphasis on two topics: (1) the similarities and differences between monotone reducibility on monotone sets (i.e., sets closed upward under ⩽ \leqslant ) and Wadge reducibility on arbitrary sets; and (2) the distinction (or lack thereof) between ‘monotone’ and ‘positive,’ where ‘positive’ means roughly ‘a priori monotone’ but is only defined in certain specific cases. (For example, a Σ 2 0 \Sigma _2^0 -positive set is a countable union of countable intersections of monotone clopen sets.) Among the main results are the following: Each of the six lowest Wadge degrees contains one or two monotone degrees (of monotone sets), while each of the remaining Wadge degrees contains uncountably many monotone degrees (including uncountable antichains and descending chains); and, although ‘monotone’ and ‘positive’ coincide in a number of cases, there are classes of monotone sets which do not match any notion of ‘positive.’