Monotone reducibility over the Cantor space

Author:
Randall Dougherty

Journal:
Trans. Amer. Math. Soc. **310** (1988), 433-484

MSC:
Primary 03E15; Secondary 54F05

DOI:
https://doi.org/10.1090/S0002-9947-1988-0943302-1

MathSciNet review:
943302

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Abstract: Define the partial ordering on the Cantor space by iff (this corresponds to the subset relation on the power set of ). A set is monotone reducible to a set iff there is a monotone (i.e., ) continuous function such that iff . 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 ) 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 -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.'

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0943302-1

Keywords:
Cantor space,
monotone sets,
positive sets,
Wadge reducibility,
difference hierarchy,
Borel hierarchy

Article copyright:
© Copyright 1988
American Mathematical Society