Topological aspects of the Medvedev lattice

Abstract

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space.

A Survey of Mučnik and Medvedev Degrees

“…Several other recent papers study other aspects of these lattices. [26] considers (among others) D cl s := { dg s (P ) : P ⊆ ω ω and P is closed } D de s := { dg s (P ) : P ⊆ ω ω and P is dense in ω ω } D di s := { dg s (P ) : P ⊆ ω ω and P is discrete }…”

“…No No Several other recent papers study other aspects of these lattices. Lewis, Shore and Sorbi[26] considers (among others) Each of these forms a sublattice of D s . D cl s is shown not to be dual-implicative, but this question is left open for the other structures.…”

Coding true arithmetic in the Medvedev and Muchnik degrees

“…By inspecting the definitions, one can check that if si and98 areclosed (compact) then so are 0~siUl~38 and {f®g \ f esiAg &93}. Thus the closed Medvedev degrees form a distributive sublattice of 9Ut which we denote by 3H c i, and the compact Medvedev degrees form a distributive sublattice of 0)1 (and of Wl d ) which we denote by M §, both as in [8] (the "01" notation is explained below). Both 9Jt c i and OJtJ?/ inherit the least element and the greatest element from Tt.…”

“…For example, Bianchini and Sorbi [1] studied the filter (in QJl) generated by the nonminimum closed degrees. Lewis, Shore, and Sorbi [8] have made a recent study of topologically-defined collections of Medvedev degrees.…”

The degree structure of Weihrauch-reducibility

“…In a related fashion, the question whether certain reducibilities induce Brouwer or Heyting algebras have been studied in the literature, e.g. in [10,14,19,23].…”