Author:
ANDREWS URI,BELIN DANIEL F.,SAN MAURO LUCA
Abstract
Abstract
We examine the degree structure
$\operatorname {\mathrm {\mathbf {ER}}}$
of equivalence relations on
$\omega $
under computable reducibility. We examine when pairs of degrees have a least upper bound. In particular, we show that sufficiently incomparable pairs of degrees do not have a least upper bound but that some incomparable degrees do, and we characterize the degrees which have a least upper bound with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees are definable in
$\operatorname {\mathrm {\mathbf {ER}}}$
. We show that every equivalence relation has continuum many self-full strong minimal covers, and that
$\mathbf {d}\oplus \mathbf {\operatorname {\mathrm {\mathbf {Id}}}_1}$
needn’t be a strong minimal cover of a self-full degree
$\mathbf {d}$
. Finally, we show that the theory of the degree structure
$\operatorname {\mathrm {\mathbf {ER}}}$
as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second-order arithmetic.
Publisher
Cambridge University Press (CUP)
Cited by
3 articles.
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