Abstract
We prove that the existential theory of the Turing degrees, in the language with Turing reduction, 0, and unary relations for the classes in the generalized high/low hierarchy, is decidable.
We also show that every finite poset labeled with elements of 
(where is the partition of 
induced by the generalized high/low hierarchy) can be embedded in preserving the labels. Note that no condition is imposed on the labels.
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Montalbán, A. There is no ordering on the classes in the generalized high/low hierarchies. Arch. Math. Logic 45, 215–231 (2006). https://doi.org/10.1007/s00153-005-0304-0
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DOI: https://doi.org/10.1007/s00153-005-0304-0