Computable embeddings for pairs of linear orders

We study computable embeddings for pairs of structures, i.e. for classes containing precisely two non-isomorphic structures. Surprisingly, even for some pairs of simple linear orders, computable embeddings induce a non-trivial degree structure. Our main result shows that $\{\omega \cdot k,\omega^\star \cdot k\}$ is computably embeddable in $\{\omega \cdot t, \omega^\star \cdot t\}$ iff $k$ divides $t$.Comment: 20 page

A Total Degree Splitting Theorem and a Jump Inversion Splitting Theorem

Abstract. We propose a meta-theorem from which some splitting theorem for total e-degrees can be derived.

Initial segments of the Σ0 2 enumeration degrees

Using properties of $\mathcal{K}$-pairs of sets, we show that every nonzero enumeration degree $\mathbf{a}$ bounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump as $\mathbf{a}$. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low enumeration degree can be capped by degrees of any possible local jump (i.e., any jump that can be realized by enumeration degrees of the local structure); every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degree $\mathbf{a}$ can be capped below $\mathbf{a}$

A note on the enumeration degrees of 1-generic sets

We show that every nonzero Delta(0)(2) enumeration degree bounds the enumeration degree of a 1-generic set. We also point out that the enumeration degrees of 1-generic sets, below the first jump, are not downwards closed, thus answering a question of Cooper