Bi-embeddability spectra and bases of spectra

TL;DR

This paper investigates the degree spectra of structures under bi-embeddability, introducing new concepts like bi-embeddable triviality and bases, and characterizes spectra for linear orderings and certain graphs.

Contribution

It introduces the notions of bi-embeddable triviality and bases, and characterizes bi-embeddability spectra for linear orderings and strongly locally finite graphs.

Findings

Several known degree families are bi-embeddability spectra.

Characterization of spectra for linear orderings.

Analysis of bases for spectra of certain graphs.

Abstract

We study degree spectra of structures with respect to the bi-embeddability relation. The bi-embeddability spectrum of a structure is the family of Turing degrees of its bi-embeddable copies. To facilitate our study we introduce the notions of bi-embeddable triviality and basis of a spectrum. Using bi-embeddable triviality we show that several known families of degrees are bi-embeddability spectra of structures. We then characterize the bi-embeddability spectra of linear orderings and study bases of bi-embeddability spectra of strongly locally finite graphs.