A note on $\Delta_2^0$-spectra of linear orderings and degree spectra of the successor relation

In this paper we construct linear orderings whose $\Delta_2^0$-spectra coincide with classes of all high$_0$ and high$_1$ degrees, respectively. We also prove that there exists a computable linear ordering such that its degree spectrum of the successor relation coincides with a fixed nonempty class of degrees which represents a $\Sigma_1^0$-spectrum of some $\emptyset'$-computable linear ordering.