In this paper we show that there are various pseudo-jump operators definable over inadmissible $J_{\beta}$ that relate to the failure of admissiblity and to non-regularity. We will use these ideas to construct some intermediate degrees.

If α is a singular cardinal (either real or fake) in L, I exhibit many natural α-re subsets, defined uniformly from the ▵ 1 subsets of α. If α is a true cardinal this provides an uppersemilattice (usl) embedding from the lattice of ▵ 1 subsets of α into the usl of α-re-degrees. It will also be shown that this embedding cannot be extended to the Σ 1 subsets of α.

If $\alpha$ is a singular cardinal (either real or fake) in $L$, I exhibit many natural $\alpha$-re subsets, defined uniformly from the $\triangle_1$ subsets of $\alpha$. If $\alpha$ is a true cardinal this provides an uppersemilattice (usl) embedding from the lattice of $\triangle_1$ subsets of $\alpha$ into the usl of $\alpha$-re-degrees. It will also be shown that this embedding cannot be extended to the $\Sigma_1$ subsets of $\alpha$.

I consider the projectum of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\beta$\end{document}-r.e. set. It is shown that there are tame r.e. sets with small projecta and that there are tame r.e. sets with large projecta.