Abstract

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 $\alpha$