$\mathbf{c}$-quasiminimal enumeration degrees

The notion of a $C$-quasiminimal set, with $C$ an arbitrary subset of the naturals, was introduced by Sasso and presents a relativization of the well-known notion of quasiminimal set which was first constructed by Medvedev for proving the existence of nontotal enumeration degrees. In this article we study the local properties of the partially ordered set of the enumeration degrees containing $C$-quasiminimal sets. In particular, we prove for arbitrary enumeration degrees $\mathbf{c}$ and $\mathbf{a}$ that if $\mathbf{c}<\mathbf{a}$ and $\mathbf{a}$ is a total $e$-degree then each at most countable partial order embeds isomorphically into the partially ordered set of $\mathbf{c}$-quasiminimal $e$-degrees lying below $\mathbf{a}$.