Let $\mathcal C, \mathcal D\subseteq 2^\omega$.
Let
$$
\DeclareMathOperator{\Either}{Either}
 \Either(\mathcal C,\mathcal D)=\{A\oplus B: \text{either }A\in \mathcal C, B\in\mathcal D\text{, or }B\in \mathcal C, A\in\mathcal D\}
$$
Has this operation been named and studied in the context of Medvedev degrees (i.e., strong reducibility of mass problems)?


Its interest comes from the fact that from an element of $\Either(\mathcal C,\mathcal D)$ we cannot necessarily compute an element of $\mathcal C$ (or $\mathcal D$) uniformly.



Join-like operation and Medvedev reducibility
Kojiro Higuchi and Takayuki Kihara have studied operations of this flavour in their papers "Inside the Muchnik degrees" I+II ([doi Part 1](https://doi.org/10.1016/j.apal.2014.01.003),[doi Part 2](https://doi.org/10.1016/j.apal.2014.03.001)). It has been a few years since I read those, and I do not remember whether this particular operation plays a role. However, the idea of combining sets in ways that foil uniform reductions features very heavily in this work.


