Texts that talk about multi-valued functions on the reals and complexes

I am interested in texts that discuss multi-valued functions, especially on $\mathbb{R}$ and $\mathbb{C}$. For example, in many precalculus and calculus texts, they define the n-th root function. It is properly not an actual function, but a multi-valued function. They also talk about inverse trigonometric functions, which are again actually multi-valued functions. I also want to know which properties of single-valued functions hold for multi-valued functions. For example, while the commutative and associative properties hold, the distributive property doesn't. To give just one example, if we let $s$ be the square root multi-valued function, then $s(x) + s(x)$ is not in general equal to $2*s(x)$, because, for instance, $s(1) + s(1) = \{1,-1\} + \{1,-1\} = \{2,0,-2\}$, while $2*s(1)=2*\{1,-1\}=\{2,-2\}$. Also, I am interested in literature on clones of multi-valued functions, analogous to clones of functions in universal algebra. Clones of multi-valued functions would consist of compositions of multi-valued functions, similar to how clones in universal algebra consist of compositions of single-valued functions.

Solution 1:

To the best of my knowledge, a hyperstructure is an algebraic structure where the operations may be multivalued. Hypergroups, hyperrings, and hyperfields have been studied due to their use in class field theory, matroid theory, and algebraic geometry.

See for example:

Connes and Consani. The hyperring of adele classes. Journal of Number Theory. 2011.

Baker and Bowler. Matroids over partial hyperstructures. Advances in Mathematics. 2019.

Jun. Algebraic geometry over hyperrings. Advances in Mathematics. 2018.

Last I knew their was no influential or highly-regarded research on general hyperstructures, but searching for phrases similar to 'universal hyperalgebra' should yield some results.