Is there any algebraic structure equipped with multi-valued map(s)? A group is a non-empty set $G$ with a map $m: G \times G \mapsto G$ where the map is associative, has an identity element and has an inverse element for all $ g \in G$.
Similarly in other algebraic structures, we equip them with one or more mapping that are always single-valued i.e. the map involved is always like the following: $m: S_1 \times S_2\times \ldots \times S_n \mapsto S_a$.
Why is it never like the following: $m: (S_1 \times S_2) \mapsto (S_1 \times S_2)$ or like $m: (S_1 \times S_2\times \ldots \times S_n) \mapsto (S_1 \times S_2\times \ldots \times S_n)$
Is there any algebraic structure with such type of map?
PS: I am confused about whether I should use the word map here. Cause almost everywhere the word map is used to denote a single-valued assignment. While my interest here is multiple-valued assignment.
 A: "Mapping to multiple elements": Yes, these structures are called "hyper" and the concept of hypergroups (careful: nowadays the definition is a bit different) was first introduced by Krasner in 1934 if I recall correctly. The main difference is a the use of $n$-ary hypermaps, which are maps of the form $S^n \rightarrow \mathcal{P}(S) \setminus \lbrace \emptyset \rbrace$. Choosing multiple values is formalized by mapping to the subset of the wanted multiple values.
This is for example used to produce one model of $\mathbb{F}_1$-geometry by doing algebraic geometry with respect to hyperrings. Let me actually (without mentioning the precise definition) give an example of a hyperring (actually a hyperfield):
The so called Krasner hyperfield $\mathbb{K} = \lbrace 0,1 \rbrace$ is given by setting $1 + 1 = \lbrace 0,1 \rbrace$ and using the addition and multiplication from $\mathbb{F}_2$ otherwise. This hyperfield captures the arithmetic of a $0$-element with non-zero elements in the following sense:
The sum of $0$ and $0$ is $0$ and the sum of a $0$-element and non-zero elements: is non-zero while the sum of two non-zero elements could be either $0$ or something non-zero, which is why we want it to be the set $\lbrace 0,1 \rbrace$.
