Is there a generalization of universal algebra that can express multi-valued functions? Is there a generalization of universal algebra that can express multi-valued functions?
I've seen mentions of generalizations of universal algebra that add conditionals and have heard about generalizations that use $\le$ instead of $\approx$ (as a topic within order theory, I think). This question asks about extending universal algebra with multiple sorts.
This answer to a question I asked a month ago defines a new quasi-operation $*$ that can be thought of as a superposition of $+$ and $\times$. This comment on that answer specifically asks about a generalization of ordinary universal algebra.
Are there already-studied generalizations of universal algebra that can handle quasi-operations like $*$?

One example of a potential generalization might be replacing $\approx$ with $\subset$, examples below.
$$ a + b \subset a * b $$
$$ a \times b \subset a * b $$
This generalization might not be expressive enough though; it is just an example. Without the ability to insist on additional rules (such as requiring certain functions to be single-valued or requiring functions to be non-zero-valued), we always have an "empty structure" where the functions always return the empty set and a "full structure" where they always return the entire universe.
 A: I'm not an expert (indeed I'm the above-linked commenter), but let me make a quick observation which suggests that this topic is probably subsumed by the theory of partially-orderd algebras (which didn't occur to me at the time of my original comment).
Suppose $(X,\mathcal{F})$ is a "multivalued algebra," that is, $X$ is a set and $\mathcal{F}$ is a collection of functions from Cartesian powers of $X$ to $\mathcal{P}_{\not=\emptyset}(X)$ (or $\mathcal{P}(X)$ if we want to allow partiality). This induces a collection of singlevalued algebras in the same signature as follows:

*

*We start by turning $\mathcal{P}(X)$ into a partially ordered single-valued algebra (which I'll conflate with $\mathcal{P}(X)$ itself): for each $n$-ary $f\in\mathcal{F}$ we get a corresponding $\hat{f}: \mathcal{P}(X)^n\rightarrow\mathcal{P}(X)$ given by $$(S_1,...,S_n)\mapsto\{x\in X: \exists s_1\in S_1,...,s_n\in S_n(x\in f(s_1,...,s_n))\}.$$ As for the partial ordering, we use $\subseteq$ as expected.


*Now we look at the collection $\mathbb{A}_{X,\mathcal{F}}$ of sub-partially-ordered-algebras of $\mathcal{P}(X)$ which contain all the singletons. Containing the singletons amounts to "building off of $X$" in some sense; omitting a singleton would be an odd thing to do. Note that $\mathcal{P}(X)$ has lots of subalgebras which might be of more interest than the whole (e.g. its smallest subalgebra containing all singletons, or the subalgebra consisting of the nonempty sets, or ...), so it's reasonable to consider this whole collection rather than fixating on $\mathcal{P}(X)$.
All the natural questions about multivalued algebra I can think of offhand can be translated directly to questions about either the $\mathbb{A}_{X,\mathcal{F}}$s or "canonical" elements of those collections (e.g. their smallest elements). As an outsider, this feels to me like good heuristic evidence that we're still solidly within the context of single-valued partially ordered algebras.
