What is a standard name for a "relation" as a subset of $X\times \mathcal P(X)$ rather than of $X\times X$

Question

(Binary) relations on $X$ are formalized as subsets of $X\times X$.

But there are also times when a "relation" is a subset of $X\times \mathcal P(X)$. For example, in topology, we may say that $x$ is in the closure of $X$, and consider this as a relation, and write $x<X$.

Is there a standard name for such a point-to-set "relation"?

Answer 1

I don't know that there's a standard name for them. But you could call them unary quantifiers. To explain why will take a bit of exposition. You may want to make yourself a cup of tea and get comfortable, as this will take a some time.

Part 0. If $Y$ is a set, we can speak of the quantifiers on $Y$, which I define as the elements of $\mathcal{P}(\mathcal{P}(Y))$. The two most well-known quantifier families are:

$$\forall_Y := \{A \subseteq Y : \forall y \in Y(y \in A)\}, \qquad \exists_Y := \{A \subseteq Y : \exists y \in Y(y \in A)\}.$$

The reason the above definitions are natural are because they gives us the following equivalences:

$$\forall(y \in Y) P(y) \iff \{y \in Y : P(y)\} \in \forall_Y$$

$$\exists(y \in Y) P(y) \iff \{y \in Y : P(y)\} \in \exists_Y$$

But there's heaps of other examples, of course. For example, a topology on $X$ is a quantifier on $X$ satisfying two axioms.

We can use this observation to invent our own quantifiers. For example, suppose $\tau$ is the standard topology on the real line. Then, simulating the above equivalences, we can write: $$\tau(y \in \mathbb{R}) P(y) \iff \{y \in \mathbb{R} : P(y)\} \in \tau$$

For example, the statement $$\forall(a,b \in \mathbb{R}) \tau(y \in \mathbb{R}) : a < y < b$$ is basically saying that the interval $(a,b)$ is an open set.

Part 1. If $Y$ is a set, we can speak of the elements of $Y$, which are basically the same thing as functions $1 \rightarrow Y$. More generally, we can speak of function $X \rightarrow Y$, where $X$ is an index set on the element of $Y$ depends. If so, then maybe having two words for "function" and "element" is somehow a mistake. A function is just a dependent element. An element is a just a function whose domain of dependence is trivial. They're gesturing at exactly the same concept; why have two words?

Similarly, if $Y$ is a set, we can also speak of the subsets of $Y$, which are basically the same thing as functions $1 \rightarrow \mathcal{P}(Y)$. More generally, we can speak of subsets that are dependent on an index set $X$. These are functions $X \rightarrow \mathcal{P}(Y)$, where $X$ is the domain upon which our subsets are depending. Such things are usually called relations, and often defined as elements of $\mathcal{P}(X \times Y)$. Indeed, it is a general fact that $$X \rightarrow \mathcal{P}(Y) \cong \mathcal{P}(X \times Y).$$ This may look strange, but it's not too surprising when rewritten as $$(2^Y)^X \cong 2^{X \times Y}.$$ A good exercise is to write down this bijection explicitly.

Anyway, in light of this, maybe having two words for "subset" and "relation" is a mistake. A subset of $Y$ is just a relation $1 \rightarrow Y$. A relation from $X$ to $Y$ is just a subset of $Y$ dependent on $X$. They're gesturing at exactly the same concept; why have two words?

Now, perhaps you see where I'm going with this. As mentioned previously, if $Y$ is a set, we can also speak of the quantifiers on $Y$, which are basically the same thing as functions $1 \rightarrow \mathcal{P}(\mathcal{P}(Y))$. Supposing we want to generalize this so that dependence on a set $X$ is allows. We're looking for a name to give to functions $X \rightarrow \mathcal{P}(\mathcal{P}(Y))$. Whatever are we to call them? Let's make good on our previous observations that there's really just too many words for things and just call them quantifiers.

Definition. A quantifier $X \rightarrow Y$ is a function $X \rightarrow \mathcal{P}(\mathcal{P}(Y))$.

We recover the previous definition by taking $X = 1$, and we recover the concept you're interested in by taking $X = Y$.

"Hang on", you may say. "That's not what I'm interested in. I'm interested in a name for the elements of the set $\mathcal{P}(X \times \mathcal{P}(X))$. You've essentially given me a name for the elements of the set $X \rightarrow \mathcal{P}(\mathcal{P}(X))$. These are not the same!"

"Aha!" says I. "Go and look at that bijection you wrote down earlier. It puts these two sets into natural correspondence!"

Part 2. Here's some existing terminology that's worth being aware of:

Accepted Terminology.

A nullary function on $X$ is a function $1 \rightarrow X$.

A unary function on $X$ is a function $X \rightarrow X$.

A binary function on $X$ is a function $X^2 \rightarrow X$.

etc.

It's tempting to apply this terminology to relations, too as in:

Terminology that won't work.

A nullary relation on $X$ is a function $1 \rightarrow \mathcal{P}(X)$.

A unary relation on $X$ is a function $X \rightarrow \mathcal{P}(X)$.

A binary relation on $X$ is a function $X^2 \rightarrow \mathcal{P}(X)$.

It doesn't work because what we're calling a unary relation above would actually be regarded as a binary relation by most authors. This forces us to invent a new phrase: "multivalued function". For example, we can then speak of a "unary multivalued function on $X$," which is the same thing as a binary relation on $X$.

There are no such issues with extrapolating to the world of quantifiers, however, because we just made the terminology up, so there's really no standards!

Proposed New Terminology.

A nullary quantifier on $X$ is a function $1 \rightarrow \mathcal{P}(\mathcal{P}(X))$.

A unary quantifier on $X$ is a function $X \rightarrow \mathcal{P}(\mathcal{P}(X))$.

A binary function on $X$ is a function $X^2 \rightarrow \mathcal{P}(\mathcal{P}(X))$.

etc.

Thus since a unary quantifier on $X$ is a function $X \rightarrow \mathcal{P}(\mathcal{P}(X))$, and since we a previous argument this is the same as an element of $\mathcal{P}(X \times \mathcal{P}(X))$, I propose that you can call such things unary quantifiers.