Name for the property of “being a Cartesian product of arbitrary sets”

Question

Suppose I have a set $S$ of pairs (or, in general, tuples, or objects with two or more parts/attributes/projections). This set may have the property that

$$(a, b) \in S \wedge (a', b') \in S \implies (a, b') \in S$$

which can also be described as

• $S$ is the Cartesian product of some sets.
• $S$ is closed under $f$ where $f((a, b), (a', b')) = (a, b')$.

Is there a more direct, and well-known or obvious, name for this property, or perhaps for $f$? Preferably one readily understood by those with a background in computer science.

The application is generally that $S$ is some collection of usable objects which have at least two independent attributes, and I want to talk about whether, for the values those attributes actually take on, every combination of those attributes is available, or whether $S$ is instead incomplete (not closed, not orthogonal).

More concretely, where this question arose is that I'm writing a collection of functions (in the programming-language sense) which all do “the same thing” but with different details of the input and output types, and I wanted to talk about the concept of whether or not the set of functions actually written has any obvious gaps, where my notion of “obvious” is “can be constructed by $f$” — note that $f$ is a meta-operation here, and not something the program actually deals in. So $f$ is really not associated with $S$ inherently; it's a tool for understanding, which I don't actually want to have to define in order to communicate that understanding.

“is a Cartesian product” seems fairly good, but it doesn't convey the notion that it's a product of possibly arbitrary sets, as opposed to meaningful ones. That is, suppose I have $S = \{\mathrm a, \mathrm b\} \times \{1, 2, 5\}$ which is a subset of $\mathbb \{\mathrm a, \mathrm b\} \times \mathbb Z^+$ — one could easily want $(\mathrm b, 3)$ which is not a member, so I would prefer to talk specifically about $S$ being closed under $f$, without implying that it is complete in the sense of containing everything one might want, but I don't know a good name for $f$.

Answer 1

First, let me apologise; this no real answer: I know of no name for such sets. But if there isn’t any name, I’d like to suggest a way for naming such sets.

If $X$, $Y$ are sets, then their product $X × Y$ is equipped with projection arrows $$π_X \colon X × Y → X \quad\text{and}\quad π_Y \colon X × Y → Y.$$ So they define a map $$π_X × π_Y \colon (X × Y) × (X × Y) → X × Y,~( (a,b),~(a',b') ) ↦ (a,b').$$ Your $f$ equals this $π_X × π_Y$. So you can do this in any category. (And maybe you will have luck finding names in this context based on this observation.)

You may consider $π_X × π_Y$ to be a multiplication on $M = X × Y$, so let’s call it $μ$ instead and write $(a,b)·(a',b') = (a,b')$. This multiplication, of course, turns out to be associative (any product always selects the leftmost left entry and the rightmost right entry of its factors, so the association doesn’t matter). Therefore, $M$ is a semigroup (and is a monoid if and only if both $X$ and $Y$ are singletons).

I’m pretty sure that, using the Yoneda lemma, this implies that any product in any category becomes a semigroup object this way. So maybe this observation might also help you finding an established name for it.

I know of no name for such a semigroup, but I’d call it the smash-semigroup on $X × Y$, because multiplication smashes together the outermost entries.

Then you would be asking for sub-smash-semigroups or something …