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Suppose I have two ordered pairs $(a_1, b_1)$ and $(a_2, b_2)$, each of type $A \times B$

I want to combine them into a single pair of type $A^2 \times B^2$:

$(a_1,b_1)$ OP $(a_2,b_2)$ = $( (a_1,a_2), (b_1,b_2) )$

Is there a name for this operation, or a standard operator? Someone suggested $\triangle$, but I've only ever seen that used for the Laplacian. It's called "zip" in functional programming.

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    My application uses functions on pairs of $A \times B$ pairs. I'm trying to avoid writing them all out as $(a_i,b_i)$ just to make things cleaner.2011-05-23

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I have never before seen a standard and well established notation for that.

As I usually say, in the absence of a certain notation, one can always choose a known one - or invent one, and be detailed and consistent about using it.

In this case, if you have chosen $\otimes$ just make sure this notation is not used for anything else standard in your field, and be sure to define it in your work.

Usually it works just fine.

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It seems to me more natural to write $(a_1, b_1) \times (a_2, b_2) \in (A \times B) \times (A \times B)$ than anything else. You can use this for arbitrary products, writing $a \times b \in A \times B$ by analogy with the tensor product. The reason I suggest you do this is that for sets $\times$ is obviously commutative but there are situations where we want to use something that behaves like $\times$ but is not obviously commutative, so there's no reason to needlessly switch around factors if you don't need to.

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    @ljp: okay, and probably you use $\otimes$ both for the tensor product of two vectors and for the tensor product of two vector spaces. It's a completely sensible overload.2011-05-23
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You are trying to transform an element of $(A\times B )\times (A\times B)$ into an element of $(A\times A )\times (B\times B)$.

If you view an ordered pair as a vector with two coordinates, and an ordered pair of ordered pairs as a $2\times2$ matrix, then this could be viewed as a kind of

transpose.