Terminology for combining ordered pairs? 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.
 A: 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.
A: 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.
A: 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.

