I have several sets, e.g.

$$A_1 = \{1,2,3\}$$ $$A_2 = \{a,b,c\}$$ $$\vdots$$ $$A_P = \{\alpha,\beta\}$$

And their Cartesian product: $$A_1 \times A_2 \times \dots \times A_P = \{(a_1,a_2,\dots,a_P) \mid a_1 \in A_1, a_2 \in A_2, \dots, a_P \in A_P\}$$

What would be a proper way to indicate that the following matrix is populated by the ordered elements from the Cartesian product above?

$E_{N\times P} = $

\begin{bmatrix} \epsilon_{1,1} & \epsilon_{1,2} & \dots & \epsilon_{1,P}\\ \epsilon_{2,1} & \ddots & &\epsilon_{2,P}\\ \vdots & & &\vdots\\ \epsilon_{N,1} & \epsilon_{N,2} & \dots & \epsilon_{N,P}\\ \end{bmatrix}

A concrete example with only the three sets above would lead to the following $18\times3$ matrix:

\begin{bmatrix} 1 & a & \alpha\\ 1 & a & \beta\\ 1 & b & \alpha\\ 1 & b & \beta\\ 1 & c & \alpha\\ 1 & c & \beta\\ 2 & a & \alpha\\ \vdots&\vdots&\vdots\\ 3 & c & \alpha\\ 3 & c & \beta\\ \end{bmatrix}

I'm looking either for a mathematical expression, or a simple sentence. So far I have:

The rows of $E_{N\times P}$ are populated by the ordered elements in the Cartesian product of the sets $A_p, \forall p=1,...,P$.

But I feel that's a bit heavy and perhaps incorrect (i.e. not sure how to express the idea of "ordered elements" referring to the order in the example).


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