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In this answer, formal intersection appears in this context:

Let each intersection $U_{i_1} \cap \cdots \cap U_{i_n}$ be denoted as $U_{i_1 \cdots i_n}$. Treat these sets as symbols (i.e., formal intersection), and distinguish $U_{ij}$ from $U_{ji}$ even though they are the same set.

How can the notion of "formal intersection" be made precise? I found nowhere a definition of it. I mean a similar definition as that of formal sum.

Edit: I mean a definition using only well-defined mathematical objects, such as set, function, etc, and no reference to the notation. Just like in the case of the definition of formal sum in the referenced Wikipedia article.

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When something is said to be a "formal sum" or "formal intersection" or "formal" anything, it is the same thing. It just means, treat it as a symbols only instead of as having the normal meaning.

So you are familiar with the formal sum $$ 2 + 3, $$ which means the sequence of symbols $2, +, 3$ (with "$+$" in particular having no defined meaning, and being just a symbol.) Similarly, here we have $$ U_{ij} $$ with $U$ having no defined meaning, and being just a symbol with two subscripts. (In particular, the order of the subscripts is significant.)

Caleb Stanford
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    To be clear, then, in this case a "formal intersection" really just means a finite tuple of indices $(i_1,\dots,i_n)$, and we will presumably later make use of the function assigning to such an index the actual intersection $U_{i_1}\cap\dots\cap U_{i_n}$. – Eric Wofsey Sep 20 '16 at 06:30