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Is there any common notation for concatenating two tuples?

For instance, let's assume we have two tuples $X:=(x_1, \ldots ,x_n )$ and $Y:=(y_1, \ldots ,y_m)$. I want the resulting tuple to be $Z:=(x_1, \ldots ,x_n , y_1, \ldots ,y_m)$.

For sequences, I have read here that one possibility is to use $\frown$ (\frown).

Is this used for tuples as well, such that $Z:=X\frown Y$ would work?

Firas_
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1 Answers1

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Tuples and sequences are the same thing, so it seems to me that this question is basically a duplicate of the previous one. But in any case I disagree with the accepted answer. I have never seen $\frown$ used in this way. In mathematics I would use $\times$ for the cartesian product here. I would notate individual tuples with lowercase letters and reserve uppercase letters for sets, so $x = (x_1, \dots x_n)$ and $y = (y_1, \dots y_m)$ give $x \times y = (x_1, \dots x_n, y_1, \dots y_m)$.

If the $x_i$ and $y_i$ are vectors in vector spaces you can also use $\oplus$ for the direct sum.

Qiaochu Yuan
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  • I disagree. I've seen $\frown$ or $\smallfrown$ used for concatenation before, and don't recall seeing $\times$ overloaded in this way. An internet search for, say, \frown concatenation brought me five independent sources using it that way on the first page of results alone, including two textbook references. – Mark S. Jan 17 '21 at 03:39
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    @Mark: it's a special case of what I think is a reasonably standard overloading of $\times$ to refer to the operation which given functions $f : X \to Y, g : Z \to W$ produces the function $f \times g : X \times Z \to Y \times W$. This is in turn a special case of a reasonably standard overloading of a functor to act on morphisms with the same symbol it acts on objects, which is most frequently used for the tensor product. I find $\frown$ both ugly and unintuitive but I suppose it at least has the benefit of not having other common meanings. – Qiaochu Yuan Jan 17 '21 at 03:43