# Is there a common symbol for concatenating two (finite) sequences?

Say we have two finite sequences $X = (x_0,...,x_n)$ and $Y = (y_0,...,y_n)$. Is there a more or less common notation for the concatenation of these sequences, like $\sum (X,Y) = (x_0,...,x_n,y_0,...,y_n)$?

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In fact I meant a symbol for two named sequences, where it is not clear what their content looks like. I changed the question correspondingly. –  Chiel Feb 9 '13 at 10:51
I have seen more notations for this, the easiest is $XY$.. –  Berci Feb 9 '13 at 10:54
Depending on context, I have seen $XY$, $X\cdot Y$, and $X^{\frown}Y$. –  Brian M. Scott Feb 9 '13 at 13:40
Yes, probably this $X{}^\frown Y$ is the symbol you are looking for. –  Berci Feb 9 '13 at 14:03
I have seen $X||Y$. –  Ron Gordon Feb 9 '13 at 14:33

The comments suggest the following notations for the concatenation of $X$ and $Y$:

• $X^\frown Y$ (given by X^\frown Y);
• $XY$ (given by XY);
• $X \cdot Y$ (given by X \cdot Y);
• $X \Vert Y$ (given by X \Vert Y);

of which the first seems not to be in use for other concepts, making it especially suitable.

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The same question on Tex SE.

From there, and more:

• $X \oplus Y$ (given by X \oplus Y);
• $(X,Y)$ (given by (X,Y));

I would avoid $X \times Y$, $XY$ or $X \cdot Y$ to not confuse it with any sort of multiplication / product.

And I would also not use $X \otimes Y$ because it is usually the tensor product. (See also here.)

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$\newcommand\mdoubleplus{\mathbin{+\mkern-10mu+}}$ In haskell the $\mdoubleplus$ operator is used for concatenating lists.

You can define it in latex using the command

\newcommand\mdoubleplus{\mathbin{+\mkern-10mu+}}

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If $x$ and $y$ are finite sequences, you could denote their concatenation by $xy$. Let me explain. There's at least two ways of formalizing the statement "$x$ and $y$ are finite sequences in $X$"

• $x$ and $y$ are functions of type $[\:\!n) \rightarrow X$, where $[\:\!n)$ is a shorthand for the set $\{0,\ldots,n-1\}$.

• $x$ and $y$ are elements of $X^*$, where $X^*$ is the monoid freely generated by $X$.

If you're interested in concatenating these things, then you should probably take the second perspective, in which case the concatenation of $x$ and $y$ is simply their product in the monoid $X^*$, which is denoted $xy$.

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