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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$\begingroup$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); - $X \mathbin\Vert Y$ (given by
X \mathbin\Vert Y);
of which the first seems not to be in use for other concepts, making it especially suitable.
$\endgroup$ 5 $\begingroup$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.)
Some relevant Wikipedia pages with common notations:
$\endgroup$ 3 $\begingroup$$\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+}}Computer science often uses the ⧺ (U+29FA) symbol for concatenation.
This is \doubleplus in the LaTeX package unicode-math (which requires a modern engine that supports Unicode), as well as the legacy packages stix and stix2. Or, in the modern toolchain, you can use the Unicode symbol in your source.
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 $xy$.
$\endgroup$ $\begingroup$In formal specifications, one way to concatenate two sequences is using the Haskell concatenation symbol as indicated in one of the comments above. In a $\mathrm\TeX$ editor one can type the following: X +\!\!\!+ Y. The result appears like this, $X+\!\!\!+Y$.
Starting with your defined sequences $X = (x_0, \ldots, x_n)$ and $Y=(y_0,\ldots,y_n)$, you can use the commonly accepted tuple/ordered pair notation:\begin{align} (X,Y) &= \left( \left(x_0,\ldots,x_n\right), \left(y_0,\ldots,y_n\right) \right) \\ &= \left(x_0,\ldots,x_n,y_0,\ldots,y_n\right) \end{align}
$\endgroup$ 3 $\begingroup$First of all, the $\frown$ (like the $\smile$) are used in algebraic topology for the cap (cup) product, so there is another use for this symbol. I haven't seen $\uplus$ suggested and I have searched extensively to find an existing use without success. It seems especially suited because sequences are in fact index-ordered sets which can contain duplicate elements. Concatenation is the union of such sets which preserves the order of concatenation and allows duplicates.
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