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Let $A \ne \emptyset$. Let $S = \{(a_1, a_2, \ldots , a_n) : a_i \in A$ and $ n \in \mathbb{N}\}$.

Now I'm curious if there is a more concise (and standard) way of writing this set down?

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    $\begingroup$ So your set is almost $A^{\lt\omega}$, the set of all finite sequences in $A$, except that you disallow $n=0$. I guess you could call it $A^{\lt\omega}\setminus A^0$? Or $\bigcup_{n=1}^{\infty}A^n$? $\endgroup$ – bof Nov 8 '13 at 1:18
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You could write $A^{<\omega}$ or $\bigcup_{n \in \mathbb{N}} A^n$. The former contains the empty string (and the latter does too if you admit $0 \in \mathbb{N}$).

Generally if $\alpha$ is an ordinal number then $A^{\alpha}$ denotes the set of sequences of elements of $A$ of order-type $\alpha$. When $\alpha \in \mathbb{N}$ this amounts to saying the number of $\alpha$-tuples. Then you can write things like $A^{<\alpha}$ to mean the set of sequences of order-type (length) less than $\alpha$.

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The set of all finite tuples of a given set $A$ is known as the free monoid on $A$ and in that context is commonly denoted by $A^*$. It also comes up in the context of language hierarchies (e.g., regular languages etc.) where it is also denoted by $A^*$.

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