# Is there any specific terminology to refer to an initial sequence of a sequence?

Lets say you have a sequence $S = (0, 1, 2, 3, 4, 5, 6, 7, 8)$

And another sequence $T = (0, 1, 2, 3)$

Is there any specific mathematical term that defines the relationship between $S$ and $T$, that specifically says that $S$ starts with $T$?

I thought $T$ would be called an initial sub-sequence, but this is incorrect because a sub-sequence seems to be a any subset of the elements of the sequence in the same order of the sequence (so even $(2, 4, 6, 8)$ would be a sub-sequence, while I want the prefix sub-sequence part, i.e. a sub-sequence that the sequence starts with)

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I usually use "initial segment", and "tail segment" or "end segment" or "final segment" to denote the last part of a sequence.

If you want to indicate the segment is not everything, then "proper initial segment" should suffice.

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Thanks, 'initial segment' fits more in what I am trying to express. –  jbx May 19 '13 at 14:26

In computer science (which is, at least at the beginning, maths), you call them words or strings over an alphabet $X$ which contains your letters.

For this alphabet $X$, you define $X^n$ by the usual Cartesian product and $X^*=\bigcup\limits_{n\in \Bbb N}X^n$. $X^*$ is the set of all words written with letters of $X$.

Then you can define a product $\cdot$ on $X^*$: $\left(u_1,\dots,u_p\right)\cdot \left(v_1,\dots,v_q\right)=\left(u_1,\dots,u_p,v_1,\dots,v_q\right)$

And so saying a word $u$ start with $v$ is saying : $\exists w \in X^*, u=v\cdot w$. And $v$ is therefore called a left factor of $u$, or a prefix.

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Thanks for your answer. My context is actually computer science. I have a sequence of actions and I want to refer to the first few elements of the sequence. –  jbx May 18 '13 at 15:58
Well since you can also define the length of a word (the number of factors of $X^1$ needed to write the word), you can just say "the left factor/prefix of $v$ of length $k$"). –  xavierm02 May 18 '13 at 16:00