What is an Initial Segment here?: Reidemeister-Schreier Method

[Bacle]

Hi, everyone:

I am reading an article on the Reidemeister-Schreier
method for finding a presentation of a subgroup H of
a group G, given a presentation for G , in which this
statement is made:

A Schreier transversal of a subgroup H of F, free with
basis X, is a subset T of F such that for distinct
t in T, the cosets Ht are distinct, and the union of
the Ht is F, and such that ...

** every initial segment of an element of T itself
belongs to T **

Now, I understand that the cosets of H in G
partition G, and we select a subset T of G so that
Ht=/Ht' for t,t' in T, and \/Ht =G , but I have no
idea of what an initial segment would mean in this
context; are we assuming there is some sort of ordering
in T; maybe inherited from G ,or are we using
Well-Ordering Principle some how?

I thought we may have been considering the case where H
has infinite index in G, so that we assign a well-ordering in G
so that we can use Choice to select the least element g
representing the class Hg (i.e., all g_i in G with Hg_i=Hg ), but
I am not too clear on this.

Thanks for Any Ideas.

 

[fresh_42]

I assume it is either simply a word in $F(T)$ or it refers to the enumeration process how to find a Schreier transversal by the Todd-Coxeter method. Cp. http://cocoa.dima.unige.it/conference/cocoa2013/posters/YvonneKaroske.pdf
I also found under "Fox calculus" an example where a differential operator $F(T)\longrightarrow \mathbb{Z}(F(T))$ was defined in a way that uses segments of words in $F(T)$.