On the 33th page of the book Presentations of Groups, when talking about the Nielsen's Method to prove Nielsen–Schreier theorem, the authors consider $F$ as any free group, $H$ as any subgroup of $F$, and $\ll$ as an order on $F$. Then, they write exactly this:
Given $w \in H$, define
$$H(w) := < \{h \in H | h^{\pm} \ll w^{\pm} \} >,$$
and then put
$$A := \{a \in H | a \not\in H(a) \},$$
so that A consists of just those $a \in H$ that cannot be written as words in smaller elements of $H$. Now let $B$ be a subset of $A$ obtained by throwing away exactly one of each inverse pair of elements of $A$, so that $A = B \cup B^{-1}$.
Thus, my questions are:
- Does $h^{\pm} \ll w^{\pm}$ mean $(h \ll w$ and $h^{-1} \ll w^{-1})$?
- What does "inverse pair of elements" mean? (If it was simply the inverse, then why to say "exactly one" since inverse is unique?).