PREGROUPS
3
Stallings' original theorem about pregroups implies the converse of theorem 3.
Stallings' result is harder to prove and deeper, in that it provides precise information
about the word problem in U(P). In fact if P is a finite pregroup, then Stallings'
theorem implies U(P) has solvable word problem.
Suppose P is a pregroup. Let X=(xh . . . ,xn) and
A ={\=a^a{, . . . ,an_han = \) be P-words. Suppose for i = \, . . . ,n that (afi\ ,xhat)
associates. We then define a P-word X*A = (xxax,a{xx1a1, . . . ,a~}xxn) called X
interleaved by A. Let R be the set of reduced words of P. Given P-reduced words
X,YeR, we say X ~ 7 if X*A=Y for some P-word A. The relation ~ is an
equivalence relation on R, and each —class of R represents a unique element of
U(P). Conversely, each element of \J(P) is represented by at least one —class. But
much more is true.
4. Theorem: (Stallings [1971, Theorem 3.A.4.5]) Let (P,D) be a pregroup. H X,Y
are F-reduced words, then X and Y represent the same element of \J(P) if and only
if X ~ 7 .
In the light of theorem 4, we can make the following definition of the length of
an element of U(P).
5. Definition: Let P be a pregroup and let xe\J(P). Then the P-length of x,
denoted lP(x), is the length of a reduced word representing x. In the case of 1 eP,
we insist that lP(\)=0, even though the set of reduced words of length 0 is empty.
The following corollaries of this theorem are worth examining. Let (P,D) be a
pregroup. Let (P,D) be a pregroup.
6. Corollary:
L:P-+U(P)
is injective.
Henceforth we identify xeP with
L(X)E\J(P).
In particular, if (x,y)ePxP,
then xy is in V(P) but perhaps not in P.
7. Corollary: P is a reduced word structure for U(P).
Thus theorem 3 may be viewed as an alternative definition for pregroups.
8. Corollary: (x,y)eD £= { (x,y)ePxP a n d x y e P }
= { (x,y)ePxP and lP(xy) 1 }.
9. Corollary: (P,D) is a group if D=PxP, because in this case theorem 4 implies
that U(P)~P.
We end this section with a discussion of the subgroups of a pregroup.
9a. Definition: Let (P,D) be a pregroup. A subpregroup of P is a pregroup (Q,E)
such that QcP, EcD, and identity, inversion, and multiplication are inherited
from P. A subgroup of P is a subgroup Q of U(P) whose trivial pregroup structure
(QQXQ) is a subpregroup of P.
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