Parti
Stallings's theorem about pregroups;
introduction to the pregroups of finite height
1. The definition of a pregroup
In this section we review some basic facts about pregroups. We define reduced
word structures and pregroup structures. By theorem 3, theorem 4 and corollary 7
below, these concepts are equivalent. References are given for proofs of theorems 3
and 4, but otherwise the exposition is self contained. Anticipating results to come,
we end this section with a discussion of subgroups of a pregroup.
We start by considering prees, or sets with partial multiplication table. Let P
be a set, let D cPxP, and let m\D-+P be a set map. Typically we denote m(x,y)
by xy or x*y. The composite concept (P,D,m) is called a pree. The universal
group V{P) of a pree P is determined by the presentation
(P;{
m(x,y)y~lx~l
| (x,y)eD }). The prees form a category C. Given prees
(P,D) and (Q,E), we define [P,Q]C to be the set maps fcP^Q such that
(x,y)eD =^((j){x),(t)(y))eE and ^(xy)=(j(x)())(y). Clearly there is a forgetful functor
F:G-C from groups to prees, and if G is a group, then the diagram
U(P)
I
13!
I
G
shows that
[ U ( P ) , ( J ]
G
and [P,F(G)]C are in 1-1 natural correspondence. The map
L:P-+U(P)
is the restriction to P of the natural projection F(P)-+U(P) from the free
group on the set P onto U(P). Let (P,D) be a pree. Here is some useful terminol-
ogy.
P-word X of length n: An element X = {xh . . . ,xn)ePn
X is P-reduced:\his means V 1 * " fl - 1 , (xhxi
+
i)£D.
X represents x: This means
X=L(XI)L(X2)
* * t(x„)eU(P).
Recieved by the editor July 14, 1985.
*
The author was a Sloan Foundation Doctoral Dissertation Fellow at UC Berkeley during the period
this paper was written.
1
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