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