2
RIMLINGER
(x{, . . . ,xn)D: This means V \ i n~\,(xl,xt + \)eD.
(w,x,y) associates: This means (i) (w,x,y) is a P-word, and (ii) (w,x,y)D, (w,xy)D,
and (wx,y)D.
Suppose X = {x{, . . . ,xn) is a P-word and (.X/,X/ + i)z for some \ i n. Then
7=(.Xi, . . . ,XjXi
+
i, . . . ,xn) is an elementary reduction of X. Clearly X and Y
represent the same element of U(P). If g e\J(P) is represented by some P-word X,
then by making elementary reductions we may proceed from X to a P-reduced word
Y representing g. It is interesting to note that some elements of U(P) may not have
any representation at all because P does not contain the "inverse elements." A
more serious objection to the category of prees is that the map t:P-U(P) is not
injective in general. The fatal blow for C is that there is no general algorithm for
deciding when two arbitrary P-words represent the same element of U(P). We now
introduce Stallings's category of pregroups, where all these objections disappear.
1. Definition: Let (P,D) be a pree. Suppose P contains a distinguished element
1 eP, and is endowed with an involution i\P-*P, denoted
i(x)=x~l.
Then (P,D) is
a pregroup if it satisfies properties (PI) to (P4) for all w,x,y,z eP.
(PI): (\9x)D,(x,\)D and lx=x=xl.
(P2):
(x~l,x)D, (x,x~l)D,
and
xx~l=x_1x
= l.
(P3): Suppose (w,x,y)D. Then { (wx,y)D or (w,xy)D } = {(w,x,y) associates
and (wx)y = w(xy) }.
(P4): If (w,x,y,z)D then { (w,xy)D or (xy,z)D }.
Axioms (PI) and (P2) are transparent, and (P3) captures the idea of "associa-
tivity whenever possible." In Rimlinger [ to appear] there is an alternative char-
acterization of pregroups in terms of "groups derived from reduced words," as fol-
lows.
2a. Definition: Suppose P is a pregroup, G is a group, PCG, and the inclusion map
P-*G is a pregroup morphism. Let f:U(P)-+G be the induced group homorphism.
Then P is a pregroup structure for G if /:U(P)-G is an isomorphism.
2b. Definition: Let G be a nontrivial group. Suppose PcG, P=P~\ and P gen-
erates G. Suppose D c P x P ; and suppose that (x,y)eD implies xyeP. Clearly
(P,D) is a pree. By abuse of notation, the P-words represent elements of G. Sup-
pose all the P-reduced words representing the same element of G are of the same
P-length. Then (P,P) is called a reduced word structure for G.
3. Theorem: (Rimlinger [to appear]) If P is a reduced word structure for G, then P
is a pregroup structure for G.
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