8 R.VOREADOU
OV3. if at least one of (f^fJiA -•C) and (f2,f£:B --D) is in the class
and f. = f! in case (f.,f!) is not in the class' , then (f1
1
Bf0,f.{Bf1:21'2
1 1
v
I
'
i^
^
ABB --CBD) is in the class
W4. if (f,fT :ABB -^-C) is in the class, so is (-rr(f ) ,TT(£f ) : A -•[B,C])
7TS. if at least one of (f^fJ-.A-v- B) and (f2 ,££:CBD --E) is in the
class and f. = f! in case (f.,f!) is not in the class, then
(£2(£1B1) ,£^(£j_Bl) : ([B,C]BA)BD --E) is in the class .
The theorem is given after the two lemmas below.
LEMMA 3. If f :A—•*- B is an allowable natural transformation which cannot
be written in the form and all the prime factors of A are non-constant,
then Tf cannot be written in the form .
Proof^ Suppose that Tf can be written in the form as a composite ff =
£(fxBl)x for a central graph T:A—- ([X,Y]BZ)BW and allowable graphs
(U L : Z —-X and £:YBW —•B. By hypothesis, [X,Y] is non-constant. We may as-
sume that p L is not of the form p(6Bl)o) with u) central, by an argument
like that of p.139 of [11]. Let t be the central natural transformation
with Tt = T. Then ["(f t ) = £(fiBl) . Moreover, ft cannot be written as
([X,Y]BZ)BW _*-^ ([F,G]BE)BH
r
GBH -J-*-B
with x* central and f*,g* allowable, because this would imply that f is of
type , contrary to the hypothesis. By Proposition 2, ft is of type ,
and hence so is f, a contradiction. So Tf cannot be written in form .
LEMMA 4. If m:A^- B is an allowable graph which cannot be written in the
form and [X,Y] is a prime factor of A, then X is non-constant.
Proof: Suppose X is constant. Let W be an iterated B-product of the prime
factors of A other than [X,Y]. There is an obvious central graph
u:A-^- [X,Y]BW. The "empty" graphs s:I -^ X and sf:X -^- I are allowable;
the restriction t:YBW—- B of m to v(Y)+v(W)+v(B) is allowable, since (by
checking linkages we see that) it is equal to the composite graph
YBW dS1, [X,YBX]HW [l,b(lBs')]•!,
[XY]BW
u'\
A
jn^B
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