is a modification (and common generalization) of Proposition 708 of [11]
and Proposition C of [18,§1], We first give an easy lemma.
LEMMA 1. If h:T —v S is an allowable natural transformation, which is of
type , being a composite
h:T -iU»([B,C]BA)BD fSalCBD -i^-S ,
with x central and f,g allowable, then h can be written in the form as
h:T - ^ ([X,Y]HZ)HW SB1 YBW —t-^s ,
with u central, s and t allowable and s not expressible in the form .
Proof^ If f is not of type , take u = x, s = f, t = g. If f is of type
, we imitate for allowable natural transformations the argument used in
[11,p.139] for graphs in the analogous situation.
PROPOSITION 2. Let h:([Q,M]HP)HN - S be an allowable natural transforma-
tion, with [Q,M] non-constant. Suppose that [" h is of the form 7|(SH1) for
graphs £:P —-Q , TpMHN —-S , such that £ cannot be written in the form
p(eBl)w for any graphs p,&,cj with co central. Suppose finally that h can-
not be written in the form
([Q,M]HP)HN X' ([F,G]BE)KH ftS)1 GSH - 1 ^ S ,
where xT,ff,g' are allowable natural transformations, xT is central, fT
cannot be written in the form and, via xf, [Q,M] is associated with a
prime factor of E and [F,G] is associated with a prime factor of P. Then
there are allowable natural transformations p:P —9- Q , q:MSN --S such
that Tp = £ , rq = t\ and h = q(pSl) .
?I99^i We induction on r (h) , using the cut elimination property of al-
lowable natural transformations (pp,xiv,xvi), and following the proof of
Proposition C of [18,§1] and the proof of Proposition 7.8 of [11], re-
placing the hypothesis Th^ of [18] by the last hypothesis of the present
proposition. As in [11],we may assume that P,N and S have no constant
prime factors and we distinguish cases according to the possible forms of h
The case of h being central is done as in [11],
The case of h being of the form y(fSg)x, for allowable f:A - C and
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