By the first part of Proposition 4, we can assume that G is an abelian monoid
. If we take [[summation].sub.s[member of]G] [a.sub.s][u.sub.s] and [[summation].sub.t[member of]G] [b.sub.t][u.sub.t] G in the commutant of A in A [[??].sup.[sigma].sub.[alpha]] G, then, by the second part of Proposition 4 and the fact that [alpha] is symmetric, we get that
The last requirement comes from the fact that, given only the global transition function, we want to be able to isolate the influence of one cell; that is why we demand that + have an identity element, which makes now ([SIGMA], +) an abelian monoid
. Of course, in order for all of this to be relevant, the transition function has to be a morphism.