CHAPTER 1 FORMULATION OF MAIN THEOREM Recall that we use T to denote the family of non-empty finite subsets of N. We let T^— T\3 {0}. One of the reasons that we are concerned with these objects is that in order to deal nicely with IP-sets, it is useful to see them as sequences indexed by either T or by T§ (this will be explained presently). Any sequence indexed by T (or by T$) will be called an ^"-sequence. If G is an abelian group then any ^-sequence (na)aej=- C G for which naUp = na + np whenever a D (3 = 0 will be called an IP-sequence. We will now show that IP-sets may be indexed in a natural way by T*, whereupon they become IP-sequences. Recall that the IP-set generated by the sequence G = {gi : i G N} C Z is given by r = {9h +9i2^ r- 9ik• : h *2 •- ik,k e N}. If we set rc{ii,i2,-.i*} = 9h + 9i2 + 1 " 9ik then (na)aejr is an IP-sequence (that is, naUp = na+np whenever aD/3 = 0) with the property that {na : a G J7} = T. Henceforth, whenever we will be dealing with IP-sets we will assume that they have the structure of an IP-sequence. However, in a mild abuse of terminology, we may sometimes still refer to these IP-sequences as IP-sets. Furthermore, any time we have an IP-sequence (na)aejr given, we will, if needed, extend the indexing set to T$ by letting n^ = 0. Suppose that a, (3 G T$ and suppose that for all a G a and b G (3 we have a b. In this case, we will write a f3. (Notice that 0 a 0 for all a G J7.) Of paramount importance to us is a notion of convergence for ^"-sequences which will be defined later in this section. For this mode of convergence, any T- sequence in a compact topological space will converge along a sub-sequence. In order to explain what is meant by a sub-sequence of an jF-sequence, we introduce, following [FK2], the notion of an IP-ring. Suppose that we are given a conventional sequence (a^e N C T with c\\ cx^ a3 • • •. Let ^ = {{J ai : /3S^}. Then J^1) is called an IP-ring, as is J^1} = F^ U {0}. Notice that the map f : T-*T^l\ £(*)= [Jen is bijective and structure preserving in the sense that £(a U (3) — £(a) U £(/?). In particular, since T^ has the structure of T, any sequence {ya)ae^^ indexed by 10

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