| Given a discrete topological space S, we define beta S to be the set of ultrafilters on S. In this dissertation, we will be working with the set of dyadic rational numbers. Using the notation o = {0, 1, 2, ...}, the set of dyadic rationals is defined as follows D=&cubl0;m2t : m ∈ Z and t ∈ o}. We shall consider D as a discrete topological space. Namely, we will be considering D+ , the set of positive numbers contained in D . Note that N = {1, 2, 3, ...} ⊆ D+ . Any number in D+ (and in turn in N ) has a terminating binary expansion. Thus, for any x ∈ D+ , there exists a unique finite nonempty subset F of the integers such that x = t∈F2t . We will investigate the algebraic structure of beta D+ and also derive several partition results in D+ .;There is a close knit relationship between the algebraic structure of beta N and combinatorial or Ramsey theoretical partition results in N . Our focus will lie on Milliken-Taylor systems (and variations thereof) with respect to both their combinatorial structure and algebraic structure in beta D+ . Namely, we will prove results similar to that used in the algebraic proof of the Finite Sums Theorem. The difference being that in this dissertation our ultrafilters will be sums and products of idempotents and we will be dealing with multiple sequences. We will begin with the subsets of elements of our ultrafilters being variations of Milliken-Taylor Systems. As we progress with our results, these subsets will be progressively more complex. Further, as Hindman used idempotent results in his algebraic proof of the Finite Sums Theorem, we shall use our idempotent results (as well as others) to prove analogues of the Finite Sums Theorem. |
