Strongly Summable Ultrafilters: Some Properties and Generalizations

This dissertation focuses on strongly summable ultrafilters, which are ultrafilters that are related to Hindman’s theorem in much the same way that Ramsey ultrafilters are related to Ramsey’s theorem. Recall that Hindman’s theorem states that whenever we partition the set of natural numbers into two (or any finite number of) cells, one of the cells must entirely contain a set of the form FS(X) for some infinite set X (here FS(X) is the collection of all nonrepeating sums of finitely many elements of X). A nonprincipal ultrafilter on the set of natural numbers is said to be strongly summable if it has a base of sets of the form FS(X), this is, if for every element A of p, there exists an infinite X such that FS(X) is both a subset of A and an element of p. These ultrafilters were first introduced by Hindman, and subsequently studied by people such as Blass, Eisworth, Hindman, Krautzberger, Matet, Protasov and others. Now, from the viewpoint of the definitions, there is nothing special about the semigroup of natural numbers, and analogous definitions for FS(X) and strongly summable ultrafilter can be considered for any semigroup (in the non-abelian case, one must first fix an ordering for X on order-type omega). It is not immediate, however, that the results that hold for strongly summable ultrafilters on the semigroup of natural numbers are still satisfied in general. Some of the main results of this dissertation are generalizations of these properties for all abelian groups and some non-abelian cases as well. Notably among these, a strongly summable ultrafilter p on an abelian group G has the so-called trivial sums property: whenever q, r are ultrafilters on G such that q+r=p, it must be the case that for some element g of G, q=p+g and r=−g+p (this is all in the context of the right-topological semigroup of all ultrafilters on G). The other significant result from this dissertation is a consistency result. It has long been known that the existence of strongly summable ultrafilters (on any abelian group) is not provable from the ZFC axioms, for it implies the existence of P-points. It is also known, however, that (at least on the semigroup of natural numbers) the existence of strongly summable ultrafilters follows from the restriction of Martin’s axiom to countable forcing notions. We prove here that there exist models of ZFC that satisfy the failure of this restriction of Martin's axiom, while at the same time in these models there exist strongly summable ultrafilters on all abelian groups. This can be done using iterations, both with finite or with countable support, of σ-centred forcing notions which resemble Mathias’s or Laver’s forcing.