David Fernandez: Strongly Productive Ultrafilters

04/October/2013, 13:30–15:00
Fields institute, Room 210

Speaker:  David Fernandez (York University)

Title:  Strongly Productive Ultrafilters

Abstract: The concept of a Strongly Productive Ultrafilter on a semigroup (known as a “strongly summable ultrafilter” when the semigroup is additively denoted) constitute an important concept ever since Hindman defined it, while trying to prove the theorem that now bears his name. In a 1998 paper of Hindman, Protasov and Strauss, it shown that strongly productive ultrafilters on abelian groups are always idempotent, but no further generalization of this fact had been made afterwards. In this talk I will show (at least the main ideas, anyway) the proof that this result holds on a large class of semigroups, which includes all solvable groups and the free semigroup, among others. After that, I’ll discuss a special class of strongly productive ultrafilters on the free semigroup (dubbed “very strongly productive ultrafilters” by N. Hindman and L. Jones), and show that they have the “trivial products property”. This means that (thinking of the free semigroup S as a subset of the free group G) if p is a very strongly productive ultrafilter on S, and q,r are nonprincipal ultrafilters on G such that $qr=p$, then there must be an element x of G such that $q=px$ and $r=x^{-1}p$. This answers a question of Hindman and Jones. Joint work with Martino Lupini.

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