2010 seminar talk: Groupoids of Ultrafilters
Talk held by Denis Saveliev (Moscow) at the KGRC seminar on 2010-06-29.
There exists a natural way to extend the operation of any groupoid (in fact, any universal algebra) to ultrafilters; the extended operation is right topological in the standard compact Hausdorff topology on the set of ultrafilters; the extensions of semigroups are semigroups. Semigroups of ultrafilters are used to obtain various deep results of number theory, algebra, dynamics, etc. The main tool is idempotent ultrafilters. They exist by a general theorem establishing the existence of idempotents in compact Hausdorff right topological semigroups.
Expanding this technique to non-associative groupoids, we isolate a class of formulas such that any satisfying them compact Hausdorff right topological groupoid has an idempotent, and a class of formulas that are stable under passing from a given groupoid to the groupoid of ultrafilters. If a formula belongs to both classes (like associativity), any satisfying it groupoid carries an idempotent ultrafilter. Results on semigroups following from the existence of idempotent ultrafilters (like Hindman's Finite Sums Theorem) remain true for such groupoids.
Another generalization concerns infinitary analogs of these results. The main obstacle here is that non-principal idempotent ultrafilters cannot be σ-additive. We define ultrafilters with two weaker properties (ultrafilters near κ-additive subgroupoids and κ-additive ultrafilters near subgroupoids) and show that their existence suffices to obtain desired infinitary theorems.