Algebra Seminar talk
2025-10-03
Peter Mayr
Boolean powers of finite simple Mal'cev algebras
Abstract:
Let A be a finite simple non-abelian Mal'cev algebra (e.g. a group, loop, ring), and let K be the class of finite direct powers of A. We show that K has the amalgamation property by the Foster-Pixley Theorem and the Ramsey property by the Graham-Rothschild Theorem (but is not closed under substructures in general). The generalized Fraïssé limit of K is a filtered Boolean power of A by the countable atomless Boolean algebra B. We show that the automorphism groups of filtered Boolean powers of A by B have ample generics, which gives a new proof of our previous results that these groups have the small index property, uncountable cofinality and the Bergman property. As an intermediate step, we extend a result of Kwiatkowska (2012) that the group of homeomorphisms of the Cantor space has ample generics to pointwise stabilisers in the homeomorphism group.