Algebra Seminar talk
2016-06-03
Mike Behrisch
Reconstructing the topology on monoids and clones of the rationals
Abstract:
We study the structures (ℚ,<) and (ℚ,≤) through their endomorphism
monoids and polymorphism clones. Our main result is that End(ℚ,<) and
End(ℚ,≤) have automatic homeomorphicity. That is to say, any monoid
isomorphism between the respective endomorphism monoid and any closed
transformation monoid on a countable set automatically is a
homeomorphism with respect to the natural topology induced by the
product topology if the underlying sets are equipped with the discrete
topology.
Moreover, we reveal a structural property of the endomorphism monoid that allows to extend automatic homeomorphicity to the full polymorphism clone. This method works for Pol(ℚ,≤), but fails for Pol(ℚ,<).
This is joint work with John K Truss and Edith Vargas-Garcìa (University of Leeds). Behrisch was partly supported by the Austrian Science Fund (FWF) under grant no. I836-N23, Vargas-Garcia was supported by CONACYT.