Algebra Seminar talk
Reconstructing the topology on monoids and clones of the rationals
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.