FG1 Seminar talk
One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A . On the other hand, if A is infinite, then in general one needs to take an infinite power in order to obtain a representation of B in terms of A , even if B is finite.
We show that by considering the natural topology on the functions of A and B in addition to the equations that hold between them, one can do with finite powers even for many interesting infinite algebras A. More precisely, we prove that if A and B are at most countable algebras which are oligomorphic, then the mapping which sends each function from A to the corresponding function in B preserves equations and is continuous if and only if B is a homomorphic image of a subalgebra of a finite power of A.
Our result has the following consequences in model theory and in theoretical computer science: two omega-categorical structures are primitive positive bi-interpretable if and only if their topological polymorphism clones are isomorphic. In particular, the complexity of the constraint satisfaction problem of an omega-categorical structure only depends on its topological polymorphism clone.
(Joint work with Manuel Bodirsky, Ecole Polytechnique)