Topological Birkhoff

One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra $\mathbf {B}$ satisfies all equations that hold in a finite algebra $\mathbf {A}$ of the same signature if and only if $\mathbf {B}$ is a homomorphic image of a subalgebra of a finite power of $\mathbf {A}$. On the other hand, if $\mathbf {A}$ is infinite, then in general one needs to take an infinite power in order to obtain a representation of $\mathbf {B}$ in terms of $\mathbf {A}$, even if $\mathbf {B}$ is finite.

We show that by considering the natural topology on the functions of $\mathbf {A}$ and $\mathbf {B}$ in addition to the equations that hold between them, one can do with finite powers even for many interesting infinite algebras $\mathbf {A}$. More precisely, we prove that if $\mathbf {A}$ and $\mathbf {B}$ are at most countable algebras which are oligomorphic, then the mapping which sends each term function over $\mathbf {A}$ to the corresponding term function over $\mathbf {B}$ preserves equations and is Cauchy-continuous if and only if $\mathbf {B}$ is a homomorphic image of a subalgebra of a finite power of $\mathbf {A}$.

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.