Topological Birkhoff

TL;DR

This paper extends Birkhoff's HSP theorem to certain infinite algebras by incorporating topology, showing that for oligomorphic, countable algebras, finite powers suffice to represent homomorphic images, impacting model theory and CSP complexity.

Contribution

It introduces a topological approach to Birkhoff's theorem for infinite algebras, demonstrating finite power representations under oligomorphic conditions.

Findings

Finite powers suffice for certain infinite algebras with topology.

Topological polymorphism clones determine CSP complexity.

Omega-categorical structures are bi-interpretable iff their clones are topologically isomorphic.

Abstract

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.