Name
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. |
Year | Venue | DocType |
|---|---|---|
2012 | CoRR | Journal |
Volume | Citations | PageRank |
abs/1203.1876 | 0 | 0.34 |
References | Authors | |
0 | 2 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Manuel Bodirsky | 1 | 644 | 54.63 |
| Michael Pinsker | 2 | 132 | 17.54 |