Definable isomorphism problem

TL;DR

This paper proves the decidability of the definable isomorphism problem for structures with atoms, showing that parameter-free definable isomorphisms exist if parameterized ones do, under mild assumptions.

Contribution

It establishes the decidability of the definable isomorphism problem and introduces a parameter-elimination technique for structures with atoms.

Findings

Decidability of the definable isomorphism problem under mild assumptions.

Existence of parameter-free definable isomorphisms when parameterized ones exist.

Core result: parameter-elimination in definable isomorphisms.

Abstract

We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying structure of atoms, we prove decidability of the problem. The core result is parameter-elimination: existence of an isomorphism definable with parameters implies existence of an isomorphism definable without parameters.