Authors: Khadijeh Keshvardoost ; Bartek Klin ; Sławomir Lasota ; Joanna Ochremiak ; Szymon Toruńczyk

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.