Uniform reducibility of representability problems for algebraic structures

Abstract

Given a countable algebraic structure \( \mathfrak{B} \) with no degree we find sufficient conditions for the existence of a countable structure \( \mathfrak{A} \) with the following properties: (1) for every isomorphic copy of \( \mathfrak{A} \) there is an isomorphic copy of \( \mathfrak{A} \) Turing reducible to the former; (2) there is no uniform effective procedure for generating a copy of \( \mathfrak{A} \) given a copy of \( \mathfrak{B} \) even having been enriched with an arbitrary finite tuple of constants.