Polynomial-time abelian groups

Abstract

This paper is a continuation of the authors' work (1991), where the main problem considered was whether a given recursive structure is recursively isomorphic to a polynomial-time (p-time) structure. In that paper, a recursive Abelian group was constructed which is not recursively isomorphic to any polynomial-time Abelian group. We now show that if every element of a recursive Abelian group has finite order, then the group is recursively isomorphic to a polynomial-time group. Furthermore, if the orders are bounded, then the group is recursively isomorphic to a polynomial-time group ( A, + A ) with universe A being the set of tally representations of natural numbers Tal( ω) = s{;1s}; ∗ or the set of binary representations of the natural numbers Bin( ω). We also construct a recursive Abelian group with all elements of finite order but which has elements of arbitrary large finite order which is not isomorphic to any polynomial-time group with universe Tal( ω) or Bin( ω). Similar results are obtained for structures ( A, f), where f is a permutation on the set A.