Π 01 - presentations of algebras ∗

Effectiveness issues in algebra and model theory have been investigated intensively in the last thirty years. One wishes to understand the effective content of model-theoretic and algebraic results, and the interplay between notions of computability, algebra, and model theory. A significant body of work has recently been done in the area, and this is attested by recent series of Handbooks and surveys in computable mathematics, computability, and algebra (see, e.g., [1], [4], [3]). An emphasis has been placed on the study of computable models and algebras. These are the structures whose domains are computable sets of natural numbers, and whose atomic diagrams are computable. The study of computable model theory and algebra can naturally be extended to include a wider class of structures. This can be done by postulating that the atomic diagrams or natural fragments of the atomic diagrams are in some complexity class such as Σn or Π 0 n. These classes of algebras include computably enumerable (c.e.) algebras and co-c.e. algebras which we call Σ1-algebras and Π 0 1-algebras, respectively. Roughly, Σ1-algebras are the ones whose positive atomic diagrams are computably enumerable, and Π1-algebras are the ones whose negative atomic diagrams are computably enumerable. These include finitely presented algebras (e.g. finitely presented groups or rings) and groups generated by finitely many computable permutations of ω. There has been some research on Σ1-algebras (see for example [2], [6], [8], [9]) but not much is known about Π1-algebras and their properties. The main goal of this paper is the study of the question as