Computably Enumerable Algebras, Their Expansions, and Isomorphisms

Computably enumerable algebras are the ones whose positive atomic diagrams are computably enumerable. Computable algebras are the ones whose atomic diagrams are computable. In this paper we investigate computably enumerable algebras and provide several algebraic and computable theoretic distinctions of these algebras from the class of computable algebras. We give a characterization of computably enumerable but not computable algebras in terms of congruences and effective conjunctions of Π 1 -sentences. Our characterization, for example, shows that computable conjunctions of negative atomic formulas true in a given c.e. algebra can be preserved in infinitely many of its homomorphic images. We also study questions on how expansions of algebras by finitely many new functions affect computable isomorphism types. In particular, we construct a c.e. algebra with unique computable isomorphism type but which has no finitely generated c.e. expansion.