Abstract:
In this paper we concentrate on open problems in two directions
in the development of the theory of constructive algebraic systems. The first
direction deals with universal algebras whose positive open diagrams can be
computably enumerated. These algebras are called positive algebras. Here we
emphasize the interplay between universal algebra and computability theory.
We propose a systematic study of positive algebras as a new direction in the
development of the theory of constructive algebraic systems. The second direction concerns the traditional topics in constructive model theory. First we
propose the study of constructive models of theories with few models such as
countably categorical theories, uncountably categorical theories, and Ehrenfeucht theories. Next, we propose the study of computable isomorphisms and
computable dimensions of such models. We also discuss issues related to the
computability-theoretic complexity of relations in constructive algebraic systems.