Open Problems in the Theory of Constructive Algebraic Systems

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 sys-