On computably enumerable structures

© 2014, Pleiades Publishing, Ltd. This is a short article based on the plenary lecture presented by the author at Algebra Computability and Logic conference in Kazan, June 1–6, 2014. The paper is a new approach that introduces effectiveness into the study of algebraic structures. In this approach the initial objects are the domains of the form ω/E, where E is an equivalence relation on ω. In this set-up one investigates effective algebraic structures that are admitted by domains of the formω/E. This lecture was motivated by the papers [2, 20, 21], open questions in [17], and is based on the papers [15, 16, 19, 23]