A Journey to Computably Enumerable Structures (Tutorial Lectures)

The tutorial focuses on computably enumerable (c.e.) structures. These structures form a class that properly extends the class of all computable structures. A computably enumerable (c.e.) structure is one that has computably enumerable equality relation E such that the atomic operations and relations of the structure are induced by c.e. operations and relations that respect E. Finitely presented universal algebras (e.g. groups, rings) are natural examples of c.e. structures. The first lecture gives an introduction to the theory, provides many examples, and proves several simple yet important results about c.e. structures. The second lecture addresses a particular problem about finitely presented expansions of universal algebras with an emphasis to semigroups and groups. The lecture is based on the interplay between important constructions, concepts, and results in computability (Post’s construction of simple sets), universal algebra (residual finiteness), and algebra (Golod-Shafarevich theorem). The third lecture is devoted to studying dependency of various properties of c.e. structures on their domains.