Abstract: Boris Plotkin's famous question, ``When do two algebraic structures have the same algebraic geometries?", gave rise to the concept of geometric equivalence and its generalizations, which were studied in conjunction with Plotkin's problem on criteria, necessary, and sufficient conditions for geometric equivalence. Thus, the geometric equivalence of algebraic structures implies the isomorphism of the categories of algebraic sets over them, in particular, their equivalence. Boris Plotkin termed the existence of the equivalence of these categories ``geometric compatibility". One of the directions in which Boris Plotkin developed his ideas was the transition from the category of algebraic sets to the category of logical sets, and hence logical equivalence and logical compatibility. Again, logical equivalence entails logical compatibility, that is, the equivalence of the categories of logical sets. In this talk, we will discuss the subsequent generalizations of Boris Plotkin's ideas, demonstrating that bi-interpretation between algebraic structures in arbitrary languages implies the equivalence of their categories of (projective) logical sets. Furthermore, Boris Plotkin's categorical constructions have proven to be a convenient and indispensable tool for proving a number of general assertions in the theory of interpretation.

Cite: Daniyarova E.
Boris Plotkin's Logical Geometry and Theory of Interpretations
Algebra, Geometry, Dynamics 2025 10-13 Nov 2025