Categories and functors of universal algebraic geometry. Automorphic equivalence of algebras

TL;DR

This paper explores the categorical framework of universal algebraic geometry, defining automorphic equivalence of algebras through category isomorphisms, and refines existing concepts with more elegant categorical definitions.

Contribution

It provides a new, more elegant categorical definition of automorphic equivalence of algebras and revisits related theorems in this context.

Findings

Categorical definition of automorphic equivalence refined

Theorems related to algebraic geometry categories revisited

Enhanced understanding of algebraic geometry in universal algebra

Abstract

Universal algebraic geometry allows considering of geometric properties of every universal algebra. When two algebras have same algebraic geometry? We must consider the categories of algebraic closed sets of these algebras to answer this question. The complete coincidence of these categories gives us a concept of the geometric equivalence of algebras. Some type of isomorphisms of these categories gives us a concept of the automorphic equivalence of algebras. This concept has been considered since article B. Plotkin, Algebras with the same (algebraic) geometry. Proceedings of the Steklov Institute of Mathematics. 242 (2003), 17--207. DOI: 10.1134/S0081543812070048. We will give by language of category theory one more elegant definition of this concept and recall some theorems related to this concept.