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

A. Tsurkov

TL;DR

The paper reframes automorphic equivalence in universal algebraic geometry using a category-theoretic lens, introducing and relating two core categories, $\mathfrak{Cl}_H$ and $\mathfrak{Cor}_H$, together with the factorization functor $\mathcal{FR}_H$. It proposes a new, streamlined definition of automorphic equivalence via an automorphism $\Phi$ of $\Theta^0$ and accompanying isomorphisms $\Lambda$ and $\Psi$ that render two commutative diagrams, and proves its equivalence with the classical formulation. It then analyzes how automorphic equivalence compares with geometric equivalence, shows how to compute the automorphism quotient $\mathfrak{A/Y}$ via the method of verbal operations, and provides concrete instances (e.g., $k$-linear algebras and vector spaces) to illustrate when automorphic equivalence diverges from or coincides with geometric equivalence. The work thus clarifies the structure of automorphisms in the category of finitely generated free algebras and offers tools and examples for understanding the landscape of equivalence notions in universal algebraic geometry. These insights illuminate when geometry is preserved under categorical symmetries and guide future exploration of verbal-operator effects across algebraic varieties.

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.

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

TL;DR

The paper reframes automorphic equivalence in universal algebraic geometry using a category-theoretic lens, introducing and relating two core categories, and , together with the factorization functor . It proposes a new, streamlined definition of automorphic equivalence via an automorphism of and accompanying isomorphisms and that render two commutative diagrams, and proves its equivalence with the classical formulation. It then analyzes how automorphic equivalence compares with geometric equivalence, shows how to compute the automorphism quotient via the method of verbal operations, and provides concrete instances (e.g., -linear algebras and vector spaces) to illustrate when automorphic equivalence diverges from or coincides with geometric equivalence. The work thus clarifies the structure of automorphisms in the category of finitely generated free algebras and offers tools and examples for understanding the landscape of equivalence notions in universal algebraic geometry. These insights illuminate when geometry is preserved under categorical symmetries and guide future exploration of verbal-operator effects across algebraic varieties.

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.
Paper Structure (17 sections, 27 theorems, 145 equations)

This paper contains 17 sections, 27 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Auxiliary facts from set theory and universal algebra. Some notation.
  3. Categories and functors of universal algebraic geometry
  4. The categories of $H$-closed congruences and of coordinate algebras of the algebra $H$
  5. Factorization functor
  6. When two algebras has same geometry?
  7. Automorphic equivalence of algebras
  8. Consequences of the Definition \ref{['old']}
  9. The functor $\Lambda$
  10. The functor $\Psi$
  11. Consequences of the Definition \ref{['new']}
  12. Consequences of the commutativity of the left diagram from the Definition \ref{['new']}
  13. Consequence of the commutativity of two diagrams from the Definition \ref{['new']}
  14. First corollaries
  15. Method of verbal operations for computation the group $\mathfrak{A/Y}$ and some conclusions.
  16. ...and 2 more sections

Key Result

Proposition 2.1

We consider again two algebras $A,B\in \Theta$ of the same variety $\Theta$ and two congruences $T_{A}\subseteq A\times A$, $T_{B}\subseteq B\times B$.

Theorems & Definitions (47)

  • Definition 1.1
  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.1
  • Proposition 2.2
  • Remark 3.1
  • Proposition 3.1
  • ...and 37 more