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

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.

💡 Research Summary

The paper investigates when two algebras share the same “algebraic geometry” by using categorical constructions within universal algebraic geometry. Starting with a fixed variety Θ of one‑sorted algebras, the author considers the category Θ₀ of finitely generated free algebras. For a given algebra H∈Θ, a system of equations is represented by a subset T⊆F(X)×F(X) of a free algebra F(X). The H‑closed congruences are those T that are equal to their double closure T″_H (the maximal system of equations having the same solution set in H). The collection of all H‑closed congruences on each free algebra forms a category Cl H: objects are pairs (F(X), T) with T H‑closed, and morphisms are homomorphisms φ:F(X₁)→F(X₂) satisfying φ(T₁)⊆T₂.

A second category Cor H is introduced, whose objects are the quotient algebras F(X)/T (with T H‑closed) and whose morphisms are ordinary homomorphisms between these quotients. The paper defines a “factorisation functor” FR H: Cl H→Cor H that sends (F(X), T) to F(X)/T and maps a morphism μ∈Hom(F(X₁),F(X₂)) (with μ(T₁)⊆T₂) to the unique induced homomorphism φ:F(X₁)/T₁→F(X₂)/T₂. Proposition 2.1 guarantees existence and uniqueness of φ, essentially the universal property of quotients. FR H is shown to be surjective on objects and morphisms (Proposition 3.2).

A second surjective functor FG H: Cl H→Θ₀ is defined by forgetting the congruence: (F(X), T)↦F(X) and sending a morphism to the underlying homomorphism. This functor extracts the “free‑algebra base” of an H‑closed system.

With these categorical tools, the paper formalises three notions of similarity between algebras H₁ and H₂:

1. Geometric equivalence (Definition 4.1): H₁ and H₂ are geometrically equivalent iff for every free algebra F(X) the sets of H₁‑closed and H₂‑closed congruences coincide, i.e. Cl H₁(F(X))=Cl H₂(F(X)). Proposition 4.1 proves that this condition is equivalent to the equality of the whole categories Cl H₁=Cl H₂.

2. Weak similarity (Definition 4.2): H₁ and H₂ are weakly similar iff there exists an isomorphism of categories Λ:Cl H₁→Cl H₂. This is a broader relation than geometric equivalence, allowing a structural re‑labelling of closed congruences.

3. Automorphic equivalence (Section 5): Assuming the variety satisfies Condition 5.1 (every automorphism Φ of Θ₀ maps each free algebra to an isomorphic free algebra), an automorphism Φ of Θ₀ is said to provide automorphic equivalence between H₁ and H₂ if Φ induces a category isomorphism between Cl H₁ and Cl H₂. This notion sits strictly between geometric equivalence and weak similarity: any geometrically equivalent pair is automorphically equivalent, but not conversely; any automorphically equivalent pair is weakly similar, but not necessarily geometrically equivalent.

The paper critiques earlier definitions of automorphic equivalence (e.g., Plotkin 2003, and later works) as cumbersome, and offers the above categorical formulation as more elegant and transparent. It proves basic properties: FR H and FG H are surjective functors; the induced map on morphisms is well‑defined and respects composition; and an automorphism Φ of Θ₀ yields a bijection between the objects of Cl H₁ and Cl H₂ preserving the morphism structure.

Finally, the author suggests future research directions: studying invariants under automorphic equivalence, exploring the impact on module categories, investigating the automorphism groups of the categories Cl H, and connecting these ideas with logical aspects of universal algebraic geometry (e.g., model‑theoretic types). The paper thus provides a clean categorical framework for comparing the algebraic geometry of universal algebras and clarifies the hierarchy of equivalence notions within this setting.