A Generalization and a New Proof of Plotkin’s Reduction Theorem

It is known that Plotkin’s reduction theorem is very important for his theory of universal algebraic geometry [1, 2]. It turns out that this theorem can be generalized to arbitrary categories containing two special objects and in this case its proof becomes considerable more simple. This new proof and applications are the subject of the present paper. INTRODUCTION An automorphism φ of a category C is called inner if it is isomorphic to the identity functor IdC in the category of all endofunctors of C. If an automorphism φ is inner the object φ(A) is isomorphic to A for every C-object A. It is known [2] that every automorphism satisfying the last condition is a composition of two automorphisms the first of which preserves the objects and second one is an inner automorphism. Thus it is sufficient to consider only such automorphisms φ that preserve the objects, i.e. φ(A) = A for all objects A of C. Further every automorphism satisfying the last condition induces a permutation on the set Hom(A,B) for every pair of objects A,B, particularly it induces an automorphism of the monoid End(A) = Hom(A,A) for every object A. Below I cite the Plotkin’s Reduction Theorem. The purpose of this theorem is to reduce the verification of a given automorphism of C to be inner to the same problem for the full subcategory of C determined by two special objects. Let V be a variety of universal algebras. Consider the category Θ whose objects are all algebras from V and whose morphisms are all homomorphisms This research is partially supported by THE ISRAEL SCIENCE FOUNDATION founded by The Israel Academy of Sciences and Humanities Center of Excellence Program. 1 2 GRIGORI ZHITOMIRSKI of them. Fix an infinite set X. Let Θ be the full subcategory of Θ defined by all free algebras from V over finite subsets of the set X. The following conditions are assumed: 1P) every object of Θ is a hopfian algebra; 2P) there exists an object F 0 = F (X) in Θ generating the whole variety V. Fix the object F0 = F (X0) in Θ 0 generated by a singular set X0 = {x0} and the homomorphism ν0 : F 0 → F0 induced by the constant map X 0 → X0 that is (∀x ∈ X) ν0(x) = x0. Theorem 1. Let φ : Θ → Θ be an automorphism which does not change objects. If φ induces the identity automorphism on the semigroup END(F ) and φ(ν0) = ν0, then φ is an inner automorphism. It should be mention that this theorem was first proved by Berzins [3] for the variety of commutative associative algebras over an infinite field. The purpose of the present paper is to show that some hypotheses used in this theorem are unnecessary, the statement can be generalized and the proof becomes essentially more easy. The author thanks Prof. B. Plotkin for useful discussions. 1. A generalized Reduction theorem Let C be a category with the following conditions: 1) there is a faithful functor Q : C → Set such that it is represented by an object A0 ; 2) there is an object A such that for every two objects A and B and for every bijection s : Q(A) → Q(B) one of the following two dual conditions is satisfied: if for every morphism ν : B → A (ν : A → A) there exists a morphism μ : A→ A (μ : A → A) such that Q(μ) = Q(ν) ◦ s (resp. Q(μ) = s ◦ Q(ν)) A GENERALIZATION AND A NEW PROOF OF PLOTKIN’S REDUCTION THEOREM 3 then there exists an isomorphism γ : A→ B such that Q(γ) = s. The sense of the last condition using algebraic language is: if the composition of a given bijection of A and every homomorphism from A to A is a homomorphism then this bijection is an isomorphism of A. Theorem 2. If φ : C → C is an automorphism of the category C that does not change the objects A0 and A 0 and induces the identity map on Hom(A0, A ) then φ is an inner automorphism. Proof. Let objects A and A0 existing under hypotheses be fixed. And let u : Q → Hom(A0, ?) be an isomorphisms of functors, i.e. a representation of the functor Q. Let φ be an isomorphism of the category C satisfying required conditions. Thus we have for every object A the bijection φA : Hom(A0, A) → Hom(A0, φ(A)) and therefore a bijection sA of the set Q(A) onto the set Q(φ(A)) unique defined by means of the following commutative diagram: Q(A) uA −−−→ Hom(A0, A) sA 