aa r X i v : . [ m a t h . C T ] F e b ON AUTOMORPHISMS OF CATEGORIES
G. ZHITOMIRSKIDEPARTMENT OF MATHEMATICS AND STATISTICSBAR-ILAN UNIVERSITY, 52900 RAMAT GAN, ISRAEL
Abstract.
Let V be a variety of algebras of some type Ω. An interest todescribing automorphisms of the category Θ ( V ) of finitely generated free V -algebras was inspired in connection with development of universal algebraicgeometry founded by B. Plotkin. There are a lot of results on this subject.A common method of getting such results was suggested and applied by B.Plotkin and the author ([7, 8]). The method is to find all terms in the languageof a given variety which determine such Ω-algebras that are isomorphic to agiven Θ ( V )-algebra and have the same underlying set with it. But this methodcan be applied only to automorphisms which take all objects to isomorphicones.The aim of the present paper is to suggest another method that is freefrom the mentioned restriction. This method is based on two main theorems.Let C be a category supplied with a faithful representative functor to thecategory of sets. Theorem 1 gives a general description of automorphisms of C , using a new notion of a quasi-inner automorphism. Theorem 5 shows howto obtain the full description of automorphisms of the category Θ ( V ). Theinvestigation ends with two examples. The first of them shows the preferenceof our method in a known situation (the variety of all semigroups) and thesecond one demonstrates obtaining new results (the variety of all modulesover arbitrary ring with unit). INTRODUCTION
Let V be a variety of algebras of some type Ω and Θ( V ) be the category of all V -algebras and their homomorphisms. For an infinite set X , let Θ ( V ) denotethe full subcategory of Θ( V ) which is determined by all free V -algebras over finitesubsets of the set X . The problem is to describe all automorphisms of the categoryΘ ( V ). Motivations for this research can be found in the papers [3, 4, 5, 6]. Themost important case is when automorphisms of this category are inner or close toinner in a sense.An automorphism Φ of a category C is called inner if it is isomorphic to theidentity functor Id C in the category of all endofunctors of C . It means that forevery object A of the given category there exists an isomorphism σ A : A → Φ( A ) such that for every morphism µ : A → B we have Φ( µ ) = σ B ◦ µ ◦ σ − A . Thusan automorphism Φ may be inner only in the case when there is an isomorphismbetween A and Φ( A ) for every C -object A .As regards the inverse proposition, it was shown in the papers [7, 8] that ifthe objects Φ( A ) and A are isomorphic for every C -object A , then Φ is inner orso called potentially inner. The last one means that Φ is inner in a categoryobtained by adding to C some new morphisms. More accurately, there exists afamily ( s A : A → Φ( A )) A ∈ Ob C of morphisms of an extended category such that forevery morphism µ : A → B of the category C we have Φ( µ ) = s B ◦ µ ◦ s − A . Thatway, the problem is to describe these new morphisms s A . A general method howto do it was suggested in the mentioned papers for categories C = Θ ( V ) where V is a suitable variety. Below is a summary of the essence of this method.Consider a C -algebra A free generated by the set { x , . . . , x n } of variables, where n is greater than arities of all Ω-operators. We can assume that algebras Φ( A ) and A have the common base set | A | and the same free generators. Then for every k -aryoperation symbol ω ∈ Ω we have two k -ary operations on | A | : ω A and ω Φ( A ) .It is obvious that every element of the algebra A can be considered as a termin the language corresponding to the algebra Φ( A ) and vice versa. In other words,every operation ω A is a derived operation in Φ( A ), and vice versa, every operation ω Φ( A ) is a derived operation in A . Thus the mappings s A : | A | → | A | such that s A ( x i ) = x i and s A ( ω A ( x , . . . , x k )) = ω Φ( A ) ( x , . . . , x k )) for every ω ∈ Ω determinethe automorphism Φ. The problem is reduced to fined terms ˜ ω in the algebra Φ( A )such that the derived algebra ( | A | , (˜ ω ) ω ∈ Ω ) belongs to the variety V .Some new results were obtained on this way. Particularly, A. Tsurkov success-fully applied this method to many-sorted algebras (for example in [9]). It is knownthat if a variety V satisfies IBN-property then every automorphism of the categoryΘ ( V ) takes every object to an isomorphic one. But there are varieties withoutIBN-property. If it is unknown whether A and Φ( A ) are isomorphic or not, themethod described above does not work. N AUTOMORPHISMS OF CATEGORIES 3
The aim of the current investigation is to fill up this gap. Since an automorphismmay be not inner in the general case, we use a new type of automorphisms calledthe quasi-inner automorphisms (Definition 1), and it turns out that this notion isenough to characterize arbitrary automorphisms. Further reasoning as a matterof fact follows the ideas of the papers [7, 8] when A and Φ( A ) are isomorphic,but instead of the method outlined above, a new method is proposed that to theauthor’s opinion is more successful.This method reduces the problem to the case when the underlying set | A | ofan algebra A is a subset of the underlying set of the algebra Φ( A ), and everyendomorphism µ of the algebra A is the restriction of the endomorphism Φ( µ ) ofthe algebra Φ( A ) to the set | A | and every restriction to | A | of an endomorphism ν of Φ( A ) is an endomorphism µ of A such that Φ( µ ) = ν . That circumstance givesus an opportunity to describe the action of the automorphism Φ.The main results are formulated in two theorems. Theorem 1 states that everyautomorphism of a category C supplied with a forgetful functor Q : C →
Set whichsatisfies two acceptable conditions is potentially quasi-inner. Theorem 5 statesthat every automorphism Φ of the category Θ ( V ) for an arbitrary variety V is theproduct of two functors Φ = Γ ◦ Ψ. The first of them is an inner isomorphismΨ : Θ ( V ) → D ,where D is a full subcategory of Θ ( V ) which is described. Thesecond functor Γ : D → Θ ( V ) is a so called extension functor, that is, | A | ⊆ | Γ( A ) | for every C -algebra A and µ ⊆ Γ( µ ) for every C -morphism µ .The last part of the paper contains two examples. The first of them shows thepreference of our method in a known situation (the variety of all semigroups) andthe second one presents a new result for the variety of all modules over arbitraryring with unit. As a consequence of the last result we obtain that in the case whenthe ring does not contain zero divisors all automorphisms are semi-inner. GRIGORI ZHITOMIRSKI Quasi-inner automorphisms
We consider only such categories C which are supplied with a faithful functor Q : C →
Set , where
Set is the category of all sets. We call Q the forgetful functoras it is accepted. Definition 1.
An automorphism Φ of a category C is called inner if there is anisomorphism between Φ and the identity functor Id C in the category of all endofunc-tors of C . An automorphism Φ is said to be quasi-inner if there is a bimorphism Id C → Φ . This definition means that Φ is inner if for every object A of the given categorythere exists an isomorphism σ A : A → Φ( A ) such that for every morphism µ : A → B the following diagram commutes: A σ A −−−−→ Φ( A ) µ y y Φ( µ ) B σ B −−−−→ Φ( B )Hence we get that Φ( µ ) = σ B ◦ µ ◦ σ − A .As a generalization of this notion, Φ is quasi-inner if for every object A of thegiven category there exists a bimorphism σ A : A → Φ( A ) such that for everymorphism µ : A → B the diagram above is commutative. Hence we have only thefollowing condition σ B ◦ µ = Φ( µ ) ◦ σ A but not the expression for Φ( µ ) like in thecase of inner automorphism if the morphism σ A is not invertible. Nevertheless, wehave Q ( µ ) = ( Q ( σ B )) − ◦ Q (Φ( µ )) ◦ Q ( σ A ). Lemma 1.
Let Φ be a quasi-inner automorphism of C and ( σ A ) A ∈ Ob C : Id C → Φ be a bimorphism. If σ B ◦ µ = ν ◦ σ A for µ : A → B, ν : Φ( A ) → Φ( B ) ,then ν = Φ( µ ) . Hence Φ( µ ) is uniquely determined by the following inclusion: Q ( σ B ) ◦ Q ( µ ) ◦ ( Q ( σ A )) − ⊆ Q (Φ( µ )) .Proof. It is obvious because σ A is an epimorphism. (cid:3) N AUTOMORPHISMS OF CATEGORIES 5 potentially quasi-inner automorphisms In this section, we assume that every category C under consideration is suppliedwith a forgetful functor Q such that there exist a C -object A and an element x ∈ Q ( A ) which satisfy the following conditions:1Q) Q is represented by the pair ( A , x ), i.e., for every object A of C and forevery element a ∈ Q ( A ) there is exactly one morphism α : A → A suchthat Q ( α )( x ) = a .2Q) for every C -object A there exists a morphism α : A → A such that for everyelement x ∈ Q ( A ) we have x = Q ( α )( a ) for some element a ∈ Q ( A ), inother words, Q ( α ) : Q ( A ) → Q ( A ) is surjective.Some of the simple properties of such categories need to be noted: Lemma 2. Let µ : A → B be a C -morphism. If Q ( µ ) : Q ( A ) → Q ( B ) is surjective then µ isan epimorphism. A morphism µ : A → B is a monomorphism if and only if the map Q ( µ ) : Q ( A ) → Q ( B ) is injective. Let Φ be an automorphism of the category C . There exists an epimorphism η : A → Φ( A ) . If η is an isomorphism then Φ is potentially inner.Proof.
1. It is obvious.2. It is also obvious that if Q ( µ ) is injective then µ is a monomorphism. Let µ : A → B is a monomorphism. Suppose that Q ( µ ) is not injective, i. e., there existelements a , a ∈ Q ( A ) such that a = a but Q ( µ )( a ) = Q ( µ )( a ). Accordingto property 1Q, there are morphisms α , α : A → A such that Q ( α )( x ) = a and Q ( α )( x ) = a . We have Q ( µ )( Q ( α )( x )) = Q ( µ )( Q ( α )( x )) and hence µ ◦ α = µ ◦ α . Since µ is a monomorphism we obtain that α = α whichcontradicts the supposition a = a . GRIGORI ZHITOMIRSKI
3. Consider the object Φ − ( A ). According to the property Q2 there is a morphism η : Φ − ( A ) → A such that the mapping Q ( η ) : Q (Φ − ( A )) → Q ( A ) issurjective. Thus η : Φ − ( A ) → A is an epimorphism. Hence η = Φ( η ) : A → Φ( A ) is an epimorphism too. If η is an isomorphism, then Φ is potentially inneraccording to Theorem 1 in [7]. (cid:3) Definition 2.
The surjective morphism η : Φ − ( A ) → A and the epimorphism η : A → Φ( A ) introduced by the proof of Lemma 2 are fixed as main epimorphismsconnected with Φ and are denoted by η Φ0 and η Φ correspondingly. If A = Φ( A ) we assume that η Φ = η Φ0 = 1 A . Definition 3.
Let
A, B ∈ Ob C . A mapping s : Q ( A ) → Q ( B ) is called C -bijection if s is injective and for any two C -morphisms α , α : B → C the equality Q ( α ) ◦ s = Q ( α ) ◦ s implies α = α . Roughly speaking the mapping s has the epimorphismproperty only in relation to C -morphisms. It is obvious that on the one hand the notion of C -bijection is a generalization ofthe notion of bijection and, on the other hand, if for a C -morphism σ : A → B themapping Q ( σ ) : Q ( A ) → Q ( B ) is a C -bijection then σ : A → B is a bimorphism. Definition 4.
An automorphism Φ of a category C is called potentially quasi-innerif there exists a family of C -bijections ( s A : Q ( A ) → Q (Φ( A ))) A ∈ Ob C such that forevery C -morphism µ : A → B the following diagram commutes: Q ( A ) s A −−−−→ Q (Φ( A )) Q ( µ ) y y Q (Φ( µ )) Q ( B ) s B −−−−→ Q (Φ( B )) It is easy to see that for potentially quasi-inner automorphisms the fact analogoustoLemma 1 is true too, i. e., if s B ◦ Q ( µ ) = Q ( ν ) ◦ s A for some morphism ν :Φ( A ) → Φ( B ) then ν = Φ( µ ) . Thus the morphism Φ( µ ) is uniquely determined bythe commutativity of the diagram above. Definition 5.
Let A be a C -object and a ∈ Q ( A ) . We denote by α Aa the uniquemorphism A → A such that Q ( α Aa )( x ) = a . N AUTOMORPHISMS OF CATEGORIES 7
Now we are ready to prove the first of two main theorems.
Theorem 1.
Every automorphism of a category which satisfies conditions 1Q and2Q is potentially quasi-inner.Proof.
Let Φ be an automorphism of a category C and A be a C -object. Define themapping s A : Q ( A ) → Q (Φ( A )) by the formula: s A ( a ) = Q (Φ( α Aa ) ◦ η )( x ) (1)for every a ∈ A . Here η = η Φ according to Definition 2.Start to check that s A is a C -bijection.1). Let s A ( u ) = s A ( v ) for u, v ∈ Q ( A ). Then Q (Φ( α Au ) ◦ η )( x ) = Q (Φ( α Av ) ◦ η )( x ).Hence Φ( α Au ) ◦ η = Φ( α Av ) ◦ η . Since η is a C -epimorphism we obtain that Φ( α Au ) =Φ( α Av ) which implies α Au = α Av , i. e., u = v . Hence s A is injective.2). Let Q ( γ ) ◦ s A = Q ( δ ) ◦ s A for some C -morphisms γ, δ : Φ( A ) → B . Using 1 wehave for all a ∈ Q ( A ) Q ( γ ) ◦ Q (Φ( α Aa ) ◦ η )( x ) = Q ( δ ) ◦ Q (Φ( α Aa ) ◦ η )( x )and hence γ ◦ Φ( α Aa ) ◦ η = δ ◦ Φ( α Aa ) ◦ η for all a ∈ Q ( A ).Since η is an epimorphism we obtain γ ◦ Φ( α Aa ) = δ ◦ Φ( α Aa ). Applying Φ − toboth sides of this equation we get that Φ − ( γ ) ◦ α Aa = Φ − ( δ ) ◦ α Aa for all a ∈ Q ( A ).Hence Φ − ( γ ) = Φ − ( δ ) which gives γ = δ . Thus s A is C -bijective.It remains to check that the corresponding square in Definition 4 is commutative.According to 1 we have for every a ∈ Q ( A ) Q (Φ( µ )) ◦ s A ( a ) = Q (Φ( µ )) ◦ Q (Φ( α Aa ) ◦ η )( x ) == Q (Φ( µ ) ◦ Φ( α Aa ) ◦ η )( x ) = Q (Φ( µ ◦ α Aa ) ◦ η )( x ) == Q (Φ( α B Q ( µ )( a ) ) ◦ η )( x ) = s B ( Q ( µ )( a )) = s B ◦ Q ( µ )( a ) . Thus Q (Φ( µ )) ◦ s A = s B ◦ Q ( µ ). (cid:3) GRIGORI ZHITOMIRSKI
Definition 6.
The family of C -bijections ( s A : Q ( A ) → Q (Φ( A ))) A ∈ Ob C defined by1 we call the main function corresponding to Φ . It is easy to see that this notion is a generalization of the notion of the mainfunction defined in [7]. In fact, if the main epimorphism η : A → Φ( A ) is anisomorphism we get the same formula for s A (see [7] formula 2.2).Of course the main function is not the unique function ( Q ( A ) → Q (Φ( A ))) A ∈ Ob C which makes the squares in Definition 4 commutative. For example, the mainfunction for the identity functor is equal (according to 1 and Definition 2) to s A ( a ) = Q ( α Aa ◦ A )( x ) = a , that is, s A = Q (1 A ). Hence in that case we have s B ◦ Q ( µ ) = Q ( µ ) ◦ s A for every morphism µ : A → B . Having in mind this property we use thefollowing notion (see [7, 8]). Definition 7.
The family ( c A ) A ∈ Ob C of C -bijective mappings c A : Q ( A ) → Q ( A ) having the following property: c B ◦ Q ( µ ) = Q ( µ ) ◦ c A for every morphism µ : A → B is called a central function. In other words, a function A → c A is a central functionif it determines the identity automorphism of the category C . Let Φ be an automorphism of C . Theorem 1 states that both Φ and Φ − are po-tentially quasi-inner. Therefore we have two main functions ( s Φ A : A → Φ( A )) A ∈ Ob C and ( s Φ − A : A → Φ − ( A )) A ∈ Ob C of C -bijections and two main epimorphisms η Φ : A → Φ( A ) and η Φ − : A → Φ − ( A ).Although in general case, the main functions do not satisfy many conditionswhich are valid in the case η is an isomorphism, they have some similar properties.In particular, we get the following commutative diagram for every morphism µ : A → B : Q ( A ) s Φ A −−−−→ Q (Φ( A )) s Φ − A ) −−−−→ Q ( A ) Q ( µ ) y y Q (Φ( µ )) y Q ( µ ) Q ( B ) s Φ B −−−−→ Q (Φ( B )) s Φ − B ) −−−−→ Q ( B ) N AUTOMORPHISMS OF CATEGORIES 9
We see that the family of mappings c Φ A = s Φ − Φ( A ) ◦ s Φ A : Q ( A ) → Q ( A ) is a centralfunction. This central function will be called the main central function for theautomorphism Φ.It is obvious that for every function A s A that determines an automorphismΦ and any two central functions A e A and A d A the function A ( e Φ( A ) ◦ s A ◦ d A ) determines the same automorphism. The converse proposition is also true. Proposition 1.
Let a function A t A , where A ∈ Ob C , determines an automor-phism Φ of a category C . Then there exists a central function A d A such that t A = ( c Φ − Φ( A ) ) − ◦ s Φ A ◦ d A for all C -objects A .Proof. Let d A = s Φ − Φ( A ) ◦ t A . It is obvious that the function A d A is central. Weget s ΦΦ( A ) ◦ s Φ − Φ( A ) ◦ t A = s ΦΦ( A ) ◦ d A . Since s ΦΦ( A ) ◦ s Φ − Φ( A ) = c Φ − Φ( A ) is injective, we get t A = ( c Φ − Φ( A ) ) − ◦ s Φ A ◦ d A (cid:3) The categories of free universal algebras
From now until the end of the article we deal with an arbitrary variety V ofuniversal algebras of some signature Ω and with the corresponding category Θ( V ) of all finitely generated V -algebras and their homomorphisms. This category andall its subcategories are supplied with usual forgetful functor Q that assigns toevery algebra A its underlying set | A | and to every homomorphism A → B thecorresponding mapping | A | → | B | . If there are no misunderstanding we denote analgebra by the same letter as its underlying set and do the same for homomorphisms.Let A be a monogenic algebra free generated by an element x . We considersuch full subcategories of Θ( V ) that contain the object A . It is obvious that everysuch category satisfies the conditions 1Q and 2Q formulated in Section 3 becausethe pair ( A , x ) represents the forgetful functor Q .Let C be a full subcategory of Θ( V ) and A be an object of C . Let Φ bean automorphism of C . Theorem 1 states that both Φ and Φ − are potentiallyquasi-inner. Therefore we have two main functions ( s Φ A : A → Φ( A )) A ∈ Ob C and ( s Φ − A : A → Φ − ( A )) A ∈ Ob C of C -bijections and four main epimorphisms: η Φ0 : Φ − ( A ) → A , η Φ : A → Φ( A ) ,η Φ − : Φ( A ) → A , η Φ − : A → Φ − ( A ) . By definition, η Φ = Φ( η Φ0 ). The epimorphism η Φ0 was introduced in the processof the proof of Lemma 2 as an epimorphism Φ − ( A ) → A but now we can specifyit. Let X be a basis of Φ − ( A ). Then we set η Φ0 ( x ) = x for all x ∈ X . Itis obvious that some arbitrariness remains in the definition of η Φ0 which dependson the choice of X . Specifically, if there is an isomorphism between Φ − ( A ) and A we prefer to assume that η Φ0 is equal to corresponding isomorphism. All thisconcerns the epimorphism η Φ − too.Just as in the previous section, we have the main central function ( c Φ A ) A ∈ Ob C .Now we describe the images of this function. Proposition 2.
Let A be a C -algebra and a ∈ A . Let c Φ A = s Φ − Φ( A ) ◦ s Φ A . Denote by w Φ ( x ) the term η Φ0 ◦ η Φ − ( x ) ∈ A . Then c Φ A ( a ) is an element of subalgebra of A generated by a , namely, c Φ A ( a ) = w Φ ( a ) . If w Φ ( x ) = x then all mappings c Φ A arethe identity mappings and therefore Φ is a potentially inner automorphism.Proof. We have c Φ A ( a ) = s Φ − Φ( A ) ( s Φ A ( a )) = Φ − ( α Φ( A ) s Φ A ( a ) ) ◦ η Φ − ( x ) = Φ − ( α Φ( A ) s Φ A ( a ) ◦ η Φ − )( x ) . Now let calculate the mapping that is in parentheses. Let x , ..., x n be a basis ofΦ( A ). By definition η Φ − ( x i ) = x . Thus α Φ( A ) s Φ A ( a ) ◦ η Φ − ( x i ) = α Φ( A ) s Φ A ( a ) ( x ) = s Φ A ( a ) = Φ( α Aa ) ◦ η Φ ( x ) == Φ( α Aa ◦ η Φ0 )( x ) = Φ( α Aa ◦ η Φ0 ) ◦ η Φ − ( x i ) . Since x i is an arbitrary element of basis we obtain that α Φ( A ) s Φ A ( a ) ◦ η Φ − = Φ( α Aa ◦ η Φ0 ) ◦ η Φ − . N AUTOMORPHISMS OF CATEGORIES 11
If we substitute the mapping in parentheses by the obtained value we get that c Φ A ( a ) = Φ − (Φ( α Aa ◦ η Φ0 ) ◦ η Φ − )( x ) == α Aa ◦ η Φ0 ◦ Φ − ( η Φ − )( x ) = α Aa ◦ η Φ0 ◦ η Φ − ( x ) = α Aa ( w Φ ( x )) = w Φ ( a ) . Thus c Φ A ( a ) ∈ h a i .If all mappings c Φ A are identity mappings then all s Φ A are bijections and thereforeΦ is a potentially inner automorphism. (cid:3) Proposition 3.
Let A be a C -algebra and X be a basis of A . For any central func-tion ( c A ) A ∈ Ob C , either none of elements of X belongs to c A ( A ) or c A is surjective.Proof. Let x ∈ c A ( A ) for some x ∈ X and a ∈ A . Let µ be an endomorphism of A such that µ ( x ) = a . Then a ∈ µ ( c A ( A )). Since c A is central we have a ∈ c A ( µ ( A ))that implies a ∈ c A ( A ). Thus c A is surjective because a is an arbitrary element of A . (cid:3) We show that the same fact takes place for s Φ A . Hereinafter we write s A insteadof s Φ A if it is clear what we have in mind. Proposition 4.
Let A be a C -algebra and X be a basis of Φ( A ) . Then either noneof elements of X belongs to s A ( A ) or s A is surjective.Proof. Let x ∈ s A ( A ) for some x ∈ X . Let u be an arbitrary element of Φ( A ).There is an endomorphism µ of Φ( A ) such that µ ( x ) = u . We have the equation s A ◦ Φ − ( µ ) = µ ◦ s A . Since x ∈ s A ( A ) there is a ∈ A such that x = s A ( a ). We have s A (Φ − ( µ )( a )) = µ ( s A ( a )) = µ ( x ) = u . Thus u ∈ s A ( A ). Since u is an arbitraryelement of Φ( A ), s A is surjective. (cid:3) Let X = { x , . . . , x n } be a basis of C -algebra A . Let y i = s Φ A ( x i ) for i = 1 , . . . , n .Since s Φ A is injective all elements of the set Y = { y , . . . , y n } are pairwise different.One the other hand, Proposition 4 declares that none of elements of Y belongs toany basis of Φ( A ). Nevertheless the set Y has the following interesting property. Proposition 5.
Let B be a C -algebra and β, γ : Φ( A ) → B be two homomorphisms.If β ( y i ) = γ ( y i ) for all i = 1 , . . . , n then β = γ .Proof. We have β ( y i ) = β ( s A ( x i )) = β (Φ( α x i ) ◦ η ( x )) = β ◦ Φ( α x i ) ◦ η ( x ). Thesame we have for γ : γ ( y i ) = γ ◦ Φ( α x i ) ◦ η ( x ). Under the suggestion, we have β ◦ Φ( α x i ) ◦ η ( x ) = γ ◦ Φ( α x i ) ◦ η ( x ) , which implies β ◦ Φ( α x i ) = γ ◦ Φ( α x i )and then Φ(Φ − β ) ◦ α x i ) = Φ(Φ − γ ) ◦ α x i ) . Since X is a basis of A then after some obvious steps we obtain that β = γ . (cid:3) Now our aim is to describe images of the main function, that is, the sets s Φ A ( A )for every object A . Theorem 2.
There exists a term t ( x ) containing only one variable x such thatfor every C -algebra A the following expression takes place: s Φ A ( A ) = { t ( u ) | u ∈ Φ( A ) } . Proof.
Let x be a member of some basis of the algebra Φ − ( A ). Then s ΦΦ − ( A ) ( x )is an element of A which we consider as a term t ( x ). Let a ∈ A and µ : Φ − ( A ) → A be a homomorphism such that µ ( x ) = a . We have the following commutativediagram: Φ − ( A ) s ΦΦ − A −−−−−−→ A µ y y Φ( µ ) A s Φ A −−−−→ Φ( A )Hence s Φ A ( a ) = Φ( µ )( s ΦΦ − ( A ) ( x )) = Φ( µ )( t ( x )) = t (Φ( µ )( x )). Thus there existsan element u ∈ Φ( A ) such that s Φ A ( a ) = t ( u ). It is obvious that the element t ( u )belongs to s Φ A ( A ) for every u ∈ Φ( A ). (cid:3) N AUTOMORPHISMS OF CATEGORIES 13 Full description of automorphisms of categories of free algebras.
The object of this section is to give a description of automorphisms of arbitrarycategories which are subcategories of the category Θ ( V ) and which contain theobject A . To this end, we select two kinds of functors. Because of the fact thatwe may have two different algebraic structures on the same set it makes sense touse different notations for an algebra and for its underlying set, namely, A and | A | correspondingly. Definition 8.
Let C and D be full subcategories of the category Θ ( V ) .1. An isomorphism Ψ :
C → D is called inner if there is a family of isomorphisms ( σ A : A → Ψ( A )) A ∈ Ob C such that Ψ( µ ) = σ B ◦ µ ◦ σ − A for every morphism µ : A → B of the category C . That is, Ψ is isomorphic to the embedding functor Id Θ ( V ) C : C → Θ ( V ) .2. An injective functor Γ :
C → D is called an extension functor if(i) | A | ⊆ | Γ( A ) | for every C -object A ,(ii) µ ⊆ Γ( µ ) for every C -morphism µ ,(iii) if ν : Γ( A ) → Γ( B ) , µ : A → B and µ ⊆ ν then ν = Γ( µ ) for any C -objects A, B and morphisms µ, ν . Below we show that any automorphism of a category C under consideration is aproduct of two functors the first of which is an inner isomorphism and the secondone is an extension functor.Let Φ be an automorphism of a given category C . We follow an idea used in[7, 8]. Since s Φ A : | A | → | Φ( A ) | and s Φ − A : | A | → | Φ − ( A ) | are C -bijections theydetermine algebraic structures on the sets ℑ s Φ A = s Φ A | A | ) ⊆ | Φ( A ) | and ℑ s Φ − A = s Φ − A ( | A | ) ⊆ | Φ − ( A ) | correspondingly. Let ω be a symbol of a k -ary operationand ω A be the corresponding operation of A . Define new ω -operations on ℑ s Φ A and ℑ s Φ − A by setting for all a , . . . , a k ∈ | A | ω ∗ A ( s Φ A ( a ) , . . . , s Φ A ( a k )) = s Φ A ( ω A ( a , . . . , a k )) ,ω A ( s Φ − A ( a ) , . . . , s Φ − A ( a k )) = s Φ − A ( ω A ( a , . . . , a k )) . The set ℑ s Φ A supplied with the operations ω ∗ A for all ω ∈ Ω is an Ω-algebra whichwe denote by A ∗ . It is obvious that the bijective mapping ( s Φ A ) ∗ : | A | → ℑ s Φ A which is the mapping s Φ A : | A | → | Φ( A ) | considered as the mapping onto ℑ s Φ A is anisomorphism ( s Φ A ) ∗ : A → A ∗ . Thus A ∗ and A are isomorphic. Similarly we havethe algebra A on the set ℑ s Φ − A and the isomorphism s A : A → A . Theorem 3.
Let µ : Φ( A ) → Φ( B ) be a homomorphism. The restriction of µ on A ∗ is a homomorphism A ∗ → B ∗ . Backwards, any homomorphism ν : A ∗ → B ∗ isa restriction of exactly one homomorphism µ : Φ( A ) → Φ( B ) .Proof. Let µ : Φ( A ) → Φ( B ). We have s Φ B ◦ Φ − ( µ ) = µ ◦ s Φ A . Then µ ( s Φ A ( A )) ⊆ s Φ B ( B ), and it is correct to consider the restriction of µ on | A ∗ | as a mapping | A ∗ | →| B ∗ | . Denoting this restriction by ν we get s ∗ B ◦ Φ − ( µ ) = ν ◦ s ∗ A . Consequently, ν = s ∗ B ◦ Φ − ( µ ) ◦ ( s ∗ A ) − and hence ν is a homomorphism A ∗ → B ∗ .Backwards, let ν : A ∗ → B ∗ . Then γ = ( s ∗ B ) − ◦ ν ◦ s ∗ A is a homomorphism γ : A → B . Let µ = Φ( γ ). It is obvious that µ : Φ( A ) → Φ( B ) and ν ⊆ µ . Theuniqueness of such homomorphism follows from Proposition 5. (cid:3) The next fact may be useful in some cases.
Proposition 6.
An automorphism Φ of the category C is quasi-inner if and only ifthere exists a central function ( c A ) A ∈ Ob C such that for each C -object A the mapping c A is a bimorphism A → Φ( A ) .Proof. Let Φ be quasi-inner and ( σ A : A → Φ( A )) A ∈ Ob C be the correspondingfamily of bimorphisms (Definition 1). Let c A = s Φ − Φ( A ) ◦ σ A . Obviously ( c A ) A is acentral function. Since the mapping s Φ − Φ( A ) can be considered as the isomorphism( s Φ − Φ( A ) ) : Φ( A ) → Φ( A ) , we get that c A : A → Φ( A ) is a bimorphism.Now suppose that there exists a central function ( c A ) A ∈ Ob C such that every c A is a bimorphism A → Φ( A ) . Let us define σ A : A → Φ( A ) as σ A = ( s Φ − Φ( A ) ) − ◦ c A = ( s Φ − A ) ) − ◦ c A N AUTOMORPHISMS OF CATEGORIES 15 for every C -algebra A . This definition is correct because c A ( | A | ) ⊆ | Φ( A ) | = ℑ s Φ − A ) and hence σ A : A → Φ( A ) is a bimorphism.Then we have for every µ : A → B that µ ◦ s Φ − Φ( A ) = s Φ − Φ( B ) ◦ Φ( µ ) which implies( s Φ − Φ( B ) ) − ◦ µ ◦ s Φ − Φ A ) ◦ ( s Φ − Φ A ) ) − ◦ c A = ( s Φ − Φ( B ) ) − ◦ s Φ − Φ( B ) ◦ Φ( µ ) ◦ σ A and hence σ B ◦ µ = Φ( µ ) ◦ σ A .Thus Φ is quasi-inner. (cid:3) Remark 1.
It may be that an automorphism Φ is quasi-inner but Φ − is not asuch one. It is clear that for Φ − the statement like Proposition 6 is true in which”*” is used instead of ” Now, we proceed to describing for each C -object A the algebra A ∗ , taking intoaccount that A ∗ perhaps is not an object of C .Suppose that arities of operations of our variety V are less of some number n .Let F be a C -algebra and X = { x , . . . , x n } be a basis of F . Let ω be a symbolof a k -ary operation and ω A be the corresponding operation on an algebra A . Weconsider the expression w = ω ( x , . . . , x k ) as a term in the language correspondingto the our variety. For every k -tuple ( a , . . . , a k ) of elements of an algebra A ,the value ω A ( a , . . . , a k ) is equal to θ ( ω ( x , . . . , x k )) where θ : F → A such that θ ( x i ) = a i for i = 1 , . . . , k .Now consider the element ˜ w = s A ( w ) of the free algebra Φ( F ). It turns out thatthe element ˜ w determines an ω -operation on the set s A ( | A | ) for every C -algebra A .Let ( u , . . . , u k ) be a k -tuple of elements of s A ( | A | ). Then we have an unique k -tuple( a , . . . , a k ) of elements of the algebra A such that s A ( a i ) = u i for i = 1 , . . . , k . Let θ : F → A be a homomorphism such that θ ( x i ) = a i for i = 1 , . . . , k . We define thenew operation e ω Φ( A ) by the value of the term ˜ w as follows: e ω Φ( A ) ( u , . . . , u k ) = Φ( θ )( ˜ w ) (2) On the other hand, have the commutative diagram: F s F −−−−→ Φ( F ) θ y y Φ( θ ) A s A −−−−→ Φ( A )We obtain u i = s A ( a i ) = s A ( θ ( x i )) = Φ( θ )( s F ( x i )) andΦ( θ )( e w ) = Φ( θ ) ◦ s F ( ω ( x , . . . , x k )) = s A ( ω A ( a , . . . , a k )) == ω ∗ A ( s A ( a ) , . . . , s A ( a k )) = ω ∗ A ( u , . . . , u k ) . Hence e ω Φ( A ) ( u , . . . , u k ) = ω ∗ A ( u , . . . , u k ) and the operation ω ∗ A is the derivedoperation e ω Φ( A ) on ℑ s A . By the way this result shows that the operation definedby (2) does not depend on the choice of an algebra F having the assumed property.Thus we have proved the following result. Theorem 4.
For every C -algebra A the derived operation e ω Φ ( A ) defined by (2) coincides with the induced operation ω ∗ A , that is, s A ( ω A ( a , . . . , a k )) = e ω Φ( A ) ( s A ( a ) , . . . , s A ( a k )) for all a , . . . , a k ∈ A . Now we are ready to prove the second main theorem.
Theorem 5.
Let C be a full subcategory of Θ ( V ) containing a monogenic algebra A . Let Φ be an automorphism of C . Then(i) there is a full subcategory D of Θ ( V ) such that Φ is a product Φ = Γ ◦ Ψ , where Ψ :
C → D is an inner isomorphism and
Γ :
D → C is an extension functor.(ii) there is a term t ( x ) ∈ | A | such that | Ψ( A ) | = { t ( u ) | u ∈ | Φ( A ) }| for every C -object A .Proof. Let D be the full subcategory of Θ ( V ) whose objects are all algebras A ∗ where A is an arbitrary C -object. Let Ψ( A ) = A ∗ for every C -object A and Ψ( µ ) = s ∗ B ◦ µ ◦ ( s ∗ A ) − for every C -morphism µ : A → B . It is clear that Ψ : C → D is afunctor and, what is more, Ψ is an inner isomorphism between C and D . N AUTOMORPHISMS OF CATEGORIES 17
Now let Γ = Φ ◦ Ψ − . Then Γ is a functor D → C . Of course, Γ is an isomorphismbetween these categories and Φ = Γ ◦ Ψ. We have Γ( A ∗ ) = Φ( A ) and therefore | A ∗ | ⊆ | Γ( A ∗ ) | . Let µ : A ∗ → B ∗ . According to the definition of Ψ, we have ν = Ψ − ( µ ) = ( s ∗ B ) − ◦ µ ◦ s ∗ A is a homomorphism A → B . Thus Γ( µ ) = Φ( ν ). Onthe other hand, s B ◦ ( s ∗ B ) − ◦ µ ◦ s ∗ A ◦ ( s A ) − = µ , that is, µ is the restriction of Γ( µ )on A ∗ . In view of Theorem 3 and Definition 8 we obtain that Γ is an extensionfunctor. The second statement follows from Theorem 2. (cid:3) Corollary 1.
The last theorem shows that the process of describing of an automor-phism Φ of a category C under consideration is reduced to the following steps: (1) We view the elements of A and find the general form of the term t ( x ) ∈| A | . Then we describe the subset s A ( | A | ) of an algebra Φ( A ) according tothe formula s A ( | A | ) = { t ( u ) : u ∈ | Φ( A ) |} . In the case t ( x ) = x this setis equal to the underlying set of Φ( A ) . (2) In order to describe operations of A ∗ , we use the fact that the restrictionof any endomorphism of Φ( A ) to s A ( | A | ) must be an endomorphism of thealgebra A ∗ , and vice versa, any endomorphism of A ∗ could be extended upto an endomorphism of Φ( A ) which is uniquely determined. (3) We use the fact that the correspondence described above is an isomorphismbetween semigroups
End( A ∗ ) and End(Φ( A )) .If the requirements of these steps are fulfilled, it remains to describe the kind ofembedding of A ∗ in Φ( A ) , which may be an isomorphism or some new kind of acorrespondence, for example, a mirror homomorphism or a screw-homomorphism. Examples.
1. First we will show how method suggested above can be applied in a simplecase when the result is already known, namely for the category
SEM of all finitelygenerated free semigroups. Let C be a full subcategory of SEM containing a mono-genic semigroup A . Let Φ be an automorphism of C . Any term t ( x ) ∈ A in ourcase has a form t ( x ) = x k where k ≥ F = F ( x , . . . , x n ) be a free semigroup generated by variables x , . . . , x n and A = Φ − ( F ). Thus A ∗ = { u k | u ∈ F } . Let || w || denote the length of the word w ∈ F .Let y be an element of a basis of A ∗ , then there exists an endomorphism γ of A ∗ such that x k = γ ( y ). Let the endomorphism ˜ γ of F be the extension of γ .Thus ˜ γ ( y ) = x k . Since y = u k for some u ∈ F we get k ≥ || y || ≥ k and hence y ∈ { x k , . . . , x kn } . Applying this result to A (n=1) we obtain that (Φ − ( A )) ∗ and A are isomorphic and hence Φ − ( A ) and A are isomorphic. We know (Section 4)that in this case all mappings s A are surjective and hence k = 1. Therefore ( x , x )is the common basis of semigroups F = F ( x , x ) and A ∗ which are isomorphic.Since ( x , x ) is the unique basis of F and the unique basis of A ∗ , there are onlytwo isomorphisms A ∗ → F , namely ϕ ( x ) = x , ϕ ( x ) = x or ϕ ( x ) = x , ϕ ( x ) = x . In the former case we have ϕ ( x ∗ x ) = x x and hence ϕ is the identitymapping. In the latter case, we have ϕ ( x ∗ x ) = x x and therefore ϕ mapsevery word u to the word u , where all letters are written in the reverse order. Weobtain that Φ is produced by isomorphisms, that is, Φ is inner, or all mappings s A : A → Φ( A ) are anti-isomorphisms.2. Now we apply our method to the variety of modules. As far as the authorknows there are no results in the general case. There are some essential resultsin this topic in [1] and [2]. Let R be an arbitrary ring with unit and Mod − R denote the category of all finitely generated free left R -modules. We consider a fullsubcategory C of the category Mod − R which satisfies the accepted conditions.In this case A = Rx .Let Φ be an automorphism of C . Consider a free left R -module M = ( M, + , , F )where F : R × M → M is the left action of R on M . Assume that { x , x } is a basisof M , that is, M = Rx ⊕ Rx . Let A = Φ − ( M ). We have to describe the finitegenerated free left R -module A ∗ = ( | A ∗ | , + ∗ , ∗ , F ∗ ). Since any term t ( x ) ∈ A inour case has a form t ( x ) = rx where r ∈ R, r = 0, we get | A ∗ | = rM . It is clearthat 0 ∈ | A ∗ | . N AUTOMORPHISMS OF CATEGORIES 19
On the other hand, for every endomorphism ϕ of A ∗ we have ϕ (0 ∗ ) = 0 ∗ . Thenaccording to (2) in Corollary 1 γ (0 ∗ ) = 0 ∗ for every endomorphism γ of M . Thus0 ∗ = 0. Further rx + ∗ rx = ix + jx for some i, j ∈ R . Let γ : M → M suchthat γ ( x ) = x , γ ( x ) = 0. Since restriction of γ to | A ∗ | is an endomorphism of A ∗ too, we get γ ( rx + ∗ rx ) = γ ( rx ) + ∗ γ ( rx ) = rγ ( x ) + ∗ rγ ( x ) = rx + ∗ rx . On the other hand, γ ( ix + jx ) = ix . Thus rx = ix and hence i = r . In exactlythe same way we get j = r and hence rx + ∗ rx = rx + rx .Let a , a ∈ M and γ be an endomorphism of M such that γ ( x ) = a and γ ( x ) = a . Just as above we get γ ( rx + ∗ rx ) = γ ( rx ) + ∗ γ ( rx ) = rγ ( x ) + ∗ rγ ( x ) = ra + ∗ ra . On the other other hand we get γ ( rx + rx ) = γ ( rx ) + γ ( rx ) = rγ ( x ) + rγ ( x ) = ra + ra . As a result we obtain ra + ∗ ra = ra + ra , which leads after obvious calculations to the fact that operations + ∗ and + coincideon | A ∗ | . Thus the different between structures M and A ∗ may be only in actionsof the ring R .It is obvious that for every k ∈ R there is an element k ∗ ∈ R such that k ∗ rx = rk ∗ x . Such an element k ∗ may be is not uniquely determined. But there exists afunction α : R → R such that k ∗ rx = rα ( k ) x . For two such function α and α we have rα = rα . In the same way as above we get: k ∗ ra = rα ( k ) a for every a ∈ M . Further( k k ) ∗ ra = k ∗ ( k ∗ ra ) = k ∗ rα ( k ) a = r ( α ( k ) α ( k )) a. On the other other hand we get( k k ) ∗ ra = rα ( k k ) a. Since a is an arbitrary element of M we obtain that rα ( k k ) = r ( α ( k ) α ( k )) . (3)In the same way we get that rα ( k + k ) = rα ( k ) + rα ( k ) (4)and rα (1) = r. (5)Summing up these investigations, we obtain the following description of automor-phisms of C .For every automorphism Φ of C the main function ( s A : A → Φ( A )) A ∈ Ob C satisfies the following conditions:(1) there exists an element r ∈ R such that for every a ∈ A s A ( a ) = ru for some u ∈ Φ( A ),(2) every C -bijection s A : A → Φ( A ) is an additive mapping,(3) there exist a function α : R → R such that for every k ∈ R and every a ∈ A itholds s A ( ka ) = rα ( k ) u where ru = s A ( a ) , u ∈ Φ( A ),(4) the function α satisfies the equations 3 - 5.Consider the case when the ring R does not contain zero divisors. Let B =Φ − ( Rx ). Suppose that a basis of the module B ∗ contains two different elements y and y . Let γ be the automorphism of B ∗ such that γ ( y ) = y and γ ( y ) = y and ˜ γ is the extension of γ up to an automorphism of Rx . Since γ = 1 B ∗ thesame property is valid for ˜ γ . Therefore we get ˜ γ ( x ) = x . Since ˜ γ ( x ) = kx for some k ∈ R we get that k = 1 and hence k = 1 or k = −
1. Thus y = y or y = − y that is contrary to the assumption.We obtain that Φ − ( Rx ) and ( Rx ) are isomorphic. In this case all mappings s A : A → Φ( A ) are surjective and r = 1. Therefore all automorphisms of C aresemi-inner, that is, the mappings s A : A → Φ( A ) are additive bijections and thereexists an endomorphism α of R such that s A ( ka ) = α ( k ) s A ( a ) for all k ∈ R and N AUTOMORPHISMS OF CATEGORIES 21 a ∈ A . Since the same is true for the automorphism Φ − we conclude that α is anautomorphism of the ring R . Acknowledgments
The author is pleased to thank B. Plotkin and E. Plotkin,R. Lipjansky and G. Mashevitsky for useful discussions and interesting suggestions.