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\begin{document}

\title{Automorphisms of categories of free algebras of varieties}

\author{G. Mashevitzky}
\address{Department of Mathematics, Ben Gurion University of the Negev, 84105, Israel}
\email{gmash@cs.bgu.ac.il}
\thanks{This work was supported in part by
the Israel Science Foundation founded 
by the Israel Academy of Sciences and Humanities --- Center of Excellence 
Program and by Intas grant ``Algebraic $K$-theory, groups and algebraic 
homotopy theory''.} 


\author{B. Plotkin}
\address{Institute of Mathematics,  
The Hebrew University, Jerusalem, 91904, Israel}
\email{borisov@math.huji.ac.il}

\author{E. Plotkin}
\address{Department of Mathematics,  
Bar Ilan University, Ramat Gan, 52900, Israel}
\email{plotkin@macs.biu.ac.il}



\subjclass[2000]{Primary 08A35, 08CO5, 14A22, 14A99}
\keywords{Algebraic variety, variety of algebras, 
category, free algebra, automorphisms}
\commby{Efim Zelmanov}
\date{July 4, 2001}

\begin{abstract}
Let $\Theta$ be an arbitrary variety of algebras
and let $\Theta^0$ be the category of all free
finitely generated algebras from $\Theta$. We study
automorphisms of such categories for special
$\Theta$. The cases of the varieties of all groups,
all semigroups, all modules over a noetherian ring,
all associative and commutative algebras over a field
are completely investigated.  The cases of associative and Lie algebras
are also considered. This topic relates to algebraic geometry
in arbitrary variety of algebras $\Theta$.
\end{abstract}
\maketitle


\section{Motivations}


\subsection{The main problem and  automorphisms of free objects}

We consider an arbitrary variety of algebras $\Theta$.
For any $\Theta$ denote by $\Theta^0$ the category of all 
free in $\Theta$ algebras $W=W(X)$, where $X$ is finite.
In order to avoid set-theoretic problems we view all $X$ 
as subsets of a universal infinite set $X^0$. 


Our main goal is to study automorphisms of the category $\Theta^0$ 
and the corresponding group $\Aut \Theta^0$.

The study of automorphisms
of the category $\Theta^0$ is tied to the study of automorphisms of
the semigroups $\End W$, $W\in \Ob \Theta^0$. The group of automorphisms
$\Aut W$ consists of invertible elements of the semigroup $\End W$.
There is the embedding $\Aut W\to \Aut(\End W)$. The image of $\Aut W$
is the group of all inner automorphisms of the semigroup $\End W$.

A great deal is known about the group $\Aut W$ for different varieties $\Theta$
and $W\in \Ob \Theta^0$. 
Automorphisms of free groups are
well known \cite{LS}, and the same is true for free Lie
algebras \cite{Co}, free associative algebras over a field
(when the number of generators is $\leq 2$; see \cite{Co,
ML, Cz, Na}), and some other varieties. 
For free associative 
algebras with a greater number of generators the question is still open
(see Cohn's conjecture \cite{Co}). 

The relevant question  
is how do the towers of automorphisms of free objects
look like. Let $W$ be a free object and consider the tower of groups 
$\Aut W$, $\Aut^2 W=\Aut(\Aut W),\dots , 
\Aut^n W=\Aut(\Aut^{(n-1)} W),\dots$\,.
Minimum $n$ such that every automorphism of $\Aut^n W$ is inner
is called the height of the tower. The heights are known for the variety
of semigroups, the variety of all groups \cite{DF, For},
the category of free modules over a field or over  ``good''
rings \cite{A, HR}, etc.

There is an embedding $\tau_W: \Aut(\End W)\to \Aut(\Aut W)$. 
Investigation of $\Ker \tau_W$ and $\Img \tau_W$ is of independent
interest. Formanek \cite{For} has shown that if $\Theta$ is
the variety of groups, then $\Ker \tau_W=1$, and the group    
$\Aut(\End W)$ is isomorphic to  $ \Aut(\Aut W)$. In the cases 
of modules or associative algebras the situation is more complicated.
 
Thus, there is much information related to automorphisms of individual free
objects. We note that  our aim is to study not these automorphisms but
{\it automorphisms of categories of free objects}. It turns out
that new notions have arisen which make this subject quite
natural and highly motivated. We give the corresponding explanations in the 
next two subsections.

\subsection{Geometric motivation} 

Our primary interest in automorphisms of categories has grown from the
universal algebraic geometry (see \cite{Pl7, PAA, PSib,
PIsr, BMR1, BMR2, MR2}, etc).
In order to make the exposition self-contained we recall the
necessary information. In this subsection we provide a glimpse
on the motivation, and in the next one there will be a sketch of the
subject with some precise definitions. Most of the material
from 1.2 and 1.3 is collected  in \cite{Pl7}. 


Let $\Theta$ be the variety of all associative, commutative algebras
over the infinite ground field $P$. Denote by 
$W(X)=P[X]$, $X=\{x_1,\dots, x_n\}$,
the algebra of polynomials with commuting variables, which is a free
algebra in $\Theta$. The classical algebraic geometry
is associated with this variety, and for any extension $L$ of the
ground field $P$ the algebraic sets in the affine
space $L^n$ correspond to the $L$-closed ideals in $P[X]$. Now suppose
$L_1$ and $L_2$ are two extensions of the ground field $P$.

The key question is {\it when do the geometries defined by $L_1$ and $L_2$
coincide?} Let us denote by $K_\Theta(L)$ the category of all algebraic sets in
$L^n$.  This category is regarded as an invariant which is responsible
for the geometry in $L$.

Then the question can be reformulated as follows: {\it when are the
categories of algebraic sets $K_\Theta(L_1)$ and $K_\Theta(L_2)$ 
isomorphic?}

Within last years it has been figured out that one can replace the
variety of associative commutative algebras (the so-called classical
variety) by an arbitrary variety of algebras $\Theta$ and construct
algebraic geometry in $\Theta$ with respect to a distinguished algebra
$H$ in $\Theta$. This $H$ takes the role of the field $L$. 
 Thus, let $\Theta$ be an arbitrary variety of algebras,
$H_1$, $H_2$ algebras in $\Theta$,  and $K_\Theta(H_1)$, 
$K_\Theta(H_2)$ the corresponding categories of algebraic sets.

The principal problem for the variety $\Theta$ repeats the
one for the classical case:

\begin{problem}
When do the geometries over $H_1$ and $H_2$ coincide, i.e., when
are the categories $K_\Theta(H_1)$ and $K_\Theta(H_2)$ isomorphic?
\end{problem}

There is an answer to this question \cite{Pl7}, which is formulated 
in terms of two notions: geometric equivalence and geometric similarity
of algebras (see 1.3).

Geometric similarity provides necessary and sufficient conditions
for the categories $K_\Theta(H_1)$ and $K_\Theta(H_2)$ to be isomorphic,
while geometric equivalence gives only a sufficient condition. However,
the notion of geometric equivalence is much more explicit, transparent
and well verified than the notion of geometric similarity. Thus, the main
problem is converted to the following:

\begin{problem}
For which categories $\Theta$ does the geometric similarity either
coincide with geometric equivalence or is close to it?
\end{problem}

We show in 1.3 that this problem is tied to the description
of automorphisms of the category of free algebras in $\Theta$.


\subsection{Basics of universal algebraic geometry} 
 


Fix an algebra $H$ in $\Theta$.
Any equation in $W(X)$, $|X|=n$ has  the form $w= w',$\ $w,w'\in W$.
Systems of equations in $W$ are denoted by $T.$
They can be viewed as binary relations in $W.$
The set of homomorphisms $ \Hom(W,H)$ is regarded
as an affine space. There is the
canonical  bijection $ \Hom(W,H)\simeq H^n$. A point $\mu: W\to H$
is a solution of equation  $w=w'$ if and only if $(w,w')\in \Ker \mu$.
Consider  sets of points $A\subset \Hom(W,H).$ 
The Galois correspondence between systems of equations $T$  
and sets $A$ is given by:
\[\begin{cases} T'=A=\{\mu:W\to H\bigm| T\subset \Ker \mu\}=T'_H,\\
A'=T=\bigcap\limits_{\mu\in A}\Ker\mu.\end{cases}\]


\begin{definition} Algebras $H_1$ and $H_2$ are called geometrically
equivalent if for every finite set $X$ and every system of equations
$T$ in $W=W(X)$ the equality
\[
T_{H_1}^{''}=T_{H_2}^{''}
\]
holds.
\end{definition}
\begin{definition} 
1. A set $A$ such that $A=T'$ for some $T$ is called an algebraic set.

2. A congruence $T$ in $W$ is called $H$-closed
if there exists an algebraic set $A$ such that $T=A'$.
\end{definition}
Denote by
$\Cl_H(W)$ \ the set of all $H$-closed congruences in $W.$
This gives rise to the contravariant functor $\Cl_H:\Theta^0\to \text{Set}.$
Thus, we can reformulate Definition 1.3 in terms of the functors
$\Cl_H$, i.e., the geometric equivalence of algebras $H_1$ and $H_2$ means
that the functors 
$\Cl_{H_1}$ and $\Cl_{H_2}$ coincide.

The geometric equivalence is a quite nice property,
which in many cases can be checked effectively: 
\begin{theorem}[\cite{PSib}] If algebras $H_1$ and $H_2$ are 
geometrically equivalent,
then they have the same quasiidentities.
\end{theorem}

In the classical case the if and only if statement is true. However,
for arbitrary $\Theta$ the converse statement 
is not valid; see \cite{MR2, GoSh}.
For special categories the situation is even more transparent:
\begin{theorem}[\cite{Be1}]
 Two abelian groups $H_1$ and $H_2$ are geometrically equivalent if and only if

1. They have the same exponents.

2. For every prime $p$ the exponents of Sylow subgroups $H_{1p}$ and
$H_{2p}$ coincide.
\end{theorem}

An easy, but crucial fact states that the geometric equivalence of algebras
$H_1$ and $H_2$ gives a sufficient condition for the categories 
of algebraic sets $K_\Theta(H_1)$
and $K_\Theta(H_2)$ to be isomorphic.

In order to get a necessary and sufficient condition we have to use
the notion of geometric similarity. Let $\Var(H_1)$ and  $\Var(H_2)$
be the varieties generated by $H_1$ and $H_2$, respectively; for
simplicity we assume that $\Var(H_1)=\Var(H_2)=\Theta$.

{\it Geometric similarity of algebras} means that there is an isomorphism 
\[\varphi:
\Var(H_1)^0\to\Var(H_2)^0\] 
with the commutative diagram
\begin{equation*}
\begin{xy}
\xymatrix{
\Var(H_1)^0\ar[r]^\varphi\ar[dr]_{\Cl_{H_1}}&\Var(H_2)^0\ar[d]_{
\Cl_{H_2}}\\
& \Sett\\
}
\end{xy}
\end{equation*}
Commutativity of the diagram indicates that there is the isomorphism 
(not necessarily
equality) of the functors
$\Cl_{H_1}$ and
$\Cl_{H_2}\varphi. $
This isomorphism $\alpha=\alpha(\varphi)$ depends on the isomorphism of categories
$\varphi$ and is
constructed in a special way.

The notion of geometric equivalence is a particular case of 
geometric similarity when $\varphi=1$. 
The principal observation \cite{Pl7} says that if the isomorphism $\varphi$
is isomorphic as a functor to the identity functor, then geometric similarity
implies geometric equivalence. Thus, we have come to the fact which lies in the basis
of investigation of automorphisms of categories of free algebras: 
if in the category
$\Theta^0$ every automorphism is isomorphic to the identity functor and
$\Var(H_1)=\Var(H_2)=\Theta$, then
the geometries
over algebras $H_1$ and $H_2$ coincide if and only if the algebras
$H_1$ and $H_2$ are geometrically equivalent.

\section{Definitions}


\subsection{Hereditary automorphisms of categories}

Let $C$ be an arbitrary (possibly small) category. 
Let $\End C$ be the semigroup of all covariant endofunctors
of the category $C$. We use the word ``endomorphisms'' instead of
``endofunctors''.
A functor $\varphi: C\to C$
is called an automorphism of the category $C$ if there exists
a functor $\varphi^{-1}:C \to C$ such that $\varphi\varphi^{-1}=
\varphi^{-1}\varphi=1_C$, where $1_C$ is the identity functor of $C$.
All automorphisms of the category $C$ form a group denoted by
$\Aut C $.


Two functors are called isomorphic if there exists an invertible natural
transformation of functors which takes one to the other.
Thus, the relation of isomorphism of functors is defined on the semigroup 
$\End C$. This relation turns out to be a congruence of $\End C$. 
The quotient semigroup
is denoted by $\End^0(C)$. The group of invertible elements of 
$\End^0(C)$ is denoted by $\Aut^0(C)$.
The group $\Aut^0(C)$ is the group of all autoequivalences of
the category $C$ which are considered up to an isomorphism
of functors.
There is the canonical homomorphism $\tau: \Aut C\to \Aut^0 C$.
The kernel of $\tau$ consists of automorphisms isomorphic
to the identity functor (inner automorphisms; see 2.2). It is not clear
for what categories the homomorphism $\tau$ is surjective.

\begin{definition} An automorphism $ \varphi: C\to C$ is called
hereditary if for every $A\in \Ob C$ the 
objects $A$ and $\varphi (A)$ are isomorphic.
\end{definition}

It is clear that an automorphism $\varphi: C\to C$ induces an isomorphism
of the semigroups $\End A$ and $\End \varphi(A)$, and of the groups
$\Aut A $ and $\Aut \varphi (A)$. This implies immediately that
every automorphism of the categories of finite sets or free semigroups
is hereditary.

A finitely generated free in $\Theta$ algebra $W=W(X)$
is hopfian if
every surjection $W\to W$ turns out to be an automorphism of $W$.

\begin{definition}
A variety $\Theta$ is called hopfian if every finitely generated 
free algebra in 
$\Theta$ is hopfian.
\end{definition}
Denote by $W_0=W(x_0)$ the free cyclic algebra with the generator
$x_0$.
\begin{proposition} If $\Theta$ is a hopfian variety and the
algebras $W_0$ and $\varphi(W_0)$ are isomorphic, then $\varphi$ is a
hereditary automorphism of the category $\Theta^0$.
\end{proposition}

\begin{definition} The category $C$ is called automorphic hereditary
 if each of its
automorphisms is hereditary.
\end{definition}
\begin{remark} The categories of sets, free semigroups, free groups, 
free modules over a noetherian ring,
free associative commutative algebras are automorphic hereditary. However, 
not every category is automorphic hereditary.  
\end{remark}

 
\subsection{Inner automorphisms}

Let $\varphi$ be a substitution on objects of the category $C$ such
that $A$ and $\varphi (A)$ are isomorphic for every $A\in\Ob C$.
Consider a function $s$ which for any object $A$ chooses an isomorphism
\[
s_A: A\to \varphi(A).
\] 
Define an automorphism 
$\hat s :C \to C$ by the rule:

1. $\hat s (A)=\varphi(A)$, for every object $A$.

2. For every morphism $\nu:A\to B$,
\[ \hat s (\nu)=s_B\nu s_A^{-1}:\varphi(A)\to\varphi(B).\]

\begin{definition} An automorphism  $\varphi: C\to C$ is called inner
if 

1. $\varphi $ is a hereditary automorphism.

2. For the substitution  $\varphi $ there exists a function $s$ such that
$\varphi=\hat s$.
\end{definition}

The equality $ \hat s (\nu)=\varphi (\nu)=s_B\nu s_A^{-1}$ can be written as
a commutative diagram,
\[\begin{CD} A @>\nu>> B\\ @V s_AVV @Vs_B VV\\
\varphi(A) @>\varphi(\nu)>> \varphi(B)\end{CD}\]

This diagram means that the natural transformation of functors
$s: 1_c\to \varphi$ is an isomorphism of functors. Thus, an automorphism
 $\varphi: C\to C$ is {\it inner if and only if   $\varphi$ is isomorphic to the
identity automorphism $1_c: C\to C$.}
Note that two automorphisms $\varphi_1, \varphi_2 : C\to C$ are isomorphic 
if and only
if $\varphi_1^{-1}\varphi_2$ is inner.
\begin{proposition} All inner automorphisms form a normal subgroup 
in $\Aut C$
denoted by $\Int C.$
\end{proposition}

Now one can define the group $\Out C$ of outer automorphisms of
the category $C$ by $\Out C=\Aut C/\Int C .$

\begin{definition} The category $C$ is called perfect if every
automorphism of it is inner.
\end{definition}

Thus, a category $C$ is perfect if and only if $\Out C=1$. Thus, if we consider
automorphisms $C$ up to isomorphisms, a perfect $C$ has no automorphisms
except trivial. 
\begin{proposition}
Every hereditary automorphism $\varphi$ of the category $C$ can be presented
in the form $\varphi=\varphi_1\varphi_2$, where $\varphi_1$ is an inner automorphism
and $\varphi_2$ is an automorphism which does not change objects.
\end{proposition}

Let us call an automorphism $\varphi$ which does not change objects
a {\it stable automorphism}.

Denote by $\HAut C$ the normal subgroup of all hereditary automorphisms and by
 $\St C$ the normal subgroup of all automorphisms which does not 
change objects of $C$.
Then $\HAut C=\Int C\cdot \St C$. For the automorphic hereditary categories,
$\HAut\ C=\Aut C $ and $\Aut C =\Int C\cdot  \St C $, respectively.

\subsection{Remarks} 

First of all observe that if an automorphism $\varphi$ of the 
category $C$ is stable, then it induces the automorphism $\varphi_A$ 
of the semigroup
$\End  A$ and of the groups $\Aut A $ for any object $A\in C.$ Thus, we get
homomorphisms $\St C\to \Aut (\End A)$ and $\St C\to \Aut(\Aut A)$ and a
description of lower floors of towers of automorphisms of free objects
becomes of special importance.

An object $A\in \Ob C$ is called {\it perfect} 
if every automorphism of the semigroup
$\End A$ is inner. If $\varphi\in \St C$ and $\varphi$ is 
inner, then $\varphi_A$ is
an inner automorphism of $\End A$. On the other hand, if $\psi$ is an
inner automorphism of $\End A$, then $\psi=\varphi_A$ for some 
$\varphi\in \St C$. Hence, if $A$ is a perfect object of $C$, then the homomorphism
$\St C\to \Aut(\End A)$ is surjective. 

Note that perfectness of $C$ does not imply that every
object of $C$ is perfect. On the other hand, perfectness of each object
 is an argument in favor of the perfectness of the category.

\section{The main theorem}
 

\subsection{Category $\Theta^0$}

Recall that for any variety of algebras $\Theta$, the category  
$\Theta^0$ is the category of all free finitely generated algebras in $\Theta$.

\begin{definition} A variety $\Theta$ is called automorphic hereditary if 
the category $\Theta^0$ is automorphic hereditary, i.e., if every automorphism 
$\varphi: \Theta^0\to \Theta^0$ is hereditary.

 A variety $\Theta$ is called regular if
for every $X$, $Y$ an isomorphism $W(X)\simeq W(Y)$, where
 algebras $W(X),W(Y)$ are free in $\Theta$, implies
$|X|=|Y|$.

 A variety $\Theta$ is called noetherian
if every finitely generated free algebra $W=W(X)$ is noetherian
with respect to congruences.
\end{definition}

It is clear that every noetherian variety is hopfian, and hence regular.

\begin{definition} A variety $\Theta$ is called perfect if 
the category of free algebras $\Theta^0$ is perfect, i.e., if every
automorphism $\varphi: \Theta^0\to \Theta^0$ is inner.

A variety $\Theta$ is called almost perfect if 
the group $\Out \Theta^0$ is finite.
 \end{definition}

\subsection{ Algebras with constants}
The main geometrical applications require the existence
of constants in the algebras under consideration.
In this section we introduce the corresponding
notions.
 
Let $\Theta$ be an arbitrary variety of algebras, and $G$
a distinguished nontrivial algebra in $\Theta$.
Consider the category $\Theta^G$ whose objects have the
form $h:G\to H$, where $H\in\Theta$ and $h$ is a
morphism in $\Theta$. Morphisms in $\Theta^G$ are
presented by commutative diagrams
\begin{equation*}
\begin{xy}
\xymatrix{
G\ar[r]^{h_1}\ar[dr]_{h_2}&H\ar[d]^\mu\\
&H'
}
\end{xy}
\end{equation*}
where $\mu, h_1,h_2$ are morphisms in $\Theta$. Objects
of $\Theta^G$ are called $G$-algebras and are denoted
by $(H,h)$. Elements of $G$ have the meaning of
constants in algebras from $\Theta$ and, adding them as
nullary operations to the signature of $\Theta$, we
get the variety of $G$-algebras $\Theta^G$. 

A free in $\Theta^G$ algebra $W=W(X)$ has the form
of the free product $G\ast W_0(X)$, where $W_0(X)$ is a free algebra in
$\Theta$.

\begin{example} 1. The variety of commutative associative
algebras over a field $P$ is of type $\Theta^G$, where
$\Theta$ is the variety of associative commutative rings
with 1, and $G$ is the field $P$.

2. The variety of associative algebras over a field.

3. The variety of $G$-groups. 
\end{example}

The category $\Theta^G$ is a subcategory in the category
$\Theta(G)$ with the same objects, while the morphisms
of $\Theta(G)$ are presented by the commutative squares
\[\begin{CD}
 G @>h >> H\\ @V\sigma VV @VV\mu V\\
G@>h'>> H'\end{CD}\]

where $\sigma\in \End G$.

Morphisms of the category  $\Theta(G)$ are called
{\it semimorphisms} of the initial category of algebras
with constants $\Theta^G$.

Consider the category $(\Theta^G)^0$ of free $G$-algebras.

\begin{definition}
An automorphism of $(\Theta^G)^0$ is called semiinner
if it is induced by an inner automorphism of the category
$\Theta(G)^0$.
\end{definition}

This means that a semiinner automorphism $\varphi$ of the category
$(\Theta^G)^0$ is given by a pair $(\sigma, s)$, where $\sigma$
is an automorphism of the algebra $G$, and $s$ is a function
which attaches to a finite set $X$ a semiisomorphism
$(\sigma, s_X): W(X)\to \varphi W(X)$. The automorphism $\sigma$ does not
depend on $X$.

All semiinner automorphisms of the category $(\Theta^G)^0$
constitute a subgroup in $\Aut (\Theta^G)^0$ denoted by
$\SInt (\Theta^G)^0$. If this subgroup has a finite index
in $\Aut (\Theta^G)^0$ then  the category $(\Theta^G)^0$ is called almost
semiperfect. The variety  $(\Theta^G)$ is almost semiperfect
if the category $(\Theta^G)^0$ is almost semiperfect.

\begin{remn}
The definitions above do not cover the case of the
category of free modules over a ring $R$ since there is
no canonical embedding of $R$ to a module. However, the
standard definition of semiautomorphisms  of a free
module has the same meaning.

Let $\sigma$ be an automorphism of a ring, 
and $KX=Kx_1\oplus\cdots \oplus Kx_n$ a free module.
Define $\sigma_X: KX\to KX$ by the rule
$\sigma_X(u)=\lambda_1^\sigma x_1+\cdots +
\lambda_n^\sigma x_n$, where
$u=\lambda_1 x_1+\cdots +
\lambda_n x_n$ is an element of $KX$. A pair
$(\sigma, \sigma_X)$ is called a {\it semiautomorphism}
 of $KX$.

Now, we can consider the category of modules with semimorphisms
(semilinear maps). In this category there are inner morphisms.
The morphisms of the category of modules induced by inner
morphisms of the category of modules with semimorphisms are called
{\it semiinner morphisms} of the category of modules.
\end{remn}
 
\begin{definition}
A variety $\Theta^G$ is called semiperfect if every
automorphism of the category $(\Theta^G)^0$ is semiinner.
\end{definition}

\begin{definition}
$G$-algebras $(H_1,h_1)$ and  $(H_2,h_2)$  are called
geometrically semi\-equivalent if there
exists an algebra  $(H,h)$ such that  $(H_1,h_1)$ and
  $(H,h)$ are semiisomorphic and  $(H,h)$
is geometrically equivalent to  $(H_2,h_2)$.
\end{definition}
  
\begin{theorem}[\cite{Pl7}]
If the geometric similarity of $G$-algebras  $(H_1,h_1)$ 
and $(H_2,h_2)$ is given by a semiinner automorphism,
then they are geometrically semiequivalent.
\end{theorem}


\subsection{The main theorem}

\begin{theorem}

1. The categories of sets and finite sets are perfect.

2. The variety of all  groups is perfect.

3. The variety of all  semigroups is almost perfect.

4. The variety of all  $R$-modules, where $R$ is a noetherian ring,
 is semiperfect. 

5. The variety of commutative associative algebras with unity element 
 over an infinite field, is semiperfect  \cite{Be1}.


6. The variety of $F$-groups, where $F$ is a free group, is semiperfect.
\end{theorem}


\begin{corollary}
1. Let $H_1$, $H_2$ be two groups, and let each of them generate the variety
of all groups. The categories of algebraic sets
$K_\Theta(H_1)$ and $K_\Theta(H_2)$ are isomorphic if and 
only if the groups are geometrically equivalent.

2. An $F$-group $(H,h)$ is called faithful if $h$ is a monomorphism.
 Let $H_1$, $H_2$ be two faithful $F$-groups. 
Then the corresponding categories of algebraic sets
 are isomorphic if and 
only if the $F$-groups are geometrically semiequivalent.

3. The same is true for modules over a noetherian ring $R$
and for  commutative associative algebras over an infinite
field.
 
\end{corollary}






\begin{problem}

Describe automorphisms of the categories of free associative 
and free Lie algebras.
\end{problem} 

\subsection{Sketch of the proof}

1. The result for the categories of sets and finite sets
is relatively easy and is based on the ideas from
\cite{SH}.

2. We prove that all varieties from the Main Theorem are hereditary
automorphic. This implies that we can study only stable 
automorphisms.
 It can be proven that every
such automorphism $\varphi$ is a {\it quasiinner automorphism}.
This means that there is a function $\sigma=\sigma(\varphi)$
which for every finite $X$ takes a bijection $\sigma_X:
W(X) \to W(X)$, and such that $\varphi(\nu)=\sigma_Y\nu\sigma_X^{-1}$
for every $\nu: W(X)\to W(Y)$.
 
3. Let $\Theta$ be the variety of all groups. By Formanek's theorem
\cite{For}, every automorphism of the semigroup $\End W(X)$ ,
$|X|>1$, is an inner automorphism. Using this result it can be proven
that the function $\sigma$ is presented in the form $\sigma=
s\tau$, where $s_X$ is an automorphism of the group $W(X)$, 
and $\tau$ is either the identity function or $\tau_X(a)=a^{-1}$
for every finite $X$
and every $a\in W(X)$. 
Since $\tau$ is a central function, it disappears and therefore
$\hat\sigma= \hat s$. For every
$\nu: W(X)\to W(Y)$, we have $\varphi(\nu)=s_Y\nu s_X^{-1}$.
Hence, $\varphi$ is an inner automorphism. 


4. The case of semigroups. Let $F=F(X)$ be a free semigroup and
$u=x_{i_1}x_{i_2}\cdots x_{i_{n-1}}x_{i_n}$ an element of $F$. Denote by
$\bar u$ the element $\bar u=x_{i_n}x_{i_{n-1}}\cdots x_{i_2}x_{i_1}$.
The map $u\to \bar u$ is a bijective involution on the set
$F(X)$.

Now we can define an automorphism $\mu$ of the category $\Theta^0$
of free semigroups. This automorphism does not change objects, and
for every $\nu: F(X)\to F(Y)$ we set $\mu(\nu)(x)=\overline{\nu(x)}$ for
every $x\in X$. Automorphism $\mu$ is called {\it a mirror} automorphism
of the category $\Theta^0$. It is clear that $\mu^2=id$. The mirror
automorphism of the semigroups $\End F(X)$ is defined similarly.

Using \cite{Ma} it can be proved that 
any automorphism $\varphi$ of the category $\Theta^0$ can be presented as 
the product of inner and mirror automorphisms. Obviously, 
$\Out\Theta^0$ is isomorphic to $Z_2$.



5. Let $\Theta$ be an arbitrary hopfian variety of algebras, and let
it be generated by a cyclic free algebra $W=W_0=W(x_0)$.
Consider an automorphism $\varphi$ of the category 
$\Theta^0$ which does not change objects. Denote by $\varphi_{W_0}$
the automorphism of the semigroup  $\End W(x_0)$ induced by
the  automorphism $\varphi$. The following theorem holds:

\begin{theorem}
If the automorphism $\varphi_{W_0}$ is trivial,
 then $\varphi$ is an inner automorphism of the
category $\Theta^0$.
\end{theorem}


6. Let us use the theorem above in the case of modules. 
Let $\Theta$ be the variety of modules over a noetherian
ring $R$ and $\varphi$ an automorphism which does not change
objects. Take the cyclic module $Rx_0$. It generates the whole variety
 $\Theta$. It can be proven that $\varphi$ induces an automorphism
of the ring $R$. The corresponding $\varphi_{Rx_0}$ is a semiinner
automorphism of $\End Rx_0$, which can be extended to a semiinner automorphism
$\psi$ of the category $\Theta^0$. 
The automorphism $\psi^{-1}\varphi$ acts trivially in the semigroup $\End(Rx_0)$.
Therefore  $\psi^{-1}\varphi$ is inner. Hence,  $\varphi$ is semiinner.

7. The case of associative commutative algebras follows the scheme of
item 6. The same scheme works for the situation of $F$-groups.

8. About Problem 3.9. Consider a generalization of Theorem 3.10. 

Let $\Theta$ be an arbitrary hopfian variety of algebras, and let
$\Theta$ be generated by an algebra $W^0=W(X^0)$, where $X^0$ is a fixed
finite set. Denote by $W_0=W(x_0)$ the cyclic free algebra.
Let $\nu_0: W^0\to W_0$ be a morphism defined by the condition:
$\nu_0(x)=x_0$ for every $x\in X^0$. 

\begin{theorem}
If the automorphism $\varphi: \Theta^0\to\Theta^0$ acts trivially 
on the semigroups $\End W^0$ and $\End W_0$ and $\varphi(\nu_0)=
\nu_0$, then $\varphi$ is an inner automorphism of the
category $\Theta^0$.
\end{theorem}

9. Let $\Theta$ be the variety of associative or Lie algebras over
a field, $F_0$ the free algebra with one variable, $F^0$ the free
algebra with two variables. Consider a full 
subcategory of $\Theta^0$
which has only two objects $F_0$ and $F^0$ and with morphisms induced by the
morphisms of $\Theta^0$. 

The theorem above allows us to reduce the problem on automorphisms of
the category $\Theta^0$ to studying the automorphisms of this subcategory.

We note that $\Theta$ is generated by the free 
algebra with two variables $F^0$. 

The notion of a mirror automorphism works 
in the variety of all associative algebras  $\Theta$ as well.


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\end{document}