Algebraic system, automorphism of an

An isomorphic mapping of an algebraic system onto itself. An automorphism of an $\mathrm{\Omega }$- system $\mathbf{A}=⟨A,\mathrm{\Omega }⟩$ is a one-to-one mapping $\varphi$ of the set $A$ onto itself having the following properties:

$$\begin{array}{}\text{(1)}& \varphi (F(x_1\dots x_n))=F(\varphi (x_1)\dots \varphi (x_n)),\end{array}$$

$$\begin{array}{}\text{(2)}& P(x_1\dots x_m)⟺P(\varphi (x_1)\dots \varphi (x_m)),\end{array}$$

for all $x_1,x_2\dots$ from $A$ and for all $F,P$ from $\mathrm{\Omega }$. In other words, an automorphism of an $\mathrm{\Omega }$- system $\mathbf{A}$ is an isomorphic mapping of the system $\mathbf{A}$ onto itself. Let $G$ be the set of all automorphisms of the system $\mathbf{A}$. If $\varphi \in G$, the inverse mapping ${\varphi }^{-1}$ also has the properties (1) and (2), and for this reason ${\varphi }^{-1}\in G$. The product $\alpha =\varphi \psi$ of two automorphisms $\varphi ,\psi$ of the system $\mathbf{A}$, defined by the formula $\alpha (x)=\psi (\varphi (x))$, $x\in A$, is again an automorphism of the system $\mathbf{A}$. Since multiplication of mappings is associative, $⟨G,\cdot ,{}^{-1}⟩$ is a group, known as the group of all automorphisms of the system $\mathbf{A}$; it is denoted by $\mathrm{Aut}(\mathbf{A})$. The subgroups of the group $\mathrm{Aut}(\mathbf{A})$ are simply called automorphism groups of the system $\mathbf{A}$.

Let $\varphi$ be an automorphism of the system $\mathbf{A}$ and let $\theta$ be a congruence of this system. Putting

$$(x,y)\in {\theta }_{\varphi }⟺({\varphi }^{-1}(x),{\varphi }^{-1}(y))\in \theta ,x,y\in \mathbf{A},$$

one again obtains a congruence ${\theta }_{\varphi }$ of the system $\mathbf{A}$. The automorphism $\varphi$ is known as an IC-automorphism if ${\theta }_{\varphi }=\theta$ for any congruence $\theta$ of the system $\mathbf{A}$. The set $\mathrm{IC}(\mathbf{A})$ of all IC-automorphisms of the system $\mathbf{A}$ is a normal subgroup of the group $\mathrm{Aut}(\mathbf{A})$, and the quotient group $\mathrm{Aut}(\mathbf{A})/\mathrm{IC}(\mathbf{A})$ is isomorphic to an automorphism group of the lattice of all congruences of the system $\mathbf{A}$[1]. In particular, any inner automorphism $x\to a^{-1}xa$ of a group defined by a fixed element $a$ of this group is an IC-automorphism. However, the example of a cyclic group of prime order shows that not all IC-automorphisms of a group are inner.

Let $\mathfrak{K}$ be a non-trivial variety of $\mathrm{\Omega }$- systems or any other class of $\mathrm{\Omega }$- systems comprising free systems of any (non-zero) rank. An automorphism $\varphi$ of a system $\mathbf{A}$ of the class $\mathfrak{K}$ is called an I-automorphism if there exists a term $f_{\varphi }(x_1\dots x_n)$ of the signature $\mathrm{\Omega }$, in the unknowns $x_1\dots x_n$, for which: 1) in the system $\mathbf{A}$ there exist elements $a_2\dots a_n$ such that for each element $x\in A$ the equality

$$\varphi (x)=f_{\varphi }(x,a_2\dots a_n)$$

is valid; and 2) for any system $\mathbf{B}$ of the class $\mathfrak{K}$ the mapping

$$x\to f_{\varphi }(x,x_2\dots x_n)(x\in B)$$

is an automorphism of this system for any arbitrary selection of elements $x_2\dots x_n$ in the system $\mathbf{B}$. The set $\text{ I }(\mathbf{A})$ of all I-automorphisms for each system $\mathbf{A}$ of the class $\mathfrak{K}$ is a normal subgroup of the group $\mathrm{Aut}(\mathbf{A})$. In the class $\mathfrak{K}$ of all groups the concept of an I-automorphism coincides with the concept of an inner automorphism of the group [2]. For the more general concept of a formula automorphism of $\mathrm{\Omega }$- systems, see [3].

Let $\mathbf{A}$ be an algebraic system. By replacing each basic operation $F$ in $\mathbf{A}$ by the predicate

$$R(x_1\dots x_n,y)⟺F(x_1\dots x_n)=y$$

$$(x_1\dots x_n,y\in A),$$

one obtains the so-called model ${\mathbf{A}}^{\ast }$ which represents the system $\mathbf{A}$. The equality $\mathrm{Aut}({\mathbf{A}}^{\ast })=\mathrm{Aut}(\mathbf{A})$ is valid. If the systems $\mathbf{A}=⟨A,\mathrm{\Omega }⟩$ and ${\mathbf{A}}^{\prime }=⟨A,{\mathrm{\Omega }}^{\prime }⟩$ have a common carrier $A$, and if $\mathrm{\Omega }\subset {\mathrm{\Omega }}^{\prime }$, then $\mathrm{Aut}(\mathbf{A})\supseteq \mathrm{Aut}({\mathbf{A}}^{\prime })$. If the $\mathrm{\Omega }$- system $\mathbf{A}$ with a finite number of generators is finitely approximable, the group $\mathrm{Aut}(\mathbf{A})$ is also finitely approximable (cf. [1]). Let $\mathfrak{K}$ be a class of $\mathrm{\Omega }$- systems and let $\mathrm{Aut}(\mathfrak{K})$ be the class of all isomorphic copies of the groups $\mathrm{Aut}(\mathbf{A})$, $\mathbf{A}\in \mathfrak{K}$, and let $\mathrm{SAut}(\mathfrak{K})$ be the class of subgroups of groups from the class $\mathrm{Aut}(\mathfrak{K})$. The class $\mathrm{SAut}(\mathfrak{K})$ consists of groups which are isomorphically imbeddable into the groups $\mathrm{Aut}(\mathbf{A})$, $\mathbf{A}\in \mathfrak{K}$.

The following two problems arose in the study of automorphism groups of algebraic systems.

1) Given a class $\mathfrak{K}$ of $\mathrm{\Omega }$- systems, what can one say about the classes $\mathrm{Aut}(\mathfrak{K})$ and $\mathrm{SAut}(\mathfrak{K})$?

2) Let an (abstract) class $K$ of groups be given. Does there exist a class $\mathfrak{K}$ of $\mathrm{\Omega }$- systems with a given signature $\mathrm{\Omega }$ such that $K=\mathrm{Aut}(\mathfrak{K})$ or even $K=\mathrm{SAut}(\mathfrak{K})$? It has been proved that for any axiomatizable class $\mathfrak{K}$ of models the class of groups $\mathrm{SAut}(\mathfrak{K})$ is universally axiomatizable [1]. It has also been proved [1], [4] that if $\mathfrak{K}$ is an axiomatizable class of models comprising infinite models, if $⟨B,\le ⟩$ is a totally ordered set and if $\mathbf{G}$ is an automorphism group of the model $⟨B,\le ⟩$, then there exists a model $\mathbf{A}\in \mathfrak{K}$ such that $A\supseteq B$, and for each element $g\in G$ there exists an automorphism $\varphi$ of the system $\mathbf{A}$ such that $g(x)=\varphi (x)$ for all $x\in B$. The group $G$ is called 1) universal if $G\in \mathrm{SAut}(\mathfrak{K})$ for any axiomatizable class $\mathfrak{K}$ of models comprising infinite models; and 2) a group of ordered automorphisms of an ordered group $\mathbf{H}$( cf. Totally ordered group) if $\mathbf{G}$ is isomorphic to some automorphism group of the group $\mathbf{H}$ which preserves the given total order $\le$ of this group (i.e. $a\le b\Rightarrow \varphi (a)\le \varphi (b)$ for all $a,b\in H$, $\varphi \in G$).

Let $l$ be the class of totally ordered sets $⟨M,\le ⟩$, let $\mathfrak{U}$ be the class of universal groups, let RO be the class of right-ordered groups and let OA be the class of ordered automorphism groups of free Abelian groups. Then [4], [5], [6]:

$$\mathrm{SAut}(l)=\mathfrak{U}=\mathrm{RO}=\mathrm{OA}.$$

Each group is isomorphic to the group of all automorphisms of some $\mathrm{\Omega }$- algebra. If $\mathfrak{K}$ is the class of all rings, $\mathrm{Aut}(\mathfrak{K})$ is the class of all groups [1]. However, if $\mathfrak{K}$ is the class of all groups, $\mathrm{Aut}(\mathfrak{K})\ne \mathfrak{K}$; for example, the cyclic groups ${\mathbf{C}}_3,{\mathbf{C}}_5,{\mathbf{C}}_7$ of the respective orders 3, 5 and 7 do not belong to the class $\mathrm{Aut}(\mathfrak{K})$. There is also no topological group whose group of all topological automorphisms is isomorphic to ${\mathbf{C}}_5$[7].

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