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\begin{document}
     
\title[On approximation of locally compact groups]{On approximation of 
locally compact groups by finite algebraic systems}

\author{L. Yu. Glebsky}
\address{IICO-UASLP,
Av. Karakorum 1470,
Lomas 4ta Session,
SanLuis Potosi SLP 78210, Mexico}
\email{glebsky@cactus.iico.uaslp.mx}
\thanks{The first author was supported in part by CONACyT-NSF 
Grant \#E120.0546 y PROMEP, PTC-62; the second author was supported in part 
by NSF Grant DMS-9970009}

\author{E. I. Gordon}
\address{Department of Mathematics and Computer Science,
Eastern Illinois University,
600 Lincoln Avenue,
Charleston, IL 61920-3099}
\email{cfyig@eiu.edu}

\subjclass[2000]{Primary 26E35, 03H05; Secondary 28E05, 42A38}

\date{June 16, 2003}

\commby{Efim Zelmanov}


\keywords{Approximation, group, quasigroup}

\begin{abstract}
We discuss the approximability of locally compact groups by finite 
semigroups and finite quasigroups 
(latin squares). We show that if a locally compact group $G$ is approximable 
by
finite semigroups, then it is approximable by finite groups, and thus many 
important groups are not
approximable by finite semigroups. This result implies, in particular, the 
impossibility to simulate 
the field of reals in computers by finite associative rings. We show that a 
locally compact
group is approximable by finite quasigroups iff it is unimodular.  
\end{abstract}

\maketitle

\section{Introduction}

In this paper we discuss the approximability of locally compact groups by 
finite algebraic systems with 
given properties, with respect to the following definition of approximability.


Let $G$ be a locally compact group.  We will denote by $\cdot$ the 
multiplication in $G$ and use the usual notation
\begin{gather*}
XY=\{x\cdot y\;|\;x\in X,\,y\in Y\},\\
X^{-1}=\{x^{-1}\;|\;x\in X\},\\
gX=\{g\cdot x\;|\;x\in X\}
\end{gather*}
for $X,Y\subset G,\ g\in G$.

\begin{definition} \label{gr_approx}
Let $C\subset G$ be a compact set, $U$ a relatively compact
neighborhood of
the
unity in $G$, and $(H,\odot)$ a finite universal algebra with one binary 
operation.

\begin{enumerate}
\item We say that a set $M\subset G$ is
an $U$-grid of $C$
iff $C\subset MU$.
\item A map $j:H\to G$ is called a $(C,U)$-homomorphism if
\[
\forall x,y\in H\;\left( (j(x),j(y),j(x)\cdot j(y)\in C)\Rightarrow
(j(x\odot y)\in j(x)j(y)U) \right)
\]
\item We say that the pair $\la H,j\ra$ is a $(C,U)$-approximation of $G$ if
$j(H)$ is
an $U$-grid of $C$
and
$j:H\to G$ is a $(C,U)$-homomorphism.
\item Let $\K$ be a class of finite algebras. We say that $G$ is 
approximable by the systems of the class $\K$ if for any
compact $C\subset G$ and for any neighborhood $U$ of the
unity  there exists a ($C,U$)-approximation $\la H,j\ra$ of $G$ such that 
$H\in\K$ and $j$ is an injection.
\end{enumerate}
\end{definition}

\begin{remark}
Since in item (2) the  elements $j(x\odot y)$ and $j(x)\cdot j(y)$ are $U$-close in 
the
{\it left} uniformity on $G$, it may seem that the definition of 
approximability of $G$ by systems of
$\K$ depends on which of two uniformities we consider. However, this is not 
so. Indeed it is clear from the definition that we deal only with the 
restrictions of the uniformities on compacts. But it
is well known that the restrictions of the left uniformity and of the right 
uniformity on any compact are equivalent.
\end{remark}

\begin{remark}
It is easy to see that a similar definition can be formulated for
any topological universal algebra, and it is not necessary to
assume that
approximated
algebras are finite. For example, the approximations of discrete groups by 
amenable groups have been introduced in \cite{AGG}. 
\end{remark}

This definition of approximability by finite groups was introduced by the 
second author in
\cite{Gor}, where the approximability of locally compact abelian groups by 
finite abelian groups
was investigated. The cases of discrete groups and locally compact nilpotent 
groups were 
considered in \cite{VG}. The possibility to approximate a locally compact 
group $G$ by finite groups 
implies some important corollaries. In \cite{AGKh} some new finite
dimensional approximations of pseudodifferential operators in
$L_2(G)$ for an abelian group $G$ were constructed using
approximations of $G$ by finite abelian groups. Approximations of
discrete groups have some interesting applications in the ergodic
theory of group actions \cite{VG}, \cite{AGG} and in symbolic
dynamics \cite{Gr}. In fact, approximations of discrete algebraic system 
(finite embeddability) were considered
by T.~Evans \cite{Ev1}. 

On the other hand, many results about the nonapproximabilty of locally 
compact groups by finite groups were obtained. 
For example, a discrete finitely presented group $G$ is approximable by 
finite groups in the sense of 
Definition \ref{gr_approx} iff $G$ is residually finite. So, a group with 
undecidable word problem is not approximable
by finite ones; see \cite{Ev2}. 

It was proved in \cite{Resv} that all approximable
locally compact groups are unimodular (the left and right Haar
measures coincide). This condition is not sufficient---we have
already mentioned that there exist nonapproximable discrete
groups. It was proved in \cite{AGG} that the simple Lie groups are
not approximable by finite groups as topological groups.

The nonapproximability of some important groups, such as the group $SO(3)$,
by finite groups make it of interest to investigate more general classes of 
finite universal algebras with one binary operation that approximate some 
locally compact groups nonapproximable by
finite groups.

The most important and well investigated extensions of the class of finite 
groups
are the classes of finite {\it semigroups} (see, for example, \cite{RT}) and 
the class of
finite {\it quasigroups}, which are the same as {\it latin squares} (see, 
for example, \cite{quas} and
\cite{Ryser}). 

\begin{definition} \label{semiquasi}
\begin{enumerate}
\item We say that an algebra $(A,\circ)$ is a right quasigroup 
(left quasigroup)
iff for every $a,b\in A$ the
equation $a\circ x=b$ ($x\circ a=b$) has the unique solution
$x=/(b,a)$ ($x=\backslash (b,a)$).
\item An algebra $(A,\circ)$ is a quasigroup iff it is a right quasigroup 
and left quasigroup.
A quasigroup $A$ with a unity (an element $e\in A$ such that that 
$\forall\,a\in A,\ a\circ e=e\circ a=a$) is called a loop.
\item We say that an algebra $(A,\circ)$ is a semigroup if the operation 
$\circ$ satisfies the 
law of associativity.
\end{enumerate}
\end{definition}

\section{Approximation of locally compact groups by finite semigroups}

The following theorem holds.

\begin{theorem} \label{semigroups}
A locally compact group is approximable by finite semigroups iff it is 
approximable by finite groups.
\end{theorem}


The proof of Theorem \ref{semigroups} is based on some results about the
structure of finite semigroups from \cite{RT}. We use also the language of 
nonstandard analysis 
(cf. for example \cite{Loeb}) that allows us to simplify the proofs 
essentially.

Theorem \ref{semigroups} has an interesting corollary about approximability 
of the field $\R$ by finite rings.

\begin{theorem} \label{reals}
The field $\R$ is not approximable by finite associative rings.
\end{theorem}

\begin{proof}
Let us sketch the proof of this theorem.

Consider the matrix group
\[
G=\{\left(\begin{array}{ll}a &b \\ 0 &1\end{array}\right) \ |\ a\neq 
0,b\in\R\}.
\]
It is well known that this group is nonunimodular and thus by the 
above-\-mentioned result
from \cite{Resv} is not approximable by finite groups. By Theorem 
\ref{semigroups} $G$ is not approximable by finite semigroups.
On the other hand, if we could approximate $\R$ by finite associative rings,
then $G$ would be approximable by semigroups 
of matrices of the same type with the elements of finite rings that 
approximate $\R$, a contradiction.
\end{proof}

Theorem \ref{reals} has the following interpretation.

Consider some examples of approximation of the field $\R$ in the 
signature $\s=\break\la+,\cdot\ra$.
Since any compact $C\subset\R$ is contained in the interval $[-a,a]$ for an 
appropriate $a$ and 
the sets $U_{\e}=\{x\in\R\ |\ |x|<\e\},\ \e>0$ form a base of the 
neighborhoods of zero in $\R$, it is enough to consider only the 
$\left([-a,a], U_{\e}\right)$-approximations of $\R$. We will call these 
approximations the $(a,\e)$-approximations. 



\begin{exa}
Recall that the normal (computer) form of a real $\alpha$ is its representation:
\begin{equation}
\alpha=\pm 10^p\cdot 0.a_1a_2\dots, 
\end{equation}
where $p\in\Z$, and $a_1a_2\dots$ is a finite or infinite sequence of 
decimal digits $0\leq a_n\leq 9$, and $a_1\neq 0$. 
The integer $p$ is called the exponent of $\alpha$, and 
$a_1a_2\dots$, its mantissa. 

Fix two natural numbers $P>Q$ and consider the finite set $A_{PQ}$ of reals 
in the form (1) such that
the exponent $p$ of $\alpha$ satisfies the inequality $|p|\leq P$ and its 
mantissa contains no more than 
$Q$ decimal digits. Define two binary operations $\oplus$ and $\odot$ on 
$A_{PQ}$. Let $\alpha,\beta\in
A_{PQ}$ and the normal form of $\alpha\times\beta$, where $\times$ is either $+$ or 
$\cdot$, is
\begin{equation}
\alpha\times\beta=\pm 10^r\cdot 0.c_1c_2\dots\,.
\end{equation} 
Notice that the mantissa of $\alpha\times\beta$ may contain more than $Q$ digits. Now
\[
\alpha\otimes\beta=\left\{\begin{array}{ll} \pm 10^r\cdot 0.c_1c_2\dots c_Q\ 
&\mbox{if}\ |r|\leq P, \\
\pm 10^{P}\cdot0.\underbrace{99\dots9}_{Q\ \mbox{\scriptsize digits}}\ &\mbox{if}\ r>P, \\
0\ &\mbox{if}\ r<-P.\end{array}\right.
\]
In the case that the mantissa of $\alpha\times\beta$ contains fewer than $Q$ digits 
we complete it to a 
$Q$-digit mantissa by zeros.

We will denote by $\A_{PQ}$ the universal algebra $\la A_{PQ},\s\ra$ such that the 
interpretations of the functional symbols $+$ and $\cdot$ are the functions 
$\oplus$ and $\odot$, respectively. 

It is easy to see that for any positive $a$ and $\e$ there exist natural 
numbers $P$ and $Q$ such that the universal algebra $\A_{PQ}$ is an 
$(a,\e)$-approximation of $\R$.

The described systems $\A_{PQ}$ are implemented in working computers.
What properties of addition and multiplication of reals hold for $\oplus$ 
and $\odot$?

It is easy to see that the operations $\oplus$ and $\odot$ are commutative, 
$\xi\oplus\left(-\xi\right)=0$ and $\xi+0=\xi$ for any $\xi\in A_{PQ}$.


Let $\alpha=\beta=0.60\dots 06$ and 
$\gamma=0.60\dots 5$ (with $Q$ digits after the decimal point). 
Then $\alpha\oplus\beta=\alpha\oplus\gamma$, so the cancellation law fails for $\oplus$, and 
thus the law of associativity fails for $\oplus$. 

It is easy to construct examples that show that the laws of associativity 
for $\odot$ and distributivity in $\A_{PQ}$ fail also.
\end{exa}


\begin{exa} Fix a natural number $M$ and a positive $\e$. Put 
$A'_{M\e}=\{k\e\ |\ k=-M\dots M\}$.
Let $N=2M+1$. For any $n\in\Z$ we will denote by $n(\mod N)$ the element of 
the set $\{-M,\dots,M\}$,
congruent to $n$ modulo $N$. The operations $\oplus$ and $\odot$ on 
$A'_{M,\e}$ are defined as follows:
\begin{align}
k\e\oplus m\e&=(k+m)(\mod N)\e, \\ 
k\e\odot m\e&=[km\e](\mod N)\e. 
\end{align}

We will denote by $\A'_{M,\e}$ the universal algebra in the signature $\s$ 
with the underlying set $A'_{M,\e}$ and 
the interpretation of the functional symbols defined by formulas (3) and (4).

It is easy to see that $\A'_{M\e}$ is an $(M\e,\e)$-approximation of $\R$. 


It is obvious that $\A'_{M,\e}$ is an abelian group with respect to $\oplus$ 
(see (3)). However, one can easily construct 
examples which show that for any big enough $M$ and small enough $\e$ the 
multiplication $\odot$ satisfies neither 
the law of associativity, nor the law of distributivity. 

This example shows that it is possible to implement in computers a numerical 
system that simulate reals, which is an abelian 
group with respect to addition, while by Theorem \ref{reals} it is 
impossible to implement such system that would be 
an associative ring (even noncommutative).

It is an interesting question whether it is possible to approximate $\R$ by 
any finite nonassociative rings.
\end{exa}

\section{Approximation of locally compact groups by finite quasigroups}

In this section we discuss the following theorem.

\begin{theorem} \label{Main_th}
A locally compact group $G$ is unimodular iff it is approximable by finite
quasigroups.
\end{theorem}

As far as we know, no characterization of unimodularity 
in algebraic and topological terms has been known up to now.


The sufficiency is more or less easy. Suppose that $G$ is approximable by 
left (right) quasigroups.

We will construct a left (right) invariant mean on $G$ using left (right) 
quasigroups
 that approximate $G$.

Let $\myH$ be the family of all pairs $\la C,U\ra$ such that
$C\subseteq G$ is a compact set and $U$ is a relatively compact
neighborhood of the unity on $G$. Let $\leq$ be the partial order
on $\myH$ such that
\[
\la C_1,U_1\ra\leq\la C_2,U_2\ra\Longleftrightarrow C_1\supseteq
C_2\land U_1\subseteq U_2.
\]

Given a pair $\la C,U\ra\in\myH$ let $\myH(C,U)=\{\la C',U'\ra\ |\
\la(C',U'\ra\leq\la C,U\ra\}$. It is easy to see that the family
$\M=\{H(C,U)\ |\ \la C,U\ra\in\myH\}$ of subsets of $\myH$ has the
finite intersection property. Thus there exists an ultrafilter
$\F$ on $\myH$ such that $\F\supseteq\M$. Fix an arbitrary such
ultrafilter $\F$.

Recall that if $\alpha:\myH\to X$ is an arbitrary map, $X$ is a Hausdorf
space and $a\in X$, then $\lim_{\F}\alpha(C,U)=a$ if 
$\{\la C,U\ra\ | \alpha(C,U)\in Y\}\in\F$ for any
neighborhood $Y\ni a$.
It is known that if $\alpha(H)$ is relatively compact, then the
$\lim_{\F}\alpha(C,U)$ exists.

For each $\la C,U\ra$ fix a finite algebra $H_{C,U}$ that is a
$(C,U)$-approximation of $G$. Without loss of generality we may
assume that $H_{C,U}\subset G$ as a set. Fix also a compact set
$V\subseteq G$ with  nonempty interior.

As usual, let $C_0(G)$ be the space of all continuous functions
with compact support on $G$. For an arbitrary $f\in C_0(G)$ put
\begin{equation}
\Lambda(f)=\lim_{\F}|H_{C,U}\cap V|^{-1}\sum_{h\in
H_{C,U}}f(h)    
\end{equation}
if this limit exists.

\begin{theorem} \label{mean}
If for any $\la C,U\ra\in\myH$ the algebra $H_{C,U}$ is a left (right) 
quasigroup, then the limit on 
the right hand side of formula (5) exists for all $f\in C_0(G)$. In this 
case the
functional $\Lambda: C_0(G)\to\R$ is a positive nonzero left (right) 
invariant functional on $C_0(G)$
\end{theorem}

Obviously this theorem implies the sufficiency of condition of 
Theorem~\ref{Main_th}. 

The following theorem is also true.

\begin{theorem} \label{lquasiappr}
Any locally compact group $G$ is approximable by finite $l$-quasigroups 
($r$-quasigroups).
\end{theorem}

Theorems \ref{mean} and \ref{lquasiappr} together give a proof of the existence 
of Haar measure.
This proof is close by ideas to the proof of the existence of Haar measure 
due to von Neumann,
which is based on equidistributed sets \cite{Neum}. For example, in the 
proof of the existence
of equidistributed sets as well as in the proof of Theorem \ref{lquasiappr},
the Marriage Lemma is 
used. 

The proof of necessity in Theorem \ref{Main_th} involves more 
complicated combinatorics.

For the case of discrete groups it follows immediately from the fact that 
any  
$n\times n$ latin subsquare
with $k$ distinct elements
can be completed to
an
$r\times r$ latin square, where $r=\max\{2n,k\}$.

Recall that $n\times n$-table is a latin subsquare iff all elements in each 
row and in each 
column are distinct. An $n\times n$-latin subsquare with $n$ different 
elements is said to be  
a latin square. It is easy to see that 
the operation table of a finite quasigroup is a latin square. 

Theorem \ref{Main_th} is now a corollary of the following 

\begin{Prop} \label{main_prop}
Any nondiscrete locally compact unimodular group $G$ is approximable by
finite quasigroups.
\end{Prop}

The proof of this proposition that will be discussed in the paper
is rather complicated. We outline the main ideas of this proof here only for 
the case of a compact group $G$. The case of locally compact groups requires 
some technical modifications. Nontrivial combinatorial arguments about
latin squares are involved in the proof.
These arguments are
based on a generalization of a result by
Hilton \cite{Hil} and discussed in \cite{GC}.

We assume in this section that $G$ is a nondiscrete compact group.
All subsets of $G$ we deal with
are assumed to be measurable with respect to the Haar measure $\nu$
that is assumed to be normalized, $\nu(G)=1$.
Recall that any compact group is unimodular and thus $\nu$ is left and
right invariant. Let $U$ be a neighborhood of the
unit in $G$, and $\myP$ a finite partition of $G$. We say that $\myP$ is
$U$-fine if
$\forall P\in\myP\ \exists g\in G\ (P\subseteq gU)$, and $\myP$ is equisize
if all sets in $\myP$ have the same Haar measure.



\begin{theorem} \label{Partition}
There exists a $U$-fine equisize partition for any neighborhood of
the unit $U\subseteq G$.
\end{theorem}

Let $\myP=\{P_1,\dots,P_n\}$ be a partition that satisfies the
assumptions of this theorem for some $U$. Consider the 
three-index matrix $w=\la w_{ijk}\ |\ 1\leq i,j,k\leq n\ra$, where
\[
w_{ijk}=\int\!\!\!\int_{G\times
G}\chi_i(xy^{-1})\chi_j(y)\chi_k(x)d\nu(x)d\nu(y);
\]
$\chi_m(x)=\chi_{P_m}(x)$ is the characteristic function of a set
$P_m$, $m\leq n$.

Obviously $w_{ijk}\geq 0$. Let $S=\supp\ w=\{\la i,j,k\ra\ |\ w_{ijk}>0\}$.

\begin{lemma} \label{Prop_w}
The three-index matrix $w_{ijk}$ has the following properties:
\begin{enumerate}
\item $\sum_iw_{ijk}=\sum_jw_{ijk}=
\sum_kw_{ijk}=\frac 1{n^2}$;
\item $S\subseteq\{\la i,j,k\ra\mid \nu(P_i\cdot P_j\cap P_k)>0\}$.
\end{enumerate}
\end{lemma}

To motivate the following consideration we use an analogy with two-index 
matrices. Recall that an $n\times n$ matrix $B=\|p_{ij}\|$ is 
bistochastic if $p_{ij}\geq 0$ and
$\sum_{i=1}^np_{ij}=\sum_{j=1}^np_{ij}=1$. According to a 
well-known G.~Birkhoff's theorem
(cf., for example, \cite{Ryser}) in this case $B$ is a convex hull of 
permutations---the matrices
that consist of zeros and ones and contain a unique one in each row and 
in each column.
Then there exists a permutation $T$ such that $\supp\ T\subseteq\supp\ B$. 
Assume for a moment that the similar fact
holds for the three-indexed matrices. We say that a three-index matrix is 
three-stochastic if it is nonnegative and the
sum of elements in each line is equal to one. We call a line any set $L$ of 
triples of elements of $\{1,\dots,n\}$
such that in all triples in $L$ two indexes are fixed and the third runs over 
$\{1,\dots,n\}$. Notice that if $w_{ijk}$
satisfies Lemma \ref{Prop_w}, then $n^2w_{ijk}$ is three-stochastic. So we 
assume that the following statement is true.

(A) If $w_{ijk}$ satisfies Lemma \ref{Prop_w}, then there exists a matrix 
$\delta_{ijk}$ that consists of zeros and ones,
contains a unique one in each line and such that $\supp\ 
\delta_{ijk}\subseteq\supp\
w_{ijk}$.

By the properties of $\delta_{ijk}$ it is easy to see that $\supp\ \delta_{ijk}$ is 
the graph of the
operation $\circ$ on $\{1,\dots,n\}$ such that $i\circ j=k$ iff 
$\gamma_{ijk}=1$. Denote by $Q$ the
algebra $\{1,\dots,n\}$ with the operation $\circ$. Since for any $i$ 
and $k$ there exists a
unique $j$ such that $\gamma_{ijk}=1$, and for any $j$ and $k$ there exists a unique 
$i$ such that $\gamma_{ijk}=1$,
we have that the left and right cancellation laws hold in $Q$, and thus $Q$ 
is a quasigroup.

Fix an arbitrary injection $\alpha:Q\to G$ such that for any $i\leq n$, 
$\alpha(i)\in P_i$.
Notice that if $i\circ j=k$ then $\la i,j,k\ra\in\supp\ 
\gamma_{ijk}\subseteq\supp\ w_{ijk}$ and thus
$P_i\cdot P_j\cap P_k\neq\emptyset$ by Lemma \ref{Prop_w} (2).

Thus, we have proved, under assumption (A), the following

\begin{lemma} \label{injection}
For any neighborhood of the unit $U$ of a compact group $G$ and for any 
$U$-fine equisize partition
$\myP$ of $G$ there exist a finite quasigroup $Q$ and an injection $\alpha:Q\to 
G$ such that
\begin{enumerate}
\item $\forall P\in\myP\exists q\in Q\ (\alpha(q)\in P);$
\item $\forall q_1,q_2\in Q\ (\alpha(q_1)\in P_1\in\myP\land
      \alpha(q_2)\in P_2\in\myP\land \alpha(q_1\circ q_2)\in P_3\in\myP
\Longrightarrow P_1\cdot P_2\cap P_3\neq\emptyset)$.
\end{enumerate}
\end{lemma}

It is easy to see that Proposition \ref{main_prop} follows from Lemma 
\ref{injection}.

Unfortunately the statement (A) is not true in general (see, for example,
\cite{GC}), and a proof of Lemma \ref{injection} follows from 
a weaker analogue of statement (A) due to Hilton \cite{Hil} 
(Theorem~\ref{weak-birkhoff}).

Let $Q$ be a quasigroup and $\s$ an equivalence relation on $Q$
which we will identify with the  partition of $Q$ by the equivalence
classes. Then $\s=\{Q_1,\dots, Q_n\}$. Denote by $Q/\s$  the subset
of $\{1,\dots,n\}^3$ such that $\la ijk\ra\in Q/\s$ iff there
exist $q\in Q_i$ and $q'\in Q_j$ with $q\circ q'\in Q_k$. Notice
that if $\s$ is a congruence relation on $Q$ (i.e. it preserves
the operation $\circ$), then the introduced set is exactly the
graph of the operation in the quotient quasigroup $Q$ by $\s$, and
so we will call the set $Q/\s$ a generalized quotient quasigroup
(gqq).

\begin{theorem} \label{weak-birkhoff} Let a nonnegative three-index matrix
$u=\la u_{ijk}\ |\ 1\leq i,j,k\leq n\ra$ satisfy the following
condition
\[
\sum_iu_{ijk}=\sum_ju_{ijk}=\sum_ku_{ijk}=l
\]
for some positive $l$. Then there exist a finite quasigroup $Q$
and its partition $\s=\{Q_1,\dots, Q_n\}$ such that the gqq
$Q/\s\subseteq\supp\ u$.
\end{theorem}

The case of an arbitrary locally compact group $G$ requires
some generalization of Hilton's result, which is contained in \cite{GC}.

\begin{thebibliography}{100}
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Finite dimensional approximations of operators in the spaces of functions on
locally compact abelian groups, {\it Acta Applicandae Mathematicae} {\bf 
64}(1),  33--73, October 2000.
\MR{2002f:47030}

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