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\begin{document}

\title{Growth Rates of Finite Algebras}
\author{Keith A. Kearnes}
\address[Keith Kearnes]{Department of Mathematics\\
University of Colorado\\
Boulder, CO 80309-0395\\
USA}
\email{Keith.Kearnes@Colorado.EDU}
\author{Emil W. Kiss}
\address[Emil W. Kiss]{E{\"o}tv{\"o}s University\\
Department of Algebra and Number Theory\\
1088 Budapest, M{\'u}zeum krt. 6--8\\
Hungary}
\email{ewkiss@cs.elte.hu}
\author{\'Agnes Szendrei}
\address[\'Agnes Szendrei]{Department of Mathematics\\
University of Colorado\\
Boulder, CO 80309-0395\\
USA}
\email{Agnes.Szendrei@Colorado.EDU}
\subjclass{}
\keywords{}

\begin{abstract}
We investigate the function $d_{\m a}(n)$,
which gives the size of a smallest generating
set for $\m a^n$, in the case where $\m a$ is a finite algebra.
\end{abstract}

\maketitle

\section{Introduction}\label{intro_sec}
For a finite algebra $\m a$, write $d_{\m a}(n)=g$
if $g$ is the least size of a generating set
for $\m a^n$, and write $h_{\m a}(g)=n$ if the
largest power of $\m a$ that is $g$-generated is $\m a^n$.
These functions map positive integers
to positive integers and satisfy the bi-implication
\[
d_{\m a}(n) \leq g \wrel{\Longleftrightarrow} n \leq h_{\m a}(g),
\]
which asserts that $d_{\m a}$ is the lower adjoint of $h_{\m a}$
and $h_{\m a}$ is the upper adjoint of $d_{\m a}$.
It follows that $d_{\m a}, h_{\m a}\colon \mathbb Z^+\to\mathbb Z^+$ 
are monotone functions,
which are inverse bijections between their images:
\[
\im(d_{\m a}) \xrightleftharpoons[d]{h} \im(h_{\m a});
\]
and, moreover, each determines the other.

In the literature, the notation is slightly different.
The least size of a generating
set for $\m a$ is denoted $d(\m a)$. 
The \emph{growth sequence} for $\m a$
is then defined to be the sequence
\[
\dd(\m a) := (d(\m a), d(\m a^2), d(\m a^3), \ldots).
\]
The largest $n$ such that 
$\m a^n$ is $g$-generated is denoted $h(g,\m a)$.
The relationship between the two notations
is $\dd(\m a) = (d_{\m a}(1),d_{\m a}(2),\ldots)$ and 
$h(g,\m a) = h_{\m a}(g)$. 
In this paper we will use the notation
$d_{\m a}(n)$ and $h_{\m a}(n)$ in place of 
$\dd(\m a)$ and $h(n,\m a)$, and use 
phrases ``growth rate'' or ``growth function''
in place of ``growth sequence''.

The $h$ function was the first to be studied, 
by Philip Hall in his 1936 paper \cite{hall}. Hall proves that if $A$
is a finite, simple, nonabelian group, then 
\[
h_{A}(g) = \frac{1}{|\Aut(A)|}\sum_{H\leq A} \mu(H)|H|^g,
\]
where $\mu$ is the M\"obius function of the subgroup
lattice of $A$. 
A formula for the $d$ function
of a finite, simple, nonabelian group was later given
by James Wiegold in his 1974 paper***CHECK*** \cite{wiegold1}: for large $n$,
$d_{\m a}(n)$ is one of the two or three integers
closest to 
\[
\log_{|A|}(n) + \log_{|A|}(|\Aut(A)|).
\]
Wiegold's paper initiated a program of research
into growth rates of groups
\cite{erfanian1,erfanian2,erfanian3,erfanian4,erfanian5,
erfanian_rezaei, erfanian_wiegold, 
kimmerle, lennox_wiegold, meier_wiegold, 
obraztsov, stewart_wiegold, wiegold2, wiegold3, wiegold4, 
wiegold_wilson,wise}.
\marginpar{When paper \\ is done\\ replace some\\ with range a--b}
The program expanded to include the 
investigation of growth rates of semigroups, in
\cite{pollak, wiegold_sem}, and later to 
include the investigation of
growth rates of arbitrary algebraic structures,
in \cite{glass_riedel, riedel}. Our own investigations
into growth rates of finite algebras,
on which we are reporting here,
was stimulated
by the recent appearance of \cite{quick_ruskuc}
and by the related preprint \cite{hyde_etal}.

Some of the questions being investigated 
about growth rates of
finite algebras are related to the following
theorems of Wiegold:
\begin{enumerate}
\item[(I)] 
A finite perfect group\footnote{$G$ is perfect
if $[G,G]=G$.}
has growth rate 
that is logarithmic ($d_{\m a}(n)\in \Theta(\log(n))$),
while a finite imperfect group has 
growth rate that is linear ($d_{\m a}(n)\in \Theta(n)$),
\cite{wiegold3}.
\item[(II)] 
A finite semigroup with identity
has growth rate that is logarithmic or linear, while a finite
semigroup without identity has
growth rate that is exponential 
($d_{\m a}(n)\in 2^{\Theta(n)}$), \cite{wiegold_sem}.
% Note: Quick and Ruskuc omit the logarithmic case when citing 
% (II) (page 2 of their preprint). 
\end{enumerate}

Herbert Riedel partially extended item (I) to 
congruence uniform varieties in \cite{riedel}
by proving that finite perfect algebras
in such varieties have logarithmic growth rate.
He adds that \emph{``Unfortunately, Wiegold's results 
for the imperfect case do not hold even for
modules.''} It is not clear what this sentence means.
It is not very difficult to show
that any finite imperfect algebra in a congruence uniform 
variety has linear growth, so either Riedel's remark is a mistake
or he means that Wiegold's sharp estimate on the
leading coefficient of the linear growth function
is not valid for every finite module. 
Riedel concludes his paper by asking
whether his result about perfect algebras can be extended 
from congruence uniform varieties to 
congruence permutable varieties; we will see that it does.

The paper \cite{quick_ruskuc} by 
Martyn Quick and Nikola Ru\v skuc
extends item (I)
to any variety of rings, modules, $k$-algebras
or Lie algebras, but 
%. These varieties are congruence
%uniform, but the results of \cite{quick_ruskuc} 
also fall short of extending item (I)
to arbitrary congruence uniform varieties.

The results from \cite{quick_ruskuc} 
can be presented in a stronger way:
let $\Sigma$ be a set of identities.
If $\m a$ is an algebra in a language
$\mathcal K$, then say that $\m a$
\emph{realizes} $\Sigma$ 
if there is a way to interpret the function symbols 
occurring in $\Sigma$ as $\mathcal K$-terms
in such a way that each identity in $\Sigma$ holds in $\m a$.
What is really proved in \cite{quick_ruskuc}
is that if $\Sigma_{\text{Grp}}$ is the set of identities
axiomatizing the variety of groups and $\m a$ is a finite
algebra realizing $\Sigma_{\text{Grp}}$,\footnote{The statement 
``$\m a$ realizes $\Sigma_{\text{Grp}}$''
is equivalent to what elsewhere might be expressed as
``$\m a$ is an expansion of a group'' or
``$\m a$ has underlying group structure''.}
 then $\m a$ has
a logarithmic growth rate if it is perfect
and has a linear growth rate if it is imperfect.
Hence the arguments in \cite{quick_ruskuc} can be 
used to extend
Wiegold's result (I) from finite groups to 
expansions of finite groups.
%arbitrary
%finite algebras realizing the identities of groups.

Our results are also best expressed in the language 
of algebras realizing a set of identities.
Call a term \emph{basic} if it contains at most one 
nonnullary function symbol. 
An identity $s\approx t$ is basic if the terms
on both sides are. This paper is an investigation into
the restrictions imposed on growth rates of finite algebras 
by a set $\Sigma$ of basic identities.
A new concept that emerges from this investigation is 
the notion of a pointed cube term.
If $\Sigma$ is a set of identities in 
a language $\mathcal L$, then a
$\mathcal L$-term $F(x_1,\ldots,x_m)$ is a
\emph{$p$-pointed, $k$-cube term}
if there is a $k\times m$ matrix $M$
consisting of variables and at most $p$ distinct constant symbols,
with every column of $M$ containing a symbol different 
from $x$, such that 
\begin{equation}\label{cube}
\Sigma\models 
F(M)\approx 
\left(
\begin{matrix}
x\\
\vdots\\
x
\end{matrix}
\right).
\end{equation}
(\ref{cube}) is meant to be a compact representation
of a sequence of $k$ row identities of a special kind.
For example, 
\begin{equation}\label{m}
\Sigma\models 
{m}\left(\begin{matrix}
x&y&y\\
y&y&x
\end{matrix}
\right) \approx
\left(\begin{matrix}
x\\
x
\end{matrix}
\right),
\end{equation}
which is the assertion that 
$\Sigma\models m(x,y,y)\approx x$ and 
$\Sigma\models m(y,y,x)\approx x$,
witnesses that $m(x_1,x_2,x_3)$ is a
$3$-ary, $0$-pointed, $2$-cube term.
The basic identities (\ref{m})
define what is called a \emph{Maltsev term}.
For another example, 
\begin{equation}\label{b}
\Sigma\models 
{B}\left(\begin{matrix}
1&x\\
x&1
\end{matrix}
\right) \approx
\left(\begin{matrix}
x\\
x
\end{matrix}
\right),
\end{equation}
which is the assertion that 
$\Sigma\models B(1,x)\approx x$ and 
$\Sigma\models B(x,1)\approx x$,
witnesses that $B(x_1,x_2)$ is a
$2$-ary, $1$-pointed, $2$-cube term.
As a final example, 
\begin{equation}\label{majority}
\Sigma\models 
{M}\left(\begin{matrix}
y&x&x\\
x&y&x\\
x&x&y
\end{matrix}
\right) \approx
\left(\begin{matrix}
x\\
x\\
x
\end{matrix}
\right),
\end{equation}
which is the assertion that $M$ is a majority term
for the variety axiomatized by $\Sigma$,
witnesses that $M(x_1,x_2,x_3)$ is a
$3$-ary, $0$-pointed, $3$-cube term.

To state our main results, 
let $\Sigma$ be a set of basic identities in a language
with only finitely many constant symbols.
\begin{enumerate}
\item[(1)]
If $\Sigma$ does not entail the existence of a pointed cube
term, then $\Sigma$ imposes no restriction on growth
rates of finite algebras (Theorem~\ref{nonrestrictive_thm}).
That is, for every finite algebra $\m a$ there is
a finite algebra $\m b$ realizing
$\Sigma$ such that $d_{\m b}=d_{\m a}$.\footnote{We 
call $\Sigma$ with this property \emph{nonrestrictive}.
Otherwise $\Sigma$ is \emph{restrictive}.}
\item[(2)]
If $\Sigma$ entails the existence of an $m$-ary
$p$-pointed $k$-cube term, then any finite algebra
realizing $\Sigma$ has growth rate that is bounded above
by a polynomial of degree at most $\log_w(mp)$, where 
$w = 2k/(2k-1)$ (Theorem~\ref{pointed_polynomial}).
Moreover, there exist finite algebras
with $1$-pointed cube terms whose growth rate is 
asymptotically equivalent to a polynomial 
of any prescribed degree (Theorem~\ref{example}).
\item[(3)]
If $\Sigma$ entails the existence of a $0$-pointed 
cube term, and $\m a$ realizes $\Sigma$, then the growth rate
of $\m a$ is logarithmic if $\m a$ is perfect and linear
if $\m a$ is imperfect (Theorem~\ref{???????}).
\end{enumerate}

Item (3) extends Wiegold's result (I) to a setting that 
includes, as special cases, any algebra with a Maltsev
term or any algebra with a majority term. 
To further specialize,\footnote{A finite
algebra in a congruence uniform variety has a Maltsev term, by
the results of \cite{clark_krauss} or \cite{mckenzie}.}
this includes the case of finite algebras
in congruence uniform varieties.

Concerning Wiegold's result (II), Remark 4.15 of \cite{quick_ruskuc} 
states that 
\emph{``At present no finite algebraic structure is known
for which the $d$--sequence does not have 
one of logarithmic, linear or exponential growth.''}
Item (2) on the list establishes the existence of such examples.

Item (1) 
on this list is perhaps the most striking of all. It 
implies, for example, that any function that arises
as the $d$-function of an arbitrary finite algebra must also 
arise as the $d$-function of a finite algebra in a congruence
distributive and congruence $3$-permutable variety.
This is the only item on this list that requires the
assumption that only finitely many distinct constants
appear in $\Sigma$. Example~\ref{example_constants}
shows that this assumption is necessary when dealing
with finite algebras.

In addition to our main results, we give a new proof of 
Kelly's Completeness Theorem for basic identities
(Theorem~\ref{kelly}). 
We give a procedure, based on this theorem, 
for deciding if a finite set of basic
identities implies the existence of a pointed cube term
(Corollary~\ref{decide_cor}). We also produce new
ways of showing that a finite algebra
has exponential growth rate (Section~\ref{???}).
Some of our results are valid and interesting for
infinite algebras. This is discussed in Section~\ref{???}.

\section{Preliminaries}\label{prelim_sec}

\subsection{Notation}\label{notation}

$[n]$ denotes the set $\{1,\ldots,n\}$.
A tuple in $A^n$ may be denoted 
$(a_1,\ldots,a_n)$ or $\wec{a}$.
A tuple $(a,a,\ldots,a)\in A^n$ with all coordinates equal to $a$
may be denoted $\hat{a}$.
The size of a set $A$, the 
length of a tuple $\wec{a}$, and the length of a string $\sigma$
are denoted $|A|$, $|\wec{a}|$ and $|\sigma|$.
Structures are denoted in bold face font,
e.g. $\m a$, while the universe of a structure
is denoted by the same character 
in italic font, e.g., $A$. 
The subuniverse of $\m a$ generated by a subset $G\subseteq A$
is denoted $\lb G\rb$.

We will use Big Oh notation.
If $f$ and $g$ are real-valued functions defined
on some subset of the real numbers, then 
$f\in O(g)$ and $f=O(g)$ both mean that there are constants
$M$ and $N$ such that $|f(x)|\leq M|g(x)|$ for all $x>N$.
We write $f\in \Omega(g)$ and $f=\Omega(g)$ to mean that there are constants
$M$ and $N$ such that $|f(x)|\geq M|g(x)|$ for all $x>N$.
Finally, $f\in \Theta(g)$ and $f=\Theta(g)$ mean that both
$f\in O(g)$ and $f\in \Omega(g)$ hold.

\subsection{Easy estimates}\label{estimates}

\begin{thm}\label{basic_estimates}
Let $\m a$ be a finite algebra.
\begin{enumerate}
\item[(1)] $d_{\m a^k}(n) = d_{\m a}(kn)$.
\item[(2)] If $\m b$ is a homomorphic image of 
$\m a$, then $d_{\m b}(n) \leq d_{\m a}(n)$.
\item[(3)] If $\m b$ is an expansion of 
$\m a$ (equivalently, if $\m a$ is a reduct of $\m b$), 
then $d_{\m b}(n) \leq d_{\m a}(n)$.
\item[(4)] {\rm (From \cite{quick_ruskuc}.)}
If $\m b$ is the full polynomial expansion
of $\m a$, then 
\[
d_{\m a}(n) - d_{\m a}(1)\leq d_{\m b}(n) \leq d_{\m a}(n).
\]
\end{enumerate}
\end{thm}

\begin{proof}
For $(1)$, both 
$d_{\m a^k}(n)$ and $d_{\m a}(kn)$
count the number of elements in a smallest
generating set for $(\m a^k)^n\cong \m a^{kn}$.

For $(2)$, if $\varphi\colon \m a\to \m b$ is surjective and
$G\subseteq A^n$ is a smallest generating set for $\m a^n$,
then $\varphi(G)$ is a generating set for $\m b^n$.
Hence $d_{\m b}(n)\leq |G| = d_{\m a}(n)$.

For $(3)$, if 
$G\subseteq A^n$ is a smallest generating set for $\m a^n$,
then $G$ is also a generating set for $\m b^n$.
Hence $d_{\m b}(n)\leq |G| = d_{\m a}(n)$.

For $(4)$, the right-hand
inequality $d_{\m b}(n) \leq d_{\m a}(n)$ follows from $(3)$.
Now let $G\subseteq A^n$ be a smallest generating set
for $\m b^n$ and let $H\subseteq A$ be a smallest generating set for $\m a$.
For each $a\in H$ let $\hat{a} = (a,a,\ldots,a)\in A^n$ be the associated
constant tuple, and let $\widehat{H}$ be the set of these.
Every tuple of $A^n$ is generated from $G$ by polynomial operations
of $\m a$ acting coordinatewise, hence is generated from 
$G\cup \widehat{H}$ by term operations of $\m a$ acting
coordinatewise. This proves
$
d_{\m a}(n) \leq |G| + |H| = d_{\m b}(n) + d_{\m a}(1),
$ 
from which the left-hand inequality follows.
\end{proof}

\begin{thm}\label{first_bounds}
If $\m a$ is a finite algebra of
more than one element, then 
\[
\lceil \log_{|A|}(n) \rceil\leq d_{\m a}(n) \leq |A|^n
\]
and
\[
\lfloor \log_{|A|}(n) \rfloor\leq h_{\m a}(n) \leq |A|^n.
\]

Moreover,
\begin{enumerate}
\item[(1)] $d_{\m a}(n) \in O(\log(n))$ iff 
$h_{\m a}(n) \in 2^{\Omega(n)}$.
\item[(2)] $d_{\m a}(n) \in O(n)$ iff 
$h_{\m a}(n) \in \Omega(n)$, and 
$d_{\m a}(n) \in \Omega(n)$ iff 
$h_{\m a}(n) \in O(n)$.
\item[(3)] $d_{\m a}(n) \in 2^{\Omega(n)}$ iff
$h_{\m a}(n) \in O(\log(n))$.
\end{enumerate}
\end{thm}

\begin{proof}
It follows from Theorem~\ref{basic_estimates}~(3) that,
among all algebras with universe $A$, the algebra equipped with
no operations has the smallest $d$-function and the
algebra equipped with all finitary operations has
the largest $d$-function. These two algebras are
also extremes for the $h$-function.

If $\m a$ has no operations, then every element of $A^n$
is a required generator, so $d_{\m a}(n) = |A|^n$.
In this case, $h_{\m a}(n)= \lfloor \log_{|A|}(n)\rfloor$,
since $h$ is the upper adjoint of $d$.

Now assume that $\m a$ is equipped with all finitary operations,
i.e., $\m a$ is primal.
The $n$-generated free algebra in the variety generated by $\m a$
is isomorphic to $\m a^{|A|^n}$ (Theorem 3 of \cite{foster}).
Since the largest $n$-generated algebra in this variety
is a power of $\m a$, it is also the largest $n$-generated 
power of $\m a$ in the variety; we obtain that
$h_{\m a}(n) = |A|^n$. 
In this case, $d_{\m a}(n)= \lceil \log_{|A|}(n)\rceil$,
since $d$ is the lower adjoint of $h$.

The fact that $d_{\m a}$ is the lower adjoint 
of $h_{\m a}$ suggests an asymmetry, in that
\begin{equation}\label{adjoint1}
d_{\m a}(n)\leq k\Longleftrightarrow n\leq h_{\m a}(k),
\end{equation}
relates an upper bound of $d_{\m a}$ to a lower bound
of $h_{\m a}$. But the fact that these functions
are defined between totally ordered sets allows us to
rewrite (\ref{adjoint1}) as
\begin{equation}\label{adjoint2}
h_{\m a}(k) < n\Longleftrightarrow k < d_{\m a}(n),
\end{equation}
which almost exactly reverses condition (\ref{adjoint1}) on 
$d_{\m a}$ and $h_{\m a}$.
Using this fact and the following claim, 
one easily verifies items (1)--(3).

\begin{clm}
If $f, g\colon [a,\infty)\to \mathbb R$ are increasing
functions that tend to infinity as $x$ tends to infinity,
then 
$\lfloor f(n)\rfloor < d_{\m a}(n)\leq \lceil g(n)\rceil$ 
holds for all large $n$
iff $\lfloor g^{-1}(n)\rfloor \leq h_{\m a}(n) < \lceil f^{-1}(n)\rceil$ 
holds for all large $n$.
\end{clm}

Allow ``$\forall N$'' to stand for ``for all large $n$''. We have
\[
\begin{array}{rl}
\forall N(d_{\m a}(n)\leq \lceil g(n)\rceil) 
&\Longrightarrow\;\;
\forall N(n\leq h_{\m a}(\lceil g(n)\rceil))\\
&\Longrightarrow\;\;
\forall N(\lfloor g^{-1}(n)\rfloor \leq 
h_{\m a}(\lceil g(\lfloor g^{-1}(n)\rfloor)\rceil))\\
&\Longrightarrow\;\;
\forall N(\lfloor g^{-1}(n)\rfloor \leq h_{\m a}(n)),
\end{array}
\]
because the monotonicity of $g$ guarantees that 
$\lceil g(\lfloor g^{-1}(n)\rfloor)\rceil))\leq n$.
The reverse implication is proved the same way, as are 
both implications in 
$\lfloor f\rfloor < d \Leftrightarrow h < \lceil f^{-1}\rceil$.
\end{proof}

%The claims of Theorem~\ref{first_bounds} can be found in 
%\cite{quick_ruskuc}, but they are not justified correctly there.
%(Specifically, their formula (2.4) is false.)

Recall that the \emph{free spectrum} of a variety
$\mathcal V$ is the function $f_{\mathcal V}(n):=|F_{\mathcal V}(n)|$
whose value at $n$ is the cardinality of the
$n$-generated free algebra in $\mathcal V$.

\begin{thm}\label{spec1}
If $\m a$ is a finite algebra and $f_{\mathcal V}$
is the free spectrum of the variety $\mathcal V = {\mathcal V}(\m a)$,
then 
$
%|A|^{h_{\m a}(n)}\leq f_{\mathcal V}(n).
h_{\m a}(n)\leq \log_{|A|}(f_{\mathcal V}(n)).
$
In particular,
\begin{enumerate}
\item[(1)] 
if $f_{\mathcal V}(n)\in O(n^m)$ for some fixed $m\in\mathbb Z^+$, 
then $d_{\m a}(n)\in 2^{\Theta(n)}$;
\item[(2)] 
if $f_{\mathcal V}(n)\in 2^{O(n)}$,
then $d_{\m a}(n)\in \Omega(n)$.
\end{enumerate}
\end{thm}

\begin{proof}
The algebra $\m a^{h_{\m a}(n)}$ is $n$-generated, hence
a quotient of the $n$-generated free algebra
$\m f_{\mathcal V}(n)$. This proves that 
$|A|^{h_{\m a}(n)}\leq f_{\mathcal V}(n)$, or 
$h_{\m a}(n)\leq \log_{|A|}(f_{\mathcal V}(n))$.

If $f_{\mathcal V}(n)\in O(n^m)$ for some fixed $m\in\mathbb Z^+$, 
then $\log(f_{\mathcal V}(n))\in O(\log(n))$,
hence $h_{\m a}(n)\in O(\log(n))$. It follows from 
Theorem~\ref{first_bounds}~(3) that 
$d_{\m a}(n)\in 2^{\Theta(n)}$.

If $f_{\mathcal V}(n)\in 2^{O(n)}$, then
$\log(f_{\mathcal V}(n))\in O(n)$,
hence $h_{\m a}(n)\in O(n)$. It follows from 
Theorem~\ref{first_bounds}~(2) that 
$d_{\m a}(n)\in \Omega(n)$.
\end{proof}

\begin{cor}\label{abelian_cor}
Let $\m a$ be a finite algebra and let $\m b$
be a homomorphic image of $\m a^k$ for some $k$.
\begin{enumerate}
\item[(1)] 
If $\m b$ is strongly abelian (or even just strongly rectangular), 
then $d_{\m a}(n)\in 2^{\Theta(n)}$.
\item[(2)] 
If $\m b$ is abelian, then $d_{\m a}(n)\in \Omega(n)$.
\end{enumerate}
\end{cor}

\begin{proof}
For (1), Theorem~5.3
of \cite{strnil} proves that a finite strongly rectangular
algebra generates a variety with free spectrum bounded
above by a polynomial.
By Theorem~\ref{spec1}, $d_{\m a}(n)\in 2^{\Theta(n)}$
in this case. The strong abelian property 
is more restrictive than the strong rectangular
property by Lemma~2.2~(11) of \cite{strnil}.

For (2), any finite abelian algebra generates a variety
$\mathcal V$ 
whose free spectrum satisfies 
$f_{\mathcal V}(n)\in 2^{O(n)}$, according to \cite{berman_mck},
so Theorem~\ref{spec1}~(2) completes the argument.
\end{proof}

\begin{thm}
Let $\m a$ be a finite algebra of more than one element.
\begin{enumerate}
\item[(1)] If $\m a$ is an affine algebra, then 
$d_{\m a}(n)\in \Theta(n)$.
\item[(2)] If $\m a$ is a \emph{simple} affine algebra, then 
$r(n-1)\leq h_{\m a}(n) \leq  rn + 1$, where 
$r = \log_{|A|}(R)$ and $R$ is the associated ring.
\end{enumerate}
\end{thm}

\begin{proof}
Is this really needed?
\end{proof}

\section{Kelly's Completeness Theorem}\label{Kelly_section}

In Subsection~\ref{Kelly_subsection}
we give a new proof of Kelly's Completeness Theorem
for basic identities. The proof involves
the construction of a model of a set of basic identities.
In Subsection~\ref{V_subsection} we construct a simpler model
by modifying the construction from the Completeness Theorem.
The simpler model is not adequate for proving the
Completeness Theorem, but it is exactly we need for 
our investigation of growth rates.
Strictly speaking, it would be possible and easier to 
construct only the simpler model, but the Completeness Theorem
is important testing claims about specific examples,
and no proof of the theorem
appears in the literature; hence we include one here.

\subsection{The Completeness Theorem for basic 
identities}\label{Kelly_subsection}

Let $\mathcal L$ be an algebraic language. Recall that
an $\mathcal L$-term is \emph{basic} if it 
contains at most one nonnullary function symbol.
An $\mathcal L$-identity $s\approx t$ is basic if both
$s$ and $t$ are basic terms. 
If $\Sigma\cup\{\varphi\}$ is a set of basic identities,
then $\varphi$ is a \emph{consequence} of $\Sigma$,
written $\Sigma\models \varphi$,
if every model of $\Sigma$ is a model of $\varphi$.


Let $C$ be the set of constant symbols of $\mathcal L$ and let
$X$ be a set of variables.
The \emph{weak closure of $\Sigma$ in the variables $X$}
is the smallest set $\overline{\Sigma}$
of basic identities containing $\Sigma$ for which 
\begin{enumerate}
%\item[(i)]
%$\Sigma\subseteq \overline{\Sigma}$.
\item[(i)] $(t\approx t)\in \overline{\Sigma}$ for all basic
$\mathcal L$-terms $t$ with variables from $X$.
\item[(ii)] If $(s\approx t)\in \overline{\Sigma}$, then 
$(t\approx s)\in \overline{\Sigma}$.
\item[(iii)] If $(r\approx s)\in \overline{\Sigma}$ and
$(s\approx t)\in \overline{\Sigma}$, then 
$(r\approx t)\in \overline{\Sigma}$.
\item[(iv)] 
%(Substitution of variables and constants.) 
If $(s\approx t)\in \overline{\Sigma}$
and $\gamma\colon X\to X\cup C$ is a function, then 
$(s[\gamma]\approx t[\gamma])\in \overline{\Sigma}$, where 
$s[\gamma]$ denotes the basic term obtained from 
$s$ by replacing each variable $x\in X$ with 
$\gamma(x)\in X\cup C$.
\item[(v)] 
%(Replacement of constant subterms.)
If $t$ is a basic $\mathcal L$-term
and $(c\approx d)\in \overline{\Sigma}$ for $c, d\in C$,
then $(t\approx t')\in \overline{\Sigma}$, where
$t'$ is the basic term obtained from $t$
by replacing one occurrence of $c$ with $d$.
\end{enumerate}

These closure conditions may be interpreted as the inference rules
of a proof calculus for basic identities. Therefore, 
write $\Sigma\vdash_X \varphi$ if $\varphi$ belongs
to the weak closure of $\Sigma$ in the variables $X$.
If the set $X$ is large enough, the relation
$\vdash_X$ captures $\models$ for basic identities, as we
will prove in Theorem~\ref{kelly}.
$X$ is \emph{large enough} if 
(a) $X$ contains at least 2 variables, 
(b) $|X|\geq {\rm arity}(F)$  for any
function symbol $F$ occurring in $\Sigma$, and 
(c) $|X|$ is at least as large as the
number of distinct variables occurring in
any identity in $\Sigma\cup\{\varphi\}$.
Call $\Sigma$ \emph{inconsistent relative to $X$}
if $\Sigma\vdash_X x\approx y$ for distinct 
$x, y\in X$ and large enough $X$.
Otherwise $\Sigma$ is \emph{consistent relative to $X$}.

\begin{thm} \label{kelly}
{\rm (David Kelly)}
Let $\Sigma\cup \{\varphi\}$ be a set of basic identities
and $X$ be a set of variables that is large enough.
%containing all variables
%occurring in $\Sigma\cup \{\varphi\}$ and 
%satisfying $|X|\geq {\rm arity}(F)+2$  for every
%function symbol $F$ occurring in $\Sigma$.
If $\Sigma$ is consistent relative to $X$, then 
$\Sigma\vdash_X \varphi$
if and only if 
$\Sigma\models \varphi$.
\end{thm}

Kelly's theorem is a natural restriction of 
Birkhoff's Completeness Theorem for equational logic
to the special case of basic identities.
However, it is in general undecidable 
for finite $\Sigma\cup\{\varphi\}$
whether $\Sigma\vdash \varphi$ using Birkhoff's
inference rules, while it is decidable
for basic identities using Kelly's restricted rules.\footnote{%
%Without wishing to overexplain this, t
The reason
that $\Sigma\vdash_X \varphi$ is decidable with
Kelly's inference rules when $\Sigma\cup \{\varphi\}$
is finite is that deciding $\Sigma\vdash_X \varphi$ 
amounts to generating $\overline{\Sigma}$.
If $\mathcal L$ is the language whose function
and constant symbols are those occurring in $\Sigma\cup \{\varphi\}$,
$X$ is a minimal (finite) set of variables that is large enough,
and $\mathcal T$ is defined to be the set of basic
$\mathcal L$-terms in the variables $X$,
then generating $\overline{\Sigma}$ amounts to generating
an equivalence relation on the finite set $\mathcal T$
using Kelly's inference rules.}

%No proof of Theorem~\ref{kelly} exists in the literature,
%so we include one here.
In the proof we use a variation of Kelly's Rule (iv):
rather than use functions $\gamma\colon X\to X\cup C$
for substitutions we will use functions
$\Gamma\colon X\cup C\to X\cup C$ whose restriction to $C$ is the identity.
(That is, we replace $\gamma$ with 
$\Gamma := \gamma\cup {\rm id}|_C$.)

\begin{lm}\label{kelly_lm}
If $\Sigma\vdash_X x\approx h$ for some
basic term $h$ in which $x$ does not occur, 
%e.g., for some $h\in C$,
then $\Sigma$ is inconsistent relative to any set $X$
containing a variable other than $x$.
\end{lm}

\begin{proof}
Append to a $\Sigma$-proof of $x\approx h$ 
the formulas $(y\approx h)$ for some $y\in X\setminus \{x\}$ (Rule (iv)); 
$(h\approx y)$ (Rule (ii)); and
$(x\approx y)$ (Rule (iii)).
\end{proof}

\begin{proof}[Proof of Theorem~\ref{kelly}]
Kelly's inference rules are sound, since they are
instances of Birkhoff's inference rules for equational logic.
Hence $\Sigma\vdash_X \varphi$ implies $\Sigma\models \varphi$
for any $X$.

Now assume that $\Sigma{\not{\vdash_X}}\varphi$, 
where $X$ is large enough
and $\Sigma$ is consistent relative to $X$.
We construct a model of $\Sigma\cup \{\neg\varphi\}$ to 
show that $\Sigma{\not{\models}}\varphi$.
Let $\mathcal T$ be the set of basic $\mathcal L$-terms
in the variables $X$,
and let $\equiv$ be the equivalence relation on $\mathcal T$
defined by Kelly provability: i.e., $s\equiv t$ if and only if 
$\Sigma\vdash_X s\approx t$. Write $[t]$ for the $\equiv$-class
of $t$. Now extend $\mathcal T$ to a set
${\mathcal T}_0 = {\mathcal T}\cup \{0\}$
where $0$ is a new symbol, and extend $\equiv$
to this set by taking the equivalence class
of $0$ to be $\{0\}$.

The universe of the model will be the set $M:={\mathcal T}_0/{\equiv}$
of equivalence classes of  ${\mathcal T}_0$ under $\equiv$.
We interpret a constant symbol $c$ as the element
$c^{\m m}:=[c]\in M$. Now let $F$ be an $m$-ary function symbol
for some $m>0$. The natural idea for interpreting $F$
as an $m$-ary operation on this set
is to define $F^{\m m}([a_1],\ldots,[a_m]) = [F(a_1,\ldots,a_m)]$.
However, this does not work, since $F(a_1,\ldots,a_m)$ will not
be a basic term unless all the $a_i$'s 
belong to $X\cup C$.
Nevertheless, we shall follow this idea as far as it takes
us, and when we cannot apply it to assign a value to
$F^{\m m}([a_1],\ldots,[a_m])$ we shall assign the value $[0]$.

Choose and fix a well-order $<$ of the set 
$C$ of constant symbols of $\mathcal L$.
Let $\mathcal I$ be the set of partial injective 
functions $\imath\colon M\to X\cup C$ that satisfy the following
conditions:
\begin{enumerate}
\item[(1)] If 
a class 
$[t]\in M$ in the domain of $\imath$ 
contains a constant 
symbol, $c\in C$, then
$\imath[t] = d$ where $d\in C$ is the least 
element in $[t]\cap C$ under $<$.
\item[(2)] If a class
$[t]$ in the domain of $\imath$
contains a variable, $x\in X$, then
$\imath[t] = x$.
\item[(3)] If a class $[t]$ in the domain of $\imath$
fails to contain a variable
or constant symbol, then $\imath[t]\in X$.
\end{enumerate}
According to Lemma~\ref{kelly_lm},
the consistency of $\Sigma$ implies that any class $[t]$
contains at most one variable, and if $[t]$
contains a constant symbol, then $[t]$ contains
no variable. 
Hence there is no ambiguity in conditions (1) and (2).

If $S\subseteq M$ has size at most $|X|$, then $S$
is the domain of some $\imath\in \mathcal I$.

If $S\subseteq M$ and a class $[t]\in S$
contains a variable $x$, then call $x$ a \emph{fixed} variable
of $S$. Any other variable is an \emph{unfixed} variable of $S$.

Now we define how to interpret an $m$-ary function symbol $F$
as an $m$-ary operation on the set $M$.
Choose any $([a_1],\ldots,[a_m])\in M^m$, 
then choose $\imath\in \mathcal I$
that is defined on $S:=\{[a_1],\ldots,[a_m]\}$.
Note that $f:=F(\imath[a_1],\ldots,\imath[a_m])$ is a basic term, since
it is a function symbol applied to elements of $X\cup C$. We refer
to this term to define $F^{\m m}([a_1],\ldots,[a_m])$.
\begin{enumerate}
\item[{\rm Case 1.}] 
(The class $[f]$ contains a term $h$ whose only variables
are among the fixed variables of $S$.) 
Define $F^{\m m}([a_1],\ldots,[a_m]) = [f]$.
\item[{\rm Case 2.}] 
($[f]$ contains a variable.)
If $x$ is a variable in $[f]$, 
then $\Sigma\vdash_X f\approx x$. 
Since $\Sigma$ is consistent, Lemma~\ref{kelly_lm} proves that
$x$ must occur in $f$,
i.e., $x=\imath[a_k]$ for some $k$. 
Hence 
\[
\Sigma\vdash_X F(\imath[a_1],\ldots,\imath[a_m])\approx \imath[a_k]
\]
for some $k$. In this case we define 
$F^{\m m}([a_1],\ldots,[a_m])=[a_k]\;(=\imath^{-1}(x)$.)
\item[{\rm Case 3.}] (The remaining cases.)
Define $F^{\m m}([a_1],\ldots,[a_m]) = [0]$.
\end{enumerate}

Before proceeding, we point out that there is overlap
in Cases 1 and 2, but no conflict in the definition
of $F^{\m m}([a_1],\ldots,[a_m])$.
If $[f]$ contains 
a term $h$ whose variables are fixed variables of $S$
and $[f]$ also contains
a variable $x$, then $\Sigma\vdash_X f\approx x$ and
$\Sigma\vdash_X h\approx x$.
The consistency of $\Sigma$ forces $x$ to be a common
variable of $f$ and $h$, and (since only fixed variables
of $S$ occur in $h$) to be a fixed
variable of $S$. In this situation, Case 1 defines
$F^{\m m}([a_1],\ldots,[a_m])=[f]=[x]$ while
Case 2 defines 
$F^{\m m}([a_1],\ldots,[a_m])=\imath^{-1}(x)=[x]$.

\begin{clm}\label{well-defined}
$F^{\m m}\colon M^m\to M$ is a well-defined function.
\end{clm}

Choose $([a_1],\ldots,[a_m])\in M^m$
and define $S=\{[a_1],\ldots,[a_m]\}$.
There exist elements of $\mathcal I$ defined on
$S$, because this
set has size $\leq {\rm arity}(F)\leq |X|$.
Suppose that $\imath, \jmath\in \mathcal I$
are both defined on this set. 
Let $f = F(\imath[a_1],\ldots,\imath[a_m])$ and 
$g = F(\jmath[a_1],\ldots,\jmath[a_m])$.
To show that $F^{\m m}([a_1],\ldots,[a_m])$
is uniquely defined it suffices
to show that the same value is assigned
whether we refer to the term $f$ or the term $g$.

In all cases of the definition
of $F^{\m m}([a_1],\ldots,[a_m])$,
the assigned value depends only on 
the term $f = F(\imath[a_1],\ldots,\imath[a_m])
= F(\imath|_S[a_1],\ldots,\imath|_S[a_m])$.
Thus, to complete the proof of Claim~\ref{well-defined},
we may replace both $\imath$ and $\jmath$
by $\imath|_S$ and $\jmath|_S$ and assume that
$\imath$ and $\jmath$ have domain $S$.
Now $\imath$ and $\jmath$ are injective functions 
from $S$ into $X\cup C$, and $\imath[t] = \jmath[t]$
whenever $[t]\in S$ and $[t]$ contains a constant
symbol or a fixed variable of $S$. 
When $\imath[t]\neq \jmath[t]$,
then both are unfixed variables of $S$.
In this situation, there is a function
$\Gamma\colon X\cup C\to X\cup C$ that is the identity
on $C$ and on the fixed variables of $S$
for which $\jmath = \Gamma\circ \imath$.
Hence $f[\Gamma]=g$ and, 
if $h$ is a term whose only variables are fixed
variables of $S$, then $h[\Gamma] = h$.

\begin{enumerate}
\item[{\rm Case 1.}] 
($[f]$ contains a term $h$ whose only variables
are among the fixed variables of $S$.) 
Here
$
\Sigma\vdash_X f=F(\imath[a_1],\ldots,\imath[a_m])\approx h.
$
Append to a $\Sigma$-proof of 
$f\approx h$ the formula
$f[\Gamma]\approx h[\Gamma]$ (Rule (iv)).
Since $f[\Gamma]=g$ and $h[\Gamma]=h$, this is a proof
of $g\approx h$. Next append
$h\approx g$ (Rule (ii)) and
$f\approx g$ (Rule (iii)).
We conclude that $[f]=[g]$, so the value $[f]$ assigned to 
$F^{\m m}([a_1],\ldots,[a_m])$ using $\imath$ is the same as the value
$[g]$ assigned using $\jmath$.
\item[{\rm Case 2.}] 
($[f]$ contains a variable.)
If $x\in X$ is a variable in $[f]$, then 
$x=\imath[a_k]$ for some $k$ and 
$
\Sigma\vdash_X F(\imath[a_1],\ldots,\imath[a_m])\approx \imath[a_k]
$
for this $k$. 
Append to a $\Sigma$-proof of 
$f\approx x$ the formula
$f[\Gamma]\approx x[\Gamma]$ (Rule (iv)).
Since $f[\Gamma]=g$ and $x[\Gamma]=\jmath[a_k]$, 
we conclude that 
$
\Sigma\vdash_X F(\jmath[a_1],\ldots,\jmath[a_m])\approx \jmath[a_k]
$
for the same $k$. Whether we use $\imath$ or $\jmath$ we get
$F^{\m m}([a_1],\ldots,[a_m])=[a_k]$.
\item[{\rm Case 3.}] (The remaining cases.)
In Case 1 we showed that $[f]=[g]$ while in Case 2 we showed
that if $x$ is a variable in $[f]$, then 
$x[\Gamma]$ is a variable in $[g]$; together these show that
if $[g]$ does not contain a variable nor a term
whose only variables are among the fixed variables of $S$,
then the same is true of $[f]$. This argument
works with $f$ and $g$ interchanged, so the remaining cases are
those where both $[f]$ and $[g]$ contain no variables nor terms
whose only variables are among the fixed variables of $S$.
Whether we use $\imath$ or $\jmath$, we get
$F^{\m m}([a_1],\ldots,[a_m]) = [0]$.
\end{enumerate}

$\m m$ is defined.
We now argue that $\m m$ is a model of $\Sigma$.
Choose an identity $(s\approx t)\in\Sigma$.
If $s$ is an $n$-ary function symbol $F$
followed by a sequence $\alpha\colon [n]\to X\cup C$ 
of length $n$ consisting of variables and constant symbols, 
then let $F[\alpha]$ be an abbreviation for $s$.
If $s$ is a variable or constant symbol, 
then $s$ determines a function 
$\alpha\colon [1]\to X\cup C\colon 1\mapsto s$,
so abbreviate $s$ by $\diamondsuit[\alpha]$.
We will, in fact, write $s$ as $F[\alpha]$ in either
case, but will remember that $F$ may equal 
the artificially introduced symbol $\diamondsuit$.
The identity $s\approx t$
takes the form $F[\alpha]\approx G[\beta]$.
\marginpar{Bring in $\alpha$ and $\im(\alpha)$ earlier?}

A valuation in $\m m$ is a function $v\colon X\cup C \to M$
satisfying 
$v(c)=c^{\m m}=[c]$ for each $c\in C$. To show that
$\m m$ satisfies $F[\alpha]\approx G[\beta]$
we must show that 
$F^{\m m}[v\circ \alpha]\approx G^{\m m}[v\circ \beta]$
for any valuation $v$.
Choose $\imath\in \mathcal I$ that is defined on the
set $\im(v\circ \alpha)\cup \im(v\circ \beta)$.
This is possible, since we assume that $|X|$
is at least as large as the number of distinct 
variables in the identity
$F[\alpha]\approx G[\beta]\in \Sigma$.
The values of $F^{\m m}[v\circ \alpha]$ 
and $G^{\m m}[v\circ \beta]$
are defined in reference to the terms 
$f:=F[\imath\circ v\circ \alpha]$
and 
$g:=G[\imath\circ v\circ \beta]$
respectively. 

\begin{clm}\label{f=g}
$[f]=[g]$.
\end{clm}

Observe that $(\imath\circ v)(c) = \imath[c] = d$,
where $d\in C$ is the $<$-least constant symbol in the class
$[c]$.
If $\Gamma\colon X\cup C\to X\cup C$ is a function
that agrees with $(i\circ v)$ on the variables in
$\im(\alpha)\cup \im(\beta)$,
but is the identity on $C$,
then applications of Rule (v) show that
$\Sigma\vdash_X F[\imath\circ v\circ \alpha]\approx
F[\Gamma\circ \alpha]$ and
$\Sigma\vdash_X G[\imath\circ v\circ \beta]\approx
G[\Gamma\circ \beta]$.
From  Rule~(iv), the fact that
$\Sigma\vdash_X F[\alpha]\approx G[\beta]$
implies that
$\Sigma\vdash_X F[\Gamma\circ \alpha]\approx G[\Gamma\circ \beta]$.
Hence 
\[
\Sigma\vdash_X f=F[\imath\circ v\circ \alpha]\approx 
F[\Gamma\circ \alpha]\approx
G[\Gamma\circ \beta]\approx
G[\imath\circ v\circ \beta]=g,
\] 
from which we get $[f]=[g]$.

\medskip

We conclude the argument that $\m m$ satisfies 
$F[\alpha]\approx G[\beta]$
as follows.
\begin{enumerate}
\item[{\rm Case 1.}] 
($[f]=[g]$ contains a term $h$ whose only variables
are among the fixed variables of $S$.) 
In this case 
$F^{\m m}[v\circ \alpha]=[f]=[g]=G^{\m m}[v\circ \beta]$.
\item[{\rm Case 2.}] 
($[f]=[g]$ contains a variable.)
If $[f]=[x]=[g]$, then 
$F^{\m m}[v\circ \alpha]=\imath^{-1}(x)=G^{\m m}[v\circ \beta]$. 
%$.
%Since $[g]=[f]=[x]$, the same argument shows that 
%$G^{\m m}[v\circ \beta]=\imath^{-1}(x)$. 
%Hence 
%$F^{\m m}[v\circ \alpha]=G^{\m m}[v\circ \beta]$.
\item[{\rm Case 3.}] (The remaining cases with $[f]=[g]$.)
$F^{\m m}[v\circ \alpha]=[0]=G^{\m m}[v\circ \beta]$.
%in these cases.
\end{enumerate}

To complete the proof 
of the theorem we must show that $\m m$ does not
satisfy $\varphi$. Suppose $\varphi$
has the form $F[\alpha]\approx G[\beta]$.
Define the canonical valuation to be
\[
v\colon X\cup C \to M\colon x\mapsto [x], c\mapsto [c].
\]
Choose $\imath\in\mathcal I$
that is defined on $\im(v\circ \alpha)\cup \im(v\circ \beta)$.
It follows from the definitions that 
$\imath\circ v\colon X\cup C\to X\cup C$ fixes every variable 
in $\im(\alpha)\cup \im(\beta)$. If $\Gamma$ is the identity
function on $X\cup C$, then $\Gamma$ agrees with $\imath\circ v$
on the variables in $\im(v\circ \alpha)\cup \im(v\circ \beta)$,
while $(\imath\circ v)(c) = d$ is the $<$-least constant symbol
in the class of $c = \Gamma(c)$. Just as in the proof of 
Claim~\ref{f=g}, we obtain 
$\Sigma\vdash_X f=F[\imath\circ v\circ \alpha]\approx 
F[\Gamma\circ \alpha] = F[\alpha]$ and 
$\Sigma\vdash_X g=G[\imath\circ v\circ \beta]\approx 
G[\Gamma\circ\beta]=G[\beta]$.
Now $[f]$ contains a term $h:=F[\alpha]$ 
whose only variables are among the fixed variables of $S=\im(\alpha)$,
so we are in Case 1 of the definition of $F^{\m m}$.
Hence $F^{\m m}(v\circ \alpha) = [F[\alpha]]$, and similarly
$G^{\m m}(v\circ \beta) = [G[\beta]]$.
Part of our assumption about $\varphi=(F[\alpha]\approx G[\beta])$ 
is that $\Sigma\not\vdash_X \varphi$, so 
$[F[\alpha]]$ and $[G[\beta]]$ are distinct elements
of $M$. Therefore, $v$ witnesses that
$\m m$ does not satisfy $\varphi$.
\end{proof}

Theorem~\ref{kelly} establishes that if $X$ and $Y$ are two sets of variables
that are large enough, then 
$\Sigma\vdash_X\varphi$ holds iff 
$\Sigma\vdash_Y\varphi$, and hence 
$\Sigma$ is consistent relative to
$X$ if and only if it is consistent relative to $Y$.
Now that the theorem is proved, we drop
the subscript in $\vdash_X$ and the phrase ``relative to $X$''
when writing about provability.

\subsection{The model $\m v$}\label{V_subsection}

Later in the paper we prove theorems about
finite algebras realizing a set $\Sigma$ of basic identities.
For this, we need to be able to construct finite
models of $\Sigma$. The model constructed in Theorem~\ref{kelly}
may be infinite, so we explain how to produce
finite models.

\begin{df}\label{V}
Let $\Sigma$ be a set of basic identities in a language $\mathcal L$
whose set of constant symbols is $C$.
Let $Y$ be a set of variables, $z$ a variable not in $Y$,
and $X$ a large enough set of variables containing
$Y\cup \{z\}$.
Let $V$ be the subset of the model $\m m$ constructed in the proof
of Theorem~\ref{kelly} consisting of 
\[
\{[y]\;|\;y\in Y\}\cup \{[c]\;|\;c\in C\}\cup \{[0]\}.
\]
Write $[Y]$ for $\{[y]\;|\;y\in Y\}$ and 
$[C]$ for $\{[c]\;|\;c\in C\}$.

Let $F$ be an $m$-ary function symbol of $\mathcal L$.
If $([a_1],\ldots,[a_m])\in V^n$, then 
let 
$a_i' = a_i$ if $a_i\in Y\cup C$ and 
$a_i'=z$ if $a_i=0$. 
Define
$
F^{\m v}([a_1],\ldots,[a_m]) = [t] 
$
if there exists $t\in Y\cup C$ such that
$
\Sigma\vdash F(a_1',\ldots,a_m') \approx t,
$
and define $F^{\m v}([a_1],\ldots,[a_m]) = [0]$
if there is no such $t$.

$\m v$ is the algebra with universe $V$ 
equipped with all operations of the form
$F^{\m v}$.
\end{df}


\begin{thm}\label{small_models}
$\m v$ is a model of $\Sigma$.
\end{thm}

\begin{proof}
Let $F[\alpha]\approx G[\beta]$ be an identity 
in $\Sigma$, and let $v\colon X\cup C\to V$ be a valuation.
We must show that 
$F^{\m v}(v\circ \alpha) = G^{\m v}(v\circ \beta)$.

The function $v$ is also a valuation
in $\m m$, because $V\subseteq M$. 
Since $\m m$ is a model of $\Sigma$, we get
$F^{\m m}[v\circ \alpha] = G^{\m m}[v\circ \beta]$.
Choose $\imath\in \mathcal I$ defined on the set
$\im(v\circ\alpha)\cup \im(v\circ\beta)$
such that $\imath[0]=z$, if $[0]$ is in this set.
Let $f = F[\imath\circ v\circ \alpha]$ and 
$g = G[\imath\circ v\circ \beta]$. 
As in the proof of Claim~\ref{f=g}, 
if $\Gamma\colon X\cup C\to X\cup C$ 
is the identity on $C$ and agrees with 
$\imath\circ v$ on the variables
in $\im(\alpha)\cup \im(\beta)$, then 
$\Sigma\vdash f \approx
F[\Gamma\circ \alpha]\approx G[\Gamma\circ \beta] \approx g$.

The term $F(a_1',\ldots,a_m')$ of Definition~\ref{V}
is none other than $f$. 
$F^{\m v}[v\circ \alpha] = [t]$ for some $t\in Y\cup C$
if and only if $\Sigma\vdash f = F(a_1',\ldots,a_m')\approx t$.
But since $\Sigma\vdash f\approx g$ 
%where $g$ is the corresponding term for $G[\beta]$, 
we also get 
$G^{\m v}[v\circ \beta]=[t]$. This shows that 
$F^{\m v}[v\circ \alpha]$ and $G^{\m v}[v\circ \beta]$
are equal when at least one of them is not $[0]$.
Of course, they are also equal when both of them
equal $[0]$, so $F^{\m v}(v\circ \alpha) = G^{\m v}(v\circ \beta)$.
\end{proof}


\begin{cor}\label{small_models_cor}
If $\Sigma$ is a consistent set of basic identities
in a language whose set of constant symbols is $C$, then 
$\Sigma$ has models of every cardinality strictly exceeding
$|C|$.
\end{cor}

\begin{proof}
Vary the size of $Y$ in the definition of $\m v$,
and use Theorem~\ref{small_models}.
\end{proof}

Corollary~\ref{small_models_cor}
is close to the best possible result 
about sizes of models of a set of basic identities, 
as the next example shows.

\begin{exmp}\label{example_constants}
Let $C$ be a set of constant symbols
and let ${\mathcal B} = \{B_{c,d}\;|\;c, d\in C\}$
be a set of binary function symbols. Let 
\[
\Sigma = \{B_{c,d}(c,x)\approx x, B_{c,d}(d,x)\approx d\;|\;c, d\in C\}.
\]
$\Sigma$ is a consistent set of basic identities,
since if $A$ is any set containing $C$ we can interpret
each $c\in C$ in $A$ as itself and each
$B_{c,d}$ on $A$ by letting $B_{c,d}^{\m a}(c,y)=y$
and $B_{c,d}^{\m a}(x,y)=x$ if $x\neq c$.

If $\m m$ is any model of $\Sigma$ and $c^{\m m} = d^{\m m}$
for some $c, d\in C$, then the identity function
$B^{\m m}_{c,d}(c^{\m m},x)$ equals the constant function
$B^{\m m}_{c,d}(d^{\m m},x)$, so $|M|=1$. Thus elements
of $C$ have distinct interpretations in any
nontrivial model of $\Sigma$, implying that 
nontrivial models have size at least 
$|C|$.

In the case where $C$ is infinite, $\Sigma$ does restrict
growth rates of finite algebras because $\Sigma$ has no
nontrivial finite models. On the other hand, 
$\Sigma$ does not entail the existence of a pointed cube term,
since no $B_{c,d}$ is a pointed cube term. 
\end{exmp}


\section{Restrictive $\Sigma$}\label{examples}

Recall (from footnote 3) that a set $\Sigma$ of identities
is nonrestrictive if, whenever $\m a$ is a finite algebra,
there is a finite algebra $\m b$ realizing $\Sigma$
such that $d_{\m b}(n)=d_{\m a}(n)$. Otherwise
$\Sigma$ is restrictive. At first glance this
definition might seem too mild to be of any use.
Surely the realization of any nontrivial $\Sigma$ ought
to affect growth rates?
In fact, this is not so. Only relatively strong 
$\Sigma$ will enforce any kind of restriction on
growth rates. In Subsection~\ref{partial_subsection}
we prove that the growth rates of finite partial algebras
are the same as the growth rates for finite
total algebras.
In Subsection~\ref{pointed_subsection}
we prove that if $\Sigma$ is restrictive, then
it entails the existence of a pointed
cube term. The converse is proved in 
Subsection~\ref{processing}, by showing 
that a finite algebra with a pointed cube term 
has growth rate the is bounded above by a polynomial.
In particular, it is shown that
a finite algebra $\m a$ with a $1$-pointed $k$-cube term
satisfies $d_{\m a}(n)\in O(n^{k-1})$.
In Subsection~\ref{poly_growth} we describe an example
of a $3$-element algebra with a $1$-pointed $k$-cube term whose
growth rate satisfies $d_{\m a}(n)\in \Theta(n^{k-1})$, showing
that the preceding estimate is sharp.
*******In Subsection~\ref{exp} give some ways of
recognizing when an algebra has an exponential growth rate.

\subsection{Growth rates of partial algebras}\label{partial_subsection}

A partial algebra is a set
equipped with a set of partial operations.
The definitions of functions 
$d_{\m a}$ and $h_{\m a}$ make sense when $\m a$
is a partial algebra,
as does the problem of determining growth rates 
of partial algebras. We will learn in this subsection 
that a function arises as the growth function
of a partial algebra if and only if it arises 
as the growth function of a total algebra.

Let $\m a = \lb A; P\rb$ be a partial algebra
with universe $A$ and a set $P$ of partial operations on $A$.
We define the \emph{one-point completion} of 
$\m a$ to be the total algebra whose
universe is $A_{0}:=A\cup \{0\}$, where
$0$ is some element not in $A$, and whose operations 
$P_0 = \{p_0\;|\;p\in P\}\cup \{\wedge\}$ are defined
as follows.

\begin{enumerate}
\item[(1)] If $p\in P$ is a partial 
$m$-ary operation on $A$ with domain $D\subseteq A^m$,
then the total operation $p_{0}\colon (A_{0})^m\to A_{0}$
is defined by 
\[
p_{0}(\wec{a}) = 
\begin{cases}
p(\wec{a}) & \textrm{if $\wec{a}\in D$; or}\\
0 & \textrm{otherwise.}
\end{cases}
\]
\item[(2)]
A meet operation $\wedge$ on $A_{0}$ is defined by
\[
a\wedge b = 
\begin{cases}
a & \textrm{if $a=b$;}\\
0 & \textrm{otherwise.}
\end{cases}
\]
\end{enumerate}
%Note that the meet operation in (2) is the same as the operation
%$b_%0(x,y)$ where $b$ is the partial binary operation on $A$
%defined on pairs of the form $(a,a)$ by $b(a,a) = a$ and is undefined
%on all other pairs. 

\begin{thm}\label{partial}
If $\m a$ is a partial algebra of more than one element, 
then $\m a^n$ and 
$\m a_0^n$ have the same minimal generating
sets. Hence $d_{\m a_0} = d_{\m a}$.
\end{thm}

\begin{proof}
It suffices to prove the following two statements:
(i) any generating set for $\m a^n$ is a generating set for 
$\m a_0^n$, and (ii) any \emph{minimal} generating set for
$\m a_0^n$ is a subset of $A^n$ and a generating set for $\m a^n$.

In this paragraph we prove (i).
If $G\subseteq A^n$ is a generating set for $\m a^n$, then as a subset
of $\m a_0^n$ it will generate 
(in exactly the same manner) all tuples in $A_0^n$
which have no $0$'s.
If $\wec{z}\in A_0^n$ is an arbitrary 
tuple and $a, b\in A$ are distinct, let
$\wec{z}_a$ and $\wec{z}_b$
be the tuples obtained from $\wec{z}$ by replacing
all $0$'s with $a$ and $b$, respectively. Then $\wec{z}_a, \wec{z}_b
\in A^n$, so they are generated by $G$, and 
$\wec{z} = \wec{z}_a\wedge \wec{z}_b$, so $\wec{z}$ is also generated
by $G$. Hence $G$ generates all of $\m a_0^n$. 

Now we prove (ii). Assume that 
$H\subseteq A_0^n$ is a minimal generating set for
$\m a_0^n$. If $\wec{a}\in A_0^n$, let $Z(\wec{a})\subseteq [n]$
be the \emph{zero set} of $\wec{a}$, by which we mean the set
of coordinates where $\wec{a}$ is $0$. It is easy to see
that for any basic operation $F$ of $\m a_0$ 
it is the case that 
\begin{equation}\label{generation}
Z(\wec{a}_1)\cup \cdots \cup Z(\wec{a}_m)
\subseteq Z(F(\wec{a}_1,\ldots,\wec{a}_m)),
\end{equation}
since $0$ is absorbing for every basic operation. 
If the right-hand side is empty, then the left-hand
side is empty as well; i.e., tuples with empty zero sets
can be generated only by tuples with empty zero sets.
Said a different way, 
if $H\subseteq A_0^n$ generates $\m a_0^n$,
then $H\cap A^n$ suffices to generate all tuples in $A^n$,
and, by (i), these tuples suffice
to generate $\m a_0^n$; hence $H\cap A^n$ generates
$\m A_0^n$. Therefore, if $H\subseteq A_0^n$
is a minimal generating set for $\m a_0^n$, then 
$H\subseteq A^n$. 
If you consider how $H$ generates elements of $A^n$
in the algebra $\m a_0^n$, it is clear that $H$ generates
those elements in the algebra $\m a^n$ in exactly the same way, so
$H$ is a generating set for $\m a^n$.
\end{proof}

\subsection{Restrictive $\Sigma$ forces a 
pointed cube term}\label{pointed_subsection}

Let $\Sigma$ be a set of basic identities 
in a language $\mathcal L$ 
whose set $C$ of constant symbols is finite.
Given an algebra $\m a$ in an arbitrary language,
we construct another algebra
$\m a_{\Sigma}$ which realizes $\Sigma$, where 
$\m a_{\Sigma}$ is finite if $\m a$ is.
%We call $\m a_{\Sigma}$ a \emph{$\Sigma$-expansion} of $\m a$.

For the first step, let 
$[C] = \{[c_1],\ldots,[c_p]\}$ be the 
same set of equivalence
classes denoted by $[C]$ in Definition~\ref{V}.
These classes represent the different 
$\Sigma$-provability classes of constant symbols.
If there are $p$ such classes,
then apply the one-point completion construction 
of Subsection~\ref{partial_subsection}
$p+1$ times to $\m a$ to produce 
a sequence $\m a$, $\m a_{z_1}$, $\m a_{z_1, z_2}$, \ldots,
ending at $\m a_{z_1,\ldots,z_p,0}$.
This is an algebra whose universe
is the disjoint union of $A$ and $\{z_1,\ldots,z_p, 0\}$.

$\m a_{\Sigma}$ will be an expansion of 
$\m a_{z_1,\ldots,z_p,0}$ obtained by merging the latter algebra
with the model $\m v$ introduced in 
Definition~\ref{V}.
Let $Y$ be a set of variables satisfying
$|Y|=|A|$,
and let $[Y] = \{[y]\;|\;y\in Y\}$ be the 
set of equivalence
classes also denoted by $[Y]$ in Definition~\ref{V}.
The universe of $\m v$ is the disjoint union
$V = [Y]\cup [C]\cup \{[0]\}$.

Let $\varphi\colon [Y]\to A$ be a bijection.
Extend this to a bijection from $V=[Y]\cup [C]\cup\{[0]\}$
to $A\cup \{z_1,\ldots,z_p\}\cup\{0\}$
by defining $\varphi([c_i]) = z_i$ and $\varphi([0])=0$.
Now $\varphi$ is a bijection from the 
universe of $\m V$ to the universe of $\m a_{z_1,\ldots,z_p,0}$.
Use this bijection to transfer the operations
of $\m v$ over to $\m a_{z_1,\ldots,z_p,0}$ to create $\m a_{\Sigma}$. 
Specifically, the interpretation of the constant symbol
$c_i$ in $\m a_{\Sigma}$ will be $z_i$, and 
if $F$ is an $m$-ary function symbol of $\mathcal L$,
then 
\begin{equation}\label{Ops}
F^{\m a_{\Sigma}}(x_1,\ldots,x_m):=
\varphi(F^{\m v}(\varphi^{-1}(x_1),\ldots,\varphi^{-1}(x_m)))
\end{equation}
will be the interpretation of the symbol $F$
in $\m a_{\Sigma}$.
$\m a_{\Sigma}$ 
is the expansion of 
$\m a_{z_1,\ldots,z_p,0}$ by all constant operations and
all operations
of the form (\ref{Ops}). Under this definition
the function $\varphi$ is an isomorphism from 
$\m v$ to the $\mathcal L$-reduct 
of $\m a_{\Sigma}$.

\begin{lm}\label{nonrestrictive}
Let $\m a$ be a finite algebra with more than
one element and let 
$\Sigma$ be a set of basic identities
involving finitely many constant symbols.
Let $\mathcal V$ be the variety axiomatized by $\Sigma$.
%and let $n$ be an integer greater than $1$.
The following statements about a positive integer $k$ are equivalent.
\begin{enumerate}
\item[(1)] 
$\mathcal V$ has a pointed $k$-cube term.
\item[(2)] 
For $n\geq k$, the family of minimal generating subsets of $\m a^n$ 
is different from the family of minimal generating subsets 
of $\m a^n_{\Sigma}$.
\item[(3)]
$\mathcal V$ has a pointed $k$-cube term
of the form $F(x_1,\ldots,x_m)$, 
where $m>1$, $F$ is a function
symbol occurring in $\Sigma$, and 
the variables $x_1,\ldots,x_m$ are distinct.
\end{enumerate}
\end{lm}


\begin{proof}

[$(1)\Rightarrow(2)$] 
Let ${\mathsf P}(x_1,\ldots,x_m)$ be a 
pointed $k$-cube term
of the variety axiomatized by $\Sigma$.
There is a $k\times m$ matrix $M=[y_{i,j}]$ 
of variables and 
$\mathcal L$-constant symbols, where 
every column contains a symbol different from $x$,
such that $\Sigma$ proves the identities
\begin{equation}\label{Pidentities}
{\mathsf P}(\wec{y}_1,\ldots,\wec{y}_m)=
{\mathsf P}\left(
\left[\begin{array}{c}
y_{1,1}\\
\vdots\\
y_{k,1}\\
\end{array}\right],
\cdots,
\left[\begin{array}{c}
y_{1,m}\\
\vdots\\
y_{k,m}\\
\end{array}\right]\right)
\approx
\left[\begin{array}{c}
x\\
\vdots\\
x\\
\end{array}\right].
\end{equation}

Let $G\subseteq \m a^n$ be a minimal 
generating set for some $n\geq k$, and let $\wec{a}\in G$ be a tuple.
Using the 
row identities of (\ref{Pidentities}),
solve the equation
${\mathsf P}(\wec{b}_1,\ldots,\wec{b}_m)=\wec{a}$
for the $\wec{b}_i$'s,
row by row, according to the following rules.
In the $i$-th row, 
\begin{enumerate}
\item[(a)] if $y_{i,j}=x$, then let $b_{i,j}=a_i$.
\item[(b)] if $y_{i,j} = c_r$ is a constant symbol, 
then let $b_{i,j}=z_r$ be its interpretation in $\m a_{\Sigma}$.
\item[(c)] if $y_{i,j}$ is a variable different from $x$, 
then let $b_{i,j}=0$.
\end{enumerate}
Under these choices, $\wec{b}_i\in A_{\Sigma}^n$ for all $i$
and ${\mathsf P}(\wec{b}_1,\ldots,\wec{b}_m)=\wec{a}$.
Moreover, since each column $\wec{y}_i$
in (\ref{Pidentities}) has a symbol
different from $x$, it follows from (a)--(c) that each
$\wec{b}_i$ has a coordinate value that is in the set 
$\{z_1,\ldots,z_p,0\}$. Hence  $\wec{b}_i\in A_{\Sigma}^n\setminus A^n$
for all $i$.

Since $G$ is a generating set for $\m a^n$, it is
a generating set for $\m a_{\Sigma}^n$. Since
$\wec{a}$ can be generated from 
$\{\wec{b}_1,\ldots,\wec{b}_m\}$ using $\mathsf P$, it follows that 
\[
G'=(G\setminus \{\wec{a}\})\cup \{\wec{b}_1,\ldots,\wec{b}_m\}
\]
is a generating set for $\m a_{\Sigma}$. Hence 
there exists a minimal generating subset $G''\subseteq G'$
for $\m a^n_{\Sigma}$. If $\m a^n$ and $\m a^n_{\Sigma}$
had the same minimal generating sets, then 
$G''$ would be a subset of $G'\cap A^n = G\setminus\{\wec{a}\}$.
But this is not so, since $G\setminus\{\wec{a}\}$ is a proper
subset of the minimal generating set $G$ of $\m a^n$,
hence it is not itself a generating set.

[$(2)\Rightarrow(3)$] 
By repeated application of Theorem~\ref{partial}
we know that $\m a^n$ has the same
minimal generating sets as $\m a^n_{z_1,\ldots,z_p,0}$.
Our assumption is that these minimal generating
sets are not the same as the minimal generating
sets of the expansion $\m a^n_{\Sigma}$ of 
$\m a^n_{z_1,\ldots,z_p,0}$.
Since every generating set for $\m a^n_{z_1,\ldots,z_p,0}$
is a generating set for the expansion $\m a^n_{\Sigma}$,
there must be a generating set $G$ for $\m a^n_{\Sigma}$
that is not a generating set for 
$\m a^n_{z_1,\ldots,z_p,0}$.
The subuniverse $S=\lb G\rb$ of $\m a^n_{z_1,\ldots,z_p,0}$
cannot contain all of $A^n$, since $A^n$
is a generating set for $\m a^n_{z_1,\ldots,z_p,0}$, and $G$ is not.

\begin{clm}
The set 
$H = S\cup (A^n_{z_1,\ldots,z_p,0}\setminus A^n)$
is a proper subuniverse of $\m a^n_{z_1,\ldots,z_p,0}$.
\end{clm}

%To show that $H$ is a subuniverse of 
%$\m a^n_{z_1,\ldots,z_k,0}$, 
Choose tuples
$\wec{h}_1,\ldots,\wec{h}_r\in H$
and operate on them to produce some element
$\wec{g} = E^{\m a^n_{z_1,\ldots,z_p,0}}(\wec{h}_1,\ldots,\wec{h}_r)$.
If some $\wec{h}_i$ belongs to $A^n_{z_1,\ldots,z_k,0}\setminus A^n$,
then so does $\wec{g}$, by (\ref{generation}).
Otherwise all $\wec{h}_i$ belong to $S$, 
in which case $\wec{g}$ does, since $S$ is a subuniverse.
$H = S\cup (A^n_{z_1,\ldots,z_p,0}\setminus A^n)$
is proper, since $H\cap A^n = S\cap A^n\neq A^n$.

\medskip

Since the proper subuniverse $H$ of $\m a^n_{z_1,\ldots,z_p,0}$ contains the
set $G$, which generates the algebra $\m a^n_{\Sigma}$,
and contains the interpretations of the $\mathcal L$-constants,
it cannot be closed under the the interpretations 
of the function symbols of $\mathcal L$. 
Hence there is a tuple $\wec{a}\notin H$, an $m$-ary function symbol $F$, 
and $m$ tuples $\wec{b}_1,\ldots, \wec{b}_m\in H$ 
such that $F^{\m a_{\Sigma}}(\wec{b}_1,\ldots,\wec{b}_m)=\wec{a}$.
Necessarily $\wec{a}\in A^n$.

Using the isomorphism $\varphi$ from $\m v$ to 
the $\mathcal L$-reduct of $\m a_{\Sigma}$, we obtain 
that there is a tuple $\wec{y} = \varphi^{-1}(\wec{a})\in 
\varphi^{-1}(A^n)=[Y]^n$ and tuples 
$\wec{v}_i = \varphi^{-1}(\wec{b}_i)\neq \wec{y}$ such that 
$
F^{\m v}(\wec{v}_1,\ldots,\wec{v}_m)=\wec{y}.
$
Since $\wec{v}_1\neq \wec{y}$, there is a coordinate $\ell$
where these tuples differ. In the $\ell$-th coordinate we have
$F^{\m v}([v_{{\ell},1}],\ldots,[v_{{\ell},m}])=[y_{\ell}]$ for 
some variable $y_{\ell}\in Y$ and some elements 
$v_{\ell,j}\in Y\cup C\cup \{0\}$ with $[v_{{\ell},1}]\neq [y_{\ell}]$.
By the definition of $\m v$, 
\begin{equation}\label{cube1}
\Sigma\vdash F(v_{\ell,1}',\ldots,v_{\ell,m}')\approx y_{\ell},
\end{equation}
where $v_{\ell,j}'=v_{\ell,j}$ when $v_{\ell,j}\in Y\cup C$
and $v_{\ell,j}'=z$ is a variable not in $Y$ when 
$v_{\ell,j}=0$. 
Since $[v_{\ell,1}]\neq [y_{\ell}]$, we have $v_{\ell,1}\neq y_{\ell}$.
After renaming variables, (\ref{cube1}) can be rewritten as
\[
\Sigma\vdash F(m_{1,1},\ldots,m_{1,m})\approx x,
\]
where each $m_{1,j}$ is a variable or constant and $m_{1,1}\neq x$.
Similarly, for each $i$, the fact that $\wec{v}_i\neq \wec{y}$
produces an identity
\[
\Sigma\vdash F(m_{i,1},\ldots,m_{i,m})\approx x,
\]
where each $m_{i,j}$ is a variable or constant and $m_{i,i}\neq x$.
Thus, it is a consequence of $\Sigma$ that the row identities
of 
\[
F([m_{i,j}])\approx
\left(\begin{matrix}
x\\
\vdots\\
x
\end{matrix}
\right)
\]
hold. 
Since the diagonal elements of $[m_{i,j}]$ are not $x$,
these identities make
$F$ a pointed cube term for $\mathcal V$.

[$(3)\Rightarrow(1)$] 
This is a tautology.
\end{proof}

The next result is the main result of this subsection.

\begin{thm}\label{nonrestrictive_thm}
Let $\Sigma$ be a set of basic identities involving 
finitely many constant symbols. If $\Sigma$
does not entail the existence of a pointed cube term,
then $\Sigma$ is nonrestrictive.
\end{thm}

\begin{proof}
Recall that ``$\Sigma$
is nonrestrictive'' means that 
for every finite algebra $\m a$ there is a
finite algebra $\m b$ realizing $\Sigma$
such that $d_{\m b}=d_{\m a}$, ``$\Sigma$
is restrictive'' means the opposite.

If $\Sigma$ is restrictive, then there must exist
a finite algebra $\m a$ with the property
that $d_{\m b}\neq d_{\m a}$
whenever $\m b$ is finite and realizes $\Sigma$.
For $\m b = \m a_{\Sigma}$ the fact that 
$d_{\m b}\neq d_{\m a}$ implies that $\m a^n$ and
$\m a^n_{\Sigma}$ do not have the same
minimal generating sets for some $n$, so the 
corollary follows from 
Lemma~\ref{nonrestrictive} (1)$\Leftrightarrow$(2).
\end{proof}

One unplanned consequence of Lemma~\ref{nonrestrictive}
is a procedure to decide if a strong
Maltsev condition involving only basic identities
implies the existence of a pointed cube term.

\begin{cor}\label{decide_cor}
A strong Maltsev condition defined by a set 
$\Sigma$ of basic identities entails the existence
of a pointed $k$-cube term if and only if 
it is possible to prove from $\Sigma$ that
some term of the form 
$F(x_1,\ldots,x_m)$ is a pointed 
$k$-cube term, where $m>1$, $F$ is a function
symbol occurring in $\Sigma$, and 
the variables $x_1,\ldots,x_m$ are distinct.
\end{cor}

\begin{proof}
A strong Maltsev condition defined by a set 
$\Sigma$ of identities 
entails the existence
of a pointed $k$-cube term if and only if 
the variety axiomatized by $\Sigma$ has a 
pointed $k$-cube term, so the 
corollary follows from 
Lemma~\ref{nonrestrictive} (1)$\Leftrightarrow$(3).
\end{proof}

That the property in the theorem statement
can be decided follows from Theorem~\ref{kelly}.

\subsection{Pointed cube terms enforce 
polynomially bounded growth}\label{processing}

In the preceding subsection  %\ref{pointed_subsection}
we proved that if $\Sigma$ is restrictive, then $\Sigma$
entails the existence of a pointed cube term.
We now prove the converse.
As the title of the subsection suggests,
we shall prove that if $\m a$ is a finite algebra with a pointed cube
term, then $d_{\m a}(n)$  is bounded above by a polynomial.
Since not all finite algebras have polynomially bounded
growth rates,
this is enough to show that $\Sigma$ is restrictive when it
entails the existence of a pointed cube term.

\begin{thm}\label{pointed_polynomial}
If $\m a$ is a finite algebra with an
$m$-ary, $p$-pointed,
$k$-cube term, with $p\geq 1$,
% and $k\geq 2$, 
then $d_{\m a}$ is bounded above by 
a polynomial 
%in $n$ 
of degree at most 
$\log_w(mp)$, where $w = 2k/(2k-1)$.
\end{thm}


\begin{proof}
Here is a coarse outline of the proof.
Let $\F(x_1,\ldots,x_m)$ be a $p$-pointed $k$-cube
term for $\m a$.
We shall describe a way to 
``process'' a randomly selected 
tuple $\wec{a}\in A^n$ which accomplishes
the following things. If $\wec{a}$ is not already ``fully processed'',
then processing $\wec{a}$ produces tuples
$\wec{b}_1,\ldots,\wec{b}_m\in A^n$ with the properties
that (i) $\F^{\m a}(\wec{b}_1,\ldots, \wec{b}_m) = \wec{a}$
and (ii) each of $\wec{b}_1,\ldots,\wec{b}_m$ is
recognizably ``more processed'' than $\wec{a}$. 
It will follow that $\m a^n$ can be generated by the
set of fully processed tuples. A count will show that
there are only polynomially many fully processed tuples.
%in $\m a^n$.

Let's fix some notation for the proof.
Suppose that the fact that $\F(x_1,\ldots,x_m)$
is a $p$-pointed $k$-cube term (with $p\geq 1$)
is witnessed by identities 
\[
\F(M)\approx \left(\begin{array}{c} x\\ \vdots\\x\end{array}\right),
\]
where $M$ is a $k\times m$ matrix of variables and 
constant symbols, where each column contains a symbol
that is not $x$. 
Since $p\geq 1$, we may assume that the only variable
appearing in $M$ is $x$, since we may choose a 
constant symbol $c$ appearing in $M$, replace all instances of 
other variables in $M$ by $c$ (Kelly's Rule (iv)),
thereby obtaining a matrix $R$
with no variables other than $x$
which witnesses that $\F$ is a $p$-pointed $k$-cube term.

The order of the 
$k$ rows identities, $\F(R)\approx [x,\ldots,x]^{\mathsf T}$,
is fixed once and for all.

For example, if the original cube term identities were
\[
\F(M) = 
\F\left(\begin{matrix}
1&x&2\\
x&y&3\\
\end{matrix}
\right) \approx
\left(\begin{matrix}
x\\
x\\
\end{matrix}
\right),
\]
then after replacing $y$ with, say, the constant symbol $2$,
we obtain
\begin{equation}\label{R}
\F(R) = 
\F\left(\begin{matrix}
1&x&2\\
x&2&3\\
\end{matrix}
\right) \approx
\left(\begin{matrix}
x\\
x\\
\end{matrix}
\right).
\end{equation}
Finally we fix the order of the identities.
For this example, once and for all, 
we declare the `first' cube identity 
to be 
$\F(1,x,2)\approx x$ and the `second' cube identity
to be 
$\F(x,2,3)\approx x$. There will be $k$ such identities
for a $k$-cube term.

We will need a function $\lambda\colon [m]\to [k]$
from the column indices to the row indices with the property
that if $\lambda(j)=i$, then $R$ has a constant symbol
in the $i,j$-th position. Such $\lambda$ exists
because every column of $R$ contains at least
one constant symbol. For the example in (\ref{R}),
one could take $\lambda\colon [3]\to [2]$ to be the function
$\lambda(1)=\lambda(3) = 1, \lambda(2) = 2$.
(This is not the only choice.)

The processing of tuples in $A^n$ will make use of 
an $m$-ary tree which we refer to as
the \emph{(processing) template}. We refer to nodes 
of the template by their
\emph{addresses}, which are finite strings in the alphabet
$[m] = \{1,\ldots,m\}$. The root node 
has empty address, and is denoted
$\wec{n}_{\emptyset}$. If $\wec{n}_{\sigma}$ is 
the node at address $\sigma$, then its children
are the nodes $\wec{n}_{\sigma 1},\ldots, \wec{n}_{\sigma m}$.

Each node $\wec{n}$ of the template is labeled 
by a subset $\ell(\wec{n})\subseteq [n]$.
(Recall that $n$ is the number 
appearing in the exponent of $A^n$.)
%The labels on nodes are meant to record which coordinates
%of a tuple $\wec{a}\in A^n$ have already been processed.)
To define the labeling function $\ell$
we first specify a fixed
method for partitioning some subsets $U\subseteq [n]$.
Given such $U=\{u_1,\ldots,u_r\}$, consider it to be a linearly
ordered set $u_1 < \ldots < u_r$ under the order
inherited from $[n]$. Now, define
$\pi(U) = (U_1,\ldots,U_k)$ to be the ordered partition
of $U$ into $k$ consecutive nonempty intervals that are as
equal sized as possible. That is, let 
\[
(U_1,\ldots,U_k)=
(\{u_1,u_2,\ldots,u_{i_1}\},\{u_{i_1+1},\ldots,u_{i_2}\},\ldots,
\{u_{i_{k-1}+1},\ldots,u_{i_k}=u_r\}),
\]
where 
\[
u_1<u_2<\cdots<u_{i_1}<u_{i_1+1}<\cdots<u_{i_2}<u_{i_{k-1}+1}<\cdots <
u_{i_k}=u_r
\] 
(the cells of the partition are consecutive nonempty intervals)
and $|U_1|\geq \cdots \geq |U_k|\geq |U_1|-1$
(the cells are as equal sized as possible).
The $k$ appearing here as the number of cells of the partition
is the same $k$ as the one in the assumption that
$\F$ is a $k$-cube term.
In order for $\pi(U)$ to be defined, it is necessary
that $|U|\geq k$.

As mentioned earlier, the label on node $\wec{n}_{\sigma}$ 
will be 
some subset $\ell(\wec{n}_{\sigma})\subseteq [n]$.
Recursively define the labels as follows:
\begin{enumerate}
\item $\ell(\wec{n}_{\emptyset})=\emptyset$.
\item If all nodes between 
$\wec{n}_{\sigma}$ and $\wec{n}_{\emptyset}$ are labeled, 
$V$ is the union of labels
occurring between $\wec{n}_{\sigma}$
and the root $\wec{n}_{\emptyset}$, and
$\pi([n]\setminus V) = (U_1,\ldots,U_k)$,
then $\ell(\wec{n}_{\sigma i})= U_{\lambda(i)}$.
\end{enumerate}
In (2),
if $[n]\setminus V$ has fewer than $k$ elements, then 
it is impossible to partition it into $k$ nonempty intervals,
in which case there do not exist sufficiently
many labels for potential
children. In this case, we do not include
any descendants of $\wec{n}_{\sigma}$ in the template.

Let's illustrate our progress with the example 
started back at (\ref{R}).
In this case, the reduced matrix $R$
for $\F$ is $2\times 3$, so $k=2$ and $m = 3$.
%Figure~\ref{fig1} 
The following picture 
depicts the processing template in the case
$n=5$.
\begin{center}
%\begin{figure}
\begin{picture}(300,150)
\setlength{\unitlength}{1mm}

\put(50,50){$\wec{n}_{\emptyset}$}

\put(20,35){$\wec{n}_{1}$}
\put(50,35){$\wec{n}_{2}$}
\put(80,35){$\wec{n}_{3}$}

\put(-10,20){$\wec{n}_{11}$}
\put(5,20){$\wec{n}_{12}$}
\put(20,20){$\wec{n}_{13}$}

\put(35,20){$\wec{n}_{21}$}
\put(50,20){$\wec{n}_{22}$}
\put(65,20){$\wec{n}_{23}$}

\put(80,20){$\wec{n}_{31}$}
\put(95,20){$\wec{n}_{32}$}
\put(110,20){$\wec{n}_{33}$}

\put(35,5){$\wec{n}_{221}$}
\put(50,5){$\wec{n}_{222}$}
\put(65,5){$\wec{n}_{223}$}

\put(50,50){\line(-2,-1){30}}
\put(50,50){\line(0,-1){15}}
\put(50,50){\line(2,-1){30}}

\put(20,35){\line(-2,-1){30}}
\put(20,35){\line(-1,-1){15}}
\put(20,35){\line(0,-1){15}}

\put(50,35){\line(-1,-1){15}}
\put(50,35){\line(0,-3){15}}
\put(50,35){\line(1,-1){15}}

\put(80,35){\line(2,-1){30}}
\put(80,35){\line(1,-1){15}}
\put(80,35){\line(0,-1){15}}

\put(50,20){\line(-1,-1){15}}
\put(50,20){\line(0,-3){15}}
\put(50,20){\line(1,-1){15}}

\color{red}
\put(50,35){\circle*{5}}
\color{black}

\end{picture}
%\caption{The processing template when 
%$k=2$, $m=3$, $n=5$, and $\lambda(1)=\lambda(3)=1$, $\lambda(2)=2$}\label{fig1}
%\end{figure}
\end{center}

Now we define precisely what is meant by processing.
Let $P=\{c_1,\ldots,c_p\}$ be the constant
symbols appearing in the cube identities
for $\F$.
A tuple $\wec{a}\in A^n$ is \emph{processed for node $\wec{n}_{\sigma}$} 
if there is a constant symbol $c\in P$
such that the $i$-th coordinate
of $\wec{a}$ is $c^{\m a}$ for all $i\in \ell(\wec{n}_{\sigma})$.
A tuple $\wec{a}$ is \emph{fully processed} if there is a 
path through the template from the root to a leaf
such that $\wec{a}$ is processed for each node in the path.

Before proceeding, we refine the coarse outline
given in the first paragraph of the proof.
The processing template describes, in reverse order, 
a particular way to generate tuples in $\m a^n$.
Given a tuple $\wec{a}\in A^n$, we assign it
to the root $\wec{n}_{\emptyset}$ and denote it 
$\wec{a}_{\emptyset}$. This tuple 
$\wec{a} = \wec{a}_{\emptyset}$ is already processed for
$\wec{n}_{\emptyset}$, since this is an empty requirement.
Now, for each address $\sigma$ of a node in the template, 
we will construct $\wec{a}_{\sigma 1}, \ldots, \wec{a}_{\sigma m}$
from $\wec{a}_{\sigma}$ so that 
(i)~$\F^{\m a}(\wec{a}_{\sigma 1}, \ldots, \wec{a}_{\sigma m}) = \wec{a}_{\sigma}$,
and (ii) 
each $\wec{a}_{\sigma i}$ is processed at
all nodes between $\wec{n}_{\sigma i}$ and $\wec{n}_{\emptyset}$.
Assign $\wec{a}_{\sigma i}$ to $\wec{n}_{\sigma i}$.
The original tuple $\wec{a}$ can be generated via
$\F^{\m a}$ by the fully processed tuples derived from
$\wec{a}$ in this way. The following claim is the heart of this argument.

%be assigned to the root of the template. This tuple
%``divides'' into $m$ tuples $\wec{a}_1,\ldots,\wec{a}_m$
%assigned to the children 
%$\wec{n}_1,\ldots,\wec{n}_m$ of the root, and 
%each $\wec{a}_i$ is ``processed'' at the coordinates
%in $\ell(\wec{n}_i)$.
%Division and processing continues down 
%the tree until one obtains ``fully processed''
%tuples at the leaves.\footnote{``Fully processed'' is 
%slight a misnomer,
%since the only processed coordinates of 
%$\wec{a}_{\sigma}$ are those
%that appear in labels in the branch between
%$\wec{n}_{\sigma}$ and the root. At the leaves
%a small number of coordinates remain unprocessed.}

\begin{clm}
Suppose that $\wec{n}_{\sigma}$ is an internal
node of the processing template.
Given an arbitrary tuple $\wec{a}\in A^n$, 
there exist tuples
$\wec{b}_{1}, \ldots, \wec{b}_{m}$ such that 
\begin{enumerate}
\item
$\F^{\m a}(\wec{b}_{1}, \ldots, \wec{b}_{m}) = \wec{a}$.
\item 
$\wec{b}_{i}$ is processed for node $\wec{n}_{\sigma i}$ for 
$i=1,\ldots,m$.
\item 
If $\wec{n}$ is a node between 
$\wec{n}_{\sigma}$ and 
$\wec{n}_{\emptyset}$, and $\wec{a}$ is processed
for $\wec{n}$, then each
$\wec{b}_{i}$ is also processed for $\wec{n}$
for $i=1,\ldots,m$.
\end{enumerate}
\end{clm}

Let $V$ be the union of labels on nodes
between $\wec{n}_{\sigma}$ and $\wec{n}_{\emptyset}$.
If $\pi([n]\setminus V) = (U_1,\ldots,U_k)$,
then $\{V,U_1,\ldots,U_k\}$ is a partition of $[n]$.
For simplicity of expression, reorder coordinates
so that $\wec{a}$ and $\wec{b}_i$ can be written
$[\wec{a}_V, \wec{a}_{U_1}, \ldots, \wec{a}_{U_k}]^{\mathsf T}$
and
$[\wec{b}_{i,V}, \wec{b}_{i,U_1}, \ldots, \wec{b}_{i,U_k}]^{\mathsf T}$,
with coordinates from $V$ or $U_j$ grouped together.
Given $\wec{a}$, we need to solve for $\wec{b}_{i,V}$
and $\wec{b}_{i,U_j}$ in 
\begin{equation}\label{ddddddd}
\small{
\F^{\m a}(\wec{b}_1,\ldots,\wec{b}_m)=
\F^{\m a}\left( 
\left[
\begin{array}{c}
\wec{b}_{1,V}\\
\wec{b}_{1,U_1}\\
\vdots\\
\wec{b}_{1,U_k}\\
\end{array}
\right],
\ldots,
\left[
\begin{array}{c}
\wec{b}_{m,V}\\
\wec{b}_{m,U_1}\\
\vdots\\
\wec{b}_{m,U_k}\\
\end{array}
\right]
\right) = 
\left[
\begin{array}{c}
\wec{a}_{V}\\
\wec{a}_{U_1}\\
\vdots\\
\wec{a}_{U_k}\\
\end{array}
\right]
=\wec{a}
}
\end{equation}
in order to satisfy item (1) of the claim.
We shall do using the first cube identity in the
$V$-coordinates and the $U_1$-coordinates, and the 
$i$-th cube identity in the $U_i$-coordinates.

Whether $W=V$ or $W=U_i$, to solve 
$\F^{\m a}(\wec{b}_{1,W},\ldots,\wec{b}_{m,W})=\wec{a}_{W}$
for the $\wec{b}_{i,W}$'s using a particular cube identity,
take $\wec{b}_{i,W}=\wec{a}_W$ if there is an $x$ in the $i$-th
place of $F$ in the cube identity, and take 
$\wec{b}_{i,W}=[c^{\m a},\ldots,c^{\m a}]^{\mathsf T}$
if there is a $c$ in the $i$-th place
of the cube identity. It is not hard to see that this works, and so
(1) holds.

The label on node $\wec{n}_{\sigma i}$ is $U_{\lambda(i)}$.
The element $\lambda(i)\in [k]$ is the number of a
cube identity which has some constant symbol
$c\in P$ in the $i$-th place of $F$.
Hence $\wec{b}_{i,U_{\lambda(i)}} = [c^{\m a},\ldots,c^{\m a}]^{\mathsf T}$.
Thus $\wec{b}_i$ is processed for node $\wec{n}_{\sigma i}$,
establishing (2).

If, in the first cube identity, there is an $x$ in the $i$-place
of $F$, then $\wec{b}_{i,V} = \wec{a}_V$. If there is a constant
symbol $c\in P$ in the $i$-th place of $F$, then 
$\wec{b}_{i,V} = [c^{\m a},\ldots,c^{\m a}]^{\mathsf T}$.
In the latter case, $\wec{b}$ is processed at all 
coordinates in $V$, hence at all nodes between 
$\wec{n}_{\sigma}$ and $\wec{n}_{\emptyset}$. In the former
case, $\wec{b}_i$ is processed at any
node between 
$\wec{n}_{\sigma}$ and $\wec{n}_{\emptyset}$ where $\wec{a}$
is processed, since $\wec{b}_{i,V} = \wec{a}_V$. 
In either case, (3) holds.
The claim is proved.

\medskip

%The claim shows that we can start with an arbitrary tuple
%$\wec{a}\in A^n$ and generate it from more highly
%processed tuples assigned to its children. This can be propagated
%through the template, resulting in the conclusion that 
%the set of fully processed tuples generates $\m a^n$.

Let's estimate the number of fully processed tuples
by estimating the number of descriptions of such tuples.
A tuple $\wec{a}$ is fully processed if there is a 
path through the template from the root to a leaf
such that $\wec{a}$ is processed for each node in the path.
For $\wec{a}$ to be processed at a node $\wec{n}$, 
there must be a constant symbol $c\in P$
such that the $i$-th coordinate
of $\wec{a}$ is $c^{\m a}$ for all $i\in \ell(\wec{n})$.
So, beginning at the root, descend 
through the template choosing a new child $\wec{n}_{\sigma i}$ and 
a new constant symbol to assign to the coordinates
in $\ell(\wec{n}_{\sigma i})$ at each step.
Each such step can be accomplished in $mp$ different ways,
since each node has $m$ children and there are $p$ constant symbols
to choose from.
If the length of a longest path in the template is 
$r$, we will produce at most $(mp)^r$ partial descriptions of
fully processed tuples when we arrive at the leaves of the template.
But there may be $k-1$ elements of $[n]$ that were not encountered
in the labels of the maximal path, so all but perhaps
$k-1$ coordinates of a typical fully processed
tuple have been described. 
By filling in the last $k-1$ coordinates
randomly, one sees that there are at most 
$(mp)^r|A|^{k-1}$ fully processed tuples.
What remains is to estimate $r$,
the length of the longest branch in the processing template.

Let $V_0 = \ell(\wec{n}_{\emptyset}) = \emptyset$.
This represents the set of coordinate positions that have been
processed before the processing begins, i.e., no 
coordinate positions. As we progress down a longest
branch in the template, $\wec{n}_{\emptyset}, 
\wec{n}_i, \wec{n}_{ij},\ldots, \wec{n}_{\sigma}$,
we may construct sets $V_{\sigma i} = V_{\sigma}\cup \ell(\wec{n}_{\sigma i})$,
where $V_{\sigma}$ represents the set of coordinate positions that have been
processed along this branch from $\wec{n}_{\emptyset}$ to $\wec{n}_{\sigma}$.
The unprocessed coordinate positions, $[n]\setminus V_{\sigma}$
are then divided evenly, 
$\pi([n]\setminus V_{\sigma}) = (U_1,\ldots,U_k)$, to appear
as labels of the children of $\wec{n}_{\sigma}$.
Thus, $|V_{\emptyset}|=0$ and
\begin{equation}\label{WWW}
|V_{\sigma i}| = |V_{\sigma}\cup \ell(\wec{n}_{\sigma i})|=
|V_{\sigma}|+|\ell(\wec{n}_{\sigma i})|.
\end{equation}
The useful parameter is the number 
$u_{\sigma}:=
|[n]\setminus V_{\sigma}|=n- |V_{\sigma}|$ 
of nodes that remain unprocessed 
after reaching $\wec{n}_{\sigma}$.
This parameter satisfies
$u_{\emptyset} = |[n]\setminus V_{\emptyset}|=n$ and, from (\ref{WWW}),
\begin{equation}\label{XXX}
u_{\sigma i} = (n - |V_{\sigma i}|) = 
(n-|V_{\sigma}|)-|\ell(\wec{n}_{\sigma i})| = 
u_{\sigma}-|\ell(\wec{n}_{\sigma i})|.
\end{equation}
Since 
$\pi([n]\setminus V_{\sigma}) = (U_1,\ldots,U_k)$ is an even division
of $[n]\setminus V_{\sigma}$ into $k$ sets, and  
$\ell(\wec{n}_{\sigma i})=U_{\lambda(i)}$, we get 
\begin{equation}\label{YYY}
|\ell(\wec{n}_{\sigma i})|=|U_{\lambda(i)}|\geq 
\lfloor(n-|V_{\sigma}|)/k\rfloor = 
\lfloor u_{\sigma}/k\rfloor.
\end{equation}
Combining (\ref{XXX}) and (\ref{YYY}) we have
\[
u_{\sigma i} 
\leq 
u_{\sigma} 
- \lfloor u_{\sigma}/k\rfloor = 
\left\lceil\left(\frac{k-1}{k}\right)u_{\sigma}\right\rceil.
\]
In order to avoid considering truncation error, we
use the following fact, whose proof we leave to the reader.
\begin{clm}
If $u\geq k\geq 1$, then 
$\left\lceil \left(\frac{k-1}{k}\right)u\right\rceil\leq
\left(\frac{2k-1}{2k}\right)u
$.
\end{clm}
Hence
\[
u_{\sigma i} 
\leq 
\left(\frac{2k-1}{2k}\right)u_{\sigma}
\]
for each $\sigma$, and therefore 
\[
u_{\sigma} 
\leq 
\left(\frac{2k-1}{2k}\right)^{|\sigma|}u_{\emptyset} =
\left(\frac{2k-1}{2k}\right)^{|\sigma|}n
\]
for each $\sigma$.
If, for some $r$, it happens that 
$\left(\frac{2k-1}{2k}\right)^{r}n < k$,
then there are fewer than $k$ unprocessed nodes
at address $\sigma$ for any $\sigma$
satisfying $|\sigma|\geq r$. Such an 
$r$ is an upper bound on the length of 
paths through the template.

Solving $\left(\frac{2k-1}{2k}\right)^{r}n < k$
for $r$ we obtain that any $r > \log_w(n/k)$, 
$w = \frac{2k}{2k-1}$, is an upper bound
on the length of paths in the template;
hence $r=\log_w(n/k)+1$ is such a bound.
Combining this with our earlier 
number $(mp)^r|A|^{k-1}$ estimating the number
of fully processed tuples, we get that
the number of such tuples is no more than
\[
\begin{array}{rl}
(mp)^{\log_w(n/k)+1}|A|^{k-1} &= 
(mp)^{\log_w(n/k)}(mp)|A|^{k-1}\\
&= (n/k)^{\log_w(mp)}(mp)|A|^{k-1}\\ 
&= n^{\log_w(mp)}(k^{-\log_w(mp)}(mp)|A|^{k-1})\in O(n^{\log_w(mp)}).
\end{array}
\]
\end{proof}

This theorem deals only with the case $p\geq 1$.
We describe next how to refine the estimate
in the case $p=1$ and how to derive the result
for $p=0$ from the $p=1$ case.

\begin{cor}\label{pointed_polynomial_cor}
If $\m a$ is a finite algebra with a $0$-pointed
or $1$-pointed $k$-cube term, then $d_{\m a}(n)\in O(n^{k-1})$. 
%is bounded above by 
%a polynomial in $n$ of degree at most $k-1.$
\end{cor}

\begin{proof}
Suppose that $\m a$ has a $1$-pointed $k$-cube term,
and that $c$ is the one constant that appears
among the cube identities. Then a fully processed tuple
$\wec{a}$ has $c^{\m a}$ in every processed coordinate position,
and has at most $k-1$ unprocessed coordinate positions.
Hence the set of tuples with a 
$c^{\m a}$ in all but at most $k-1$ positions 
contains all the fully processed tuples, and therefore
is a generating set for $\m a^n$. The number of such tuples
is 
\[
\binom{n}{k-1}|A|^{k-1}\in O(n^{k-1}).
\]

Now suppose that $F(x_1,\ldots,x_m)$ is a $0$-pointed $k$-cube term
of $\m a$. Suppose that the cube identities are
\begin{equation}\label{F}
\F(M)\approx \left(\begin{array}{c} x\\ \vdots\\x\end{array}\right).
\end{equation}
If $\m b$ is the polynomial expansion of $\m a$, then
we can choose an element $1\in B$ and replace all variables
other than $x$ in (\ref{F}) with $1$ to obtain 
identities witnessing that $F(x_1,\ldots,x_m)$
is a $1$-pointed $k$-cube term for $\m b$.
Hence $d_{\m b}(n)\in O(n^{k-1})$ by the earlier
part of the argument. Now 
$d_{\m a}(n)\in O(n^{k-1})$ by 
Theorem~\ref{basic_estimates}~(4).
\end{proof}

In Section~\ref{comb} we shall improve this result by showing
that Wiegold dichotomy holds for algebras with a 
$0$-pointed $k$-cube term. In particular,
this means that growth rates are at most
linear for such algebras.

Let's combine the results of this subsection 
with the results of the previous subsection.

\begin{thm}
The following are equivalent for a set 
$\Sigma$ of basic identities in which only finitely
many constant symbols occur. 
\begin{enumerate}
\item[(1)] $\Sigma$ is restrictive.
\item[(2)] The variety axiomatized by $\Sigma$
has a pointed cube term.
\item[(3)] The variety axiomatized by $\Sigma$
has a pointed cube term
of the form $F(x_1,\ldots,x_m)$, 
where $m>1$, $F$ is a function
symbol occurring in $\Sigma$, and 
the variables $x_1,\ldots,x_m$ are distinct.
\item[(4)] If $\m a$ is a finite algebra realizing $\Sigma$,
then $d_{\m a}(n)$ is bounded above by a polynomial.
\item[(5)] There is no finite algebra $\m a$ realizing
$\Sigma$ such that $d_{\m a}(n) = 2^n$ for all $n$.
\end{enumerate}
\end{thm}


\begin{proof}

[$(1)\Rightarrow(2)$] Theorem~\ref{nonrestrictive_thm}.

[$(2)\Leftrightarrow(3)$] Lemma~\ref{nonrestrictive}.

[$(2)\Rightarrow(4)$] Theorem~\ref{pointed_polynomial}
and Corollary~\ref{pointed_polynomial_cor}.

[$(4)\Rightarrow(5)$] $d_{\m a}(n) = 2^n$ is not bounded
above by a polynomial.

[$(5)\Rightarrow(1)$] There exists a finite algebra
$\m a$ with $d_{\m a}(n) = 2^n$, namely the $2$-element set
equipped with no operations.
\end{proof}


\subsection{Finite algebras with polynomial growth}\label{poly_growth}

In this subsection we prove that the bound on growth
rates for finite algebras with $1$-pointed
$k$-cube terms, which we established in
Corollary~\ref{pointed_polynomial_cor},
is sharp. 

\begin{thm}\label{example}
For each $k\geq 2$ there is a finite algebra 
with a $1$-pointed $k$-cube term whose growth rate
satisfies $d_{\m a}(n)\in \Theta(n^{k-1})$.
\end{thm}

\begin{proof}
We shall first construct a partial 
with the desired growth rate, 
then modify it slightly to obtain a total algebra
satisfying the hypotheses of the theorem.

The universe of the partial
algebra will be $A = \{a_1,\ldots,a_q,1\}$.
We equip this set with a partial
$k$-ary operation $F$ which satisfies
\[
F^{\m a}(1,x,\ldots,x,x) =
F^{\m a}(x,1,\ldots,x,x) = 
\cdots =
F^{\m a}(x,x,\ldots,x,1) = x
\]
for each $x\in A$, and which is undefined otherwise.
Thus, $F^{\m a}$ is a partial near unanimity operation
that is defined only on the nearly unanimous tuples
where the lone dissenter is $1$ and on the tuple
whose entries are unanimously $1$.
Set $\m a = \langle A; F\rangle$. \marginpar{$\langle A;\{F\}\rangle$?}

We shall prove the exact formula
\begin{equation}\label{d_formula}
d_{\m a}(n)=
\binom{n}{0} + q\binom{n}{1} + q^2\binom{n}{2} + \cdots + q^{k-1}\binom{n}{k-1}
\end{equation}
for this partial algebra,
which is a polynomial in $n$ of degree $k-1$, since 
$k = {\rm arity}(F)$ and 
$q=|A|-1$ are fixed. 
This will show that $\m a$ is an $(q+1)$-element
partial algebra with $d_{\m a}(n)\in\Theta(n^{k-1})$.
(When $q=1$ we will obtain a $2$-element partial algebra
with $d_{\m a}(n)\in\Theta(n^{k-1})$.)

Choose and fix $n$. Define
the \emph{support} of a tuple $\wec{a}\in A^n$ to be the 
subset $\supp(\wec{a})\subseteq [n]$ 
consisting of indices $s$ where $a_s\neq 1$.
The proof involves showing that the set of all tuples
whose support has size at most $k-1$
is the unique minimal generating set for $\m a^n$.
%To set up language for the argument, call a tuple 
%$\wec{b}\in A^n$ an \emph{essential generator}
%if it is contained in any generating set for $\m a^n$.
%%%This terminology was used much earlier%%%%%

\begin{clm}\label{support}
If $S\subseteq [n]$ and $G\subseteq A^n$, then let $G_S$ denote the tuples
in $G$ that have support contained in $S$.
If $\wec{a}\in \lb G\rb$, then $\wec{a}\in \lb G_S\rb$.
\end{clm}

In $\m a$, we have 
\[
F^{\m a}(x_1,\ldots,x_k) = 1\Longleftrightarrow x_1 = x_2 = \cdots = x_k = 1.
\]
Hence, in $\m a^n$, 
if $F^{\m a^n}(\wec{g}_1,\ldots,\wec{g}_k)$
is defined and equal to $\wec{b}$,
then $i\notin \supp(\wec{b})$ if and only if $i\notin \supp(\wec{g}_i)$
for any $\wec{g}_i$. Equivalently, 
\begin{equation}\label{generate}
\supp(F^{\m a^n}(\wec{g}_1,\ldots,\wec{g}_k)) = \bigcup_{i=1}^k \supp(\wec{g}_i)
\end{equation}
whenever $F^{\m a^n}(\wec{g}_1,\ldots,\wec{g}_k)$ is defined.
Now let $G(0) = G$, $G_S(0)=G_S$,
$G(j+1) = G(j)\cup F^{\m a^n}(G(j),\ldots,G(j))$, and
$G_S(j+1) = G_S(j)\cup F^{\m a^n}(G_S(j),\ldots,G_S(j))$.
By induction on $j$, using (\ref{generate}),
it can be shown that any tuple in $G(j)$ that has support
in $S$ lies in $G_S(j)$. Since $\lb G\rb = \bigcup_j G(j)$
and $\lb G_S\rb = \bigcup_j G_S(j)$, 
any tuple in $\lb G\rb$ with support in $S$ lies in $\lb G_S\rb$.
%this proves the claim.

\medskip

\begin{clm}\label{empty}
The tuple $\hat{1}=[1,1,\ldots,1]^{\mathsf T}$ of empty support
is an essential generator.
\end{clm}

This follows immediately from Claim~\ref{support}.

\medskip

\begin{clm}\label{supp}
Any tuple whose support has size at most $k-1$
is an essential generator of $\m a^n$.
\end{clm}

Let $\wec{b}\in A^n$ be a tuple of support $S$
where $1\leq |S|\leq k-1$. Without loss of generality,
$S = [\ell]  =\{1,\ldots,\ell\}$ for some $1\leq \ell\leq k-1$. 
In order to obtain a contradiction to the claim,
assume that $\wec{b}$ is not an essential generator.
Then $\wec{b}$ can be generated by elements different from
$\wec{b}$, so
the equation $F^{\m a^n}(\wec{x}_1,\ldots,\wec{x}_k) = \wec{b}$
can be solved for the $\wec{x}_i$ in such a way
that $\wec{b}\notin \{\wec{x}_1,\ldots,\wec{x}_k\}$.
Moreover, by (\ref{generate}), the $\wec{x}_i$'s must be taken
from the tuples whose support is contained in $S$.
The equation to be solved is therefore:
\begin{equation}\label{dividers}
\small{
F^{\m a^n}(\wec{x}_1,\ldots,\wec{x}_k)=
F^{\m a^n}\left( 
\left[
\begin{array}{c}
x_{1,1}\\
\vdots\\
\underline{\;\;x_{\ell,1}\;\;}\\
1\\
\vdots\\
1\\
\end{array}
\right],
%\left[
%\begin{array}{c}
%x_{1,2}\\
%\vdots\\
%\underline{\;\;x_{\ell,2}\;\;}\\
%1\\
%\vdots\\
%1\\
%\end{array}
%\right],
\ldots,
\left[
\begin{array}{c}
x_{1,k}\\
\vdots\\
\underline{\;\;x_{\ell,k}\;\;}\\
1\\
\vdots\\
1\\
\end{array}
\right]
\right) = 
\left[
\begin{array}{c}
b_1\\
\vdots\\
\underline{\;\;\;b_{\ell}\;\;\;}\\
1\\
\vdots\\
1\\
\end{array}
\right]=\wec{b}.
}
\end{equation}
We have introduced horizontal segments
as dividers separating the coordinates
in $S = [\ell]$ from the
remaining coordinates in order
to make the argument clearer.
Since $F^{\m a^n}(\wec{x}_1,\ldots,\wec{x}_k)$ is defined,
every row above the dividers
is a nearly unanimous row with 
exactly one $1$. Hence there are exactly
$\ell$ $1$'s above the dividers.
This means that there are at most $\ell$ columns which
contain a $1$ above the dividers. Since there are
$k$ such columns, and $k>\ell$, there is a column
$\wec{x}_j$ that contains no $1$ above the dividers.
Since the $i$-th row above the dividers is nearly
unanimous with majority value $b_i$, the column $\wec{x}_j$
which contains no $1$'s above the dividers
is exactly $\wec{b}$. This contradicts the assumption that
$\wec{b}\notin \{\wec{x}_1,\ldots,\wec{x}_k\}$,
showing that $\wec{b}$ is indeed an essential generator.

%\medskip
%
%The previous claim shows that every generating subset
%of $\m a^n$ contains all tuples whose support 
%has size at most $k$. The next claim proves that
%the set of these tuples is a generating
%set, hence is the unique minimal generating set for $\m a^n$.

\begin{clm}
$\m a^n$ is generated by the tuples whose support 
has size at most $k-1$.
\end{clm}

It is enough to show that if $\wec{b}$ has support
$S$ of size $\ell \geq k$, then $\wec{b}$ can be generated
from tuples whose support is properly contained in $S$.
It is enough to prove this in the case where $S = [\ell]$.
For this we must explain how to solve
\begin{equation}\label{dividers2}
\small{
F^{\m a^n}(\wec{x}_1,\ldots,\wec{x}_k)=
F^{\m a^n}\left( 
\left[
\begin{array}{c}
x_{1,1}\\
\vdots\\
\underline{\;\;x_{\ell,1}\;\;}\\
1\\
\vdots\\
1\\
\end{array}
\right],
%\left[
%\begin{array}{c}
%x_{1,1}\\
%\vdots\\
%\underline{\;\;x_{\ell,1}\;\;}\\
%1\\
%\vdots\\
%1\\
%\end{array}
%\right],
\ldots,
\left[
\begin{array}{c}
x_{1,k}\\
\vdots\\
\underline{\;\;x_{\ell,k}\;\;}\\
1\\
\vdots\\
1\\
\end{array}
\right]
\right) = 
\left[
\begin{array}{c}
b_1\\
\vdots\\
\underline{\;\;\;b_{\ell}\;\;\;}\\
1\\
\vdots\\
1\\
\end{array}
\right]=\wec{b}
}
\end{equation}
when $\ell \geq k$ in such a way that every 
column contains at least one $1$ above the dividers
and the $i$-th row above the dividers is nearly unanimously
equal to $b_i$. This is easy to do.
Set $x_{1,1} = \cdots = x_{k,k} = 1$, then 
put exactly one $1$ arbitrarily in each 
of rows $k+1$ to $\ell$, then fill
in the remaining entries above the dividers
so that the $i$-th row above the dividers is nearly unanimously
equal to $b_i$. 

\medskip

We have established up to this point that the set of tuples
of support of size at most $k-1$ is the unique minimal
generating set for $\m a^n$.
To complete the proof that the partial algebra $\m a$ has the specified
growth rate, observe that the number of tuples with 
support $S$ is $(|A|-1)^{|S|} = q^{|S|}$, so the number of tuples
whose support has size $i$ is $q^i\binom{n}{i}$.
This yields the formula $d_{\m a}(n)=\sum_{i=0}^{k-1} q^i\binom{n}{i}$. 

The one-point completion, $\m a_0$, is a total algebra
with the same growth rate as $\m a$. Let $\m b$
be the expansion of $\m a$ by one constant symbol
$1$ whose interpretation is $1^{\m b} = 1$.
The operation $F^{\m b}$ still
satisfies
\[
F^{\m b}(1,x,\ldots,x,x) =
F^{\m b}(x,1,\ldots,x,x) = 
\cdots =
F^{\m b}(x,x,\ldots,x,1) = x
\]
for each $x\in A_0$, so it is a $1$-pointed
$k$-cube term for $\m b$.

$\m a^n$ and $\m a_0^n$ have the same unique minimal generating
set, $G$, which is the set of all tuples with support at most $k-1$; 
this set contains $\hat{1}$.
%$[1,\ldots,1]^{\mathsf T}$.
The algebra $\m b$ must also have a unique minimal generating
set, namely the set obtained from $G$
by deleting $\hat{1}=1^{\m b^n}$.
%=[1,\ldots,1]^{\mathsf T}$.
Thus $d_{\m b}(n) = d_{\m a}(n)-1 = \sum_{i=1}^{k-1} q^i\binom{n}{i}\in O(n^{k-1})$. 
\end{proof}

\subsection{Exponential growth}\label{exp}


\section{Wiegold dichotomy holds for algebras with a cube term}\label{comb}

In this section we investigate growth rates of finite algebras
with a ($0$-pointed) cube term. Unlike the previous section,
where we established only an upper bound on growth rates, 
we will prove that growth rates must be one of two kinds:
if $\m a$ is a finite algebra with a cube term, then 
$d_{\m a}(n)\in \Theta(\log(n))$ if $\m a$ is perfect
and $d_{\m a}(n)\in \Theta(n)$ if $\m a$ is imperfect.

Our approach will be through the analysis of the maximal 
subuniverses of $\m A^n$. The relevance
of maximal subuniverses may be recognized from the fact
that a subset $G\subseteq A^n$
is a generating set for $\m a^n$
if and only if $G\not\subseteq M$
for any maximal subuniverse $M$ of $\m a^n$.
Our analysis of a maximal subuniverse $M$
of $\m a^n$ involves finding surjective homomorphisms
$\varphi\colon \m a^n\to \m b$ such that $\varphi(M)\neq B$.
When $\varphi$ is 
such a homomorphism, $M_{\varphi}:=\varphi(M)$ will
be a maximal subuniverse of $\m b$ and 
$M$ will equal $\varphi^{-1}(M_{\varphi})$
(since $\varphi^{-1}(M_{\varphi})$ is a proper
subuniverse of $\m a^n$ that contains $M$).
We say that $M$ is \emph{induced} by $\varphi$
in this situation.

We will consider two types of homomorphisms.
The first type of homomorphism considered
is the projection 
$\pi_U\colon \m a^n \to \m a^U$ 
onto the coordinates in $U$ for some
subset $U\subseteq [n]$.
The second type will be the 
homomorphism $\eta\colon \m a^n\to (\m a/{[1,1]})^n$
induced by the abelianization map 
%$\m a\to \m a/{[1,1]}$ 
in each coordinate.

Our first main result,
Theorem~\ref{induced},
states that if $\m a$ has a $0$-pointed
$k$-cube term, then every maximal subuniverse is induced
by either (i) a projection $\pi_U$ onto a set
of coordinates of size $|U|<\max\{3,k\}$, or (ii) the 
%abelianization 
homomorphism $\eta$. The second main result
is that there exists a set $G_{\pi}\subseteq A^n$ of size
$O(\log(n))$ that is contained
in no maximal subuniverse which can be induced by a type (i) map,
and there exists a set $G_{\eta}\subseteq A^n$ of size $O(n)$
that is contained in no maximal subuniverse 
which can be induced by a type (ii) map. 
It follows that $G_{\pi}\cup G_{\eta}$ is contained in no
maximal subuniverse at all, so it
is a generating set of size $O(n)$ for $\m a^n$.
When $\m a$ is imperfect, the $O(n)$-estimate is 
asymptotically optimal, since it is proved
in Corollary~\ref{abelian_cor}~(2) that 
$d_{\m a}(n)\in \Omega(n)$ when $\m a$ is imperfect.
When $\m a$ is perfect, $\eta$ is constant, 
hence no maximal subuniverse of $\m a^n$ can be induced by $\eta$.
It follows that $G_{\pi}$ is a generating set of size $\Theta(\log(n))$
for $\m a^n$, which is asymptotically optimal by Theorem~\ref{first_bounds}.

***ref relevant theorems****

\subsection{Maximal subuniverses}\label{max_sec}

In this subsection we relate arbitrary maximal subuniverses
of $\m a^n$ to critical maximal subuniverses.

\begin{dfs} 
\text{\cite{parallelogram}}
A \emph{compatible $n$-ary relation} of $\m a$
is a subuniverse of $\m a^n$.

A compatible relation $R$ is \emph{critical} if it is completely
$\cap$-irreducible in the subalgebra lattice of $\m a^n$ and
directly indecomposable as a relation. 
(The latter means that $R$ is not of the form 
$S\times T$ for subsets $S\subseteq A^U$
and $T\subseteq A^V$, where $\{U,V\}$ is a partition
of $[n]$ into two cells.)
\end{dfs}

Any maximal subuniverse $M$ of $\m a^n$ is 
completely $\cap$-irreducible in the subalgebra
lattice of $\m a^n$, so a
critical maximal subuniverse of $\m a^n$
is just a maximal subuniverse
that is directly indecomposable as a relation.
%These can be described in terms of the following
%notion.
%To express the relationship between arbitrary
%and critical maximal subuniverses we introduce ...

\begin{df}
If $M$ is a subuniverse of $\m a^n$, then a 
\emph{support} of $M$ is a subset $U\subseteq [n]$
%that is minimal under inclusion for the property
such that $\pi_U(M)\neq A^U$.
%where ${\rm pr}_S\colon \m a^n\to \m a^S$ is the projection map.
\end{df}

\begin{lm}
If $M$ is a maximal subuniverse of $\m a^n$, then $M$
has a unique minimal support. If $U$ is the 
minimal support of $M$,
then $M_U:=\pi_U(M)$ is a critical maximal subuniverse
of $\m a^U$, $M=M_U\times A^V$ for $V=[n]\setminus U$, and
$M$ is induced by the projection $\pi_U$.
In particular, $M$ itself
is critical if and only if its 
unique support is $[n]$.
\end{lm}

\begin{proof}
Assume that $U, V\subseteq [n]$ are distinct minimal supports
of the maximal subuniverse $M\leq \m a^n$.
$U$ and $V$ must be incomparable under inclusion.
Let $M_U=\pi_U(M)$ and $M_V=\pi_V(M)$.

We shall view elements of $\m a^n$ as functions
from $[n]$ to $A$. In this language, 
$M$ is a proper subset of the set of all functions,
$M_U$ is the set of restrictions to $U$ of the functions
in $M$, and 
$M_V$ is the set of restrictions to $V$ of the functions
in $M$. 
Since $M_U\neq A^U$, there is a function $f\colon U\to A$ 
that is not in $M_U$. Since $V$ is a minimal support
and $U\cap V$ is properly contained in $V$, it follows that every 
function from $U\cap V$ to $A$ is the restriction
of some function in $M$. In particular, 
$f|_{U\cap V}=g|_{U\cap V}$ for some $g\in M$.
Let $h\in A^n$ be any function that agrees with
$f$ on $U$ and $g$ on $V$. 
Then $h|_U = f\notin M_U$, so $h\notin M$.
Yet $h|_V = g|_V\in M_V$, so $h\in M$, a contradiction.
This shows that $M$ has a unique minimal support.

If $U$ is the  minimal support of $M$
and $V=[n]\setminus U$, then $M=M_U\times A^V$ is induced by $\pi_U$.
(This does not require minimality of $U$, 
only that $U$ is a support and that $M$ is a maximal
subuniverse.)
To show that $M_U$ is a critical maximal subuniverse
of $\m a^U$, observe that $M_U\leq \m a^U$ is a maximal subuniverse
since it is the image of one under the surjective homomorphism $\pi_U$.
If $M_U = S\times T$, where
$S\leq \m a^X$ and $T\leq \m a^Y$ for some partition
$\{X,Y\}$ of $U$, then either 
$A^X\neq \pi^{\m a^U}_X(M_U) = \pi^{\m a^n}_X(M)$, or 
$A^Y\neq \pi^{\m a^U}_Y(M_U) = \pi^{\m a^n}_Y(M)$.
Either way, one obtains that $X$ or $Y$ is a proper
subset of $U$ that is a support of $M$, contradicting
the minimality of $U$.

For the final statement of the lemma, if the minimal support of 
$M$ is $[n]$, then $\pi_{[n]}(M)=M$ is critical by the second
statement of the lemma.
Conversely, assume that $M$ is critical and $U\subseteq [n]$
is its minimal support. Since $M = M_U\times A^V$
and $M$ is directly indecomposable
as a relation, we get $V = \emptyset$, equivalently $[n]=U$.
\end{proof}





\subsection{The parallelogram property for critical relations}\label{parallel}

In the preceding subsection we showed that all maximal subuniverses
of $\m a^n$ are induced by critical maximal subuniverses
on projections $\m a^U$ of $\m a^n$. In this section
we show that the critical maximal subuniverses of $\m a^U$
have a special structure when $\m a$ has a $0$-pointed
$k$-cube term.

\begin{df} 
\text{\cite{parallelogram}}
Given a partition $\{S, T\}$ of $[n]$ into two cells,
write $\wec{x}\wec{y}$ for a tuple in $A^n$ to mean that 
$\wec{x}\in A^S$ and $\wec{y}\in A^T$.
A compatible $n$-ary relation $R$ satisfies the 
\emph{parallelogram property} if, for any partition
$\{S, T\}$ of $[n]$, 
$\wec{a}\wec{u}, \wec{a}\wec{v}, \wec{b}\wec{v}\in R$
implies $\wec{b}\wec{u}\in R$.
\end{df}

Theorem~3.5 and Theorem~3.6~(3) of \cite{parallelogram} together prove the
following theorem.

\begin{thm}
A variety $\mathcal V$ has a $0$-pointed $k$-cube term
if and only if every member $\m a\in \mathcal V$
has the property that any critical relation of $\m a$
of arity at least $k$ has the parallelogram property.
\end{thm}

It follows from this theorem that if $\m a$ has 
a $0$-pointed $k$-cube term, and $M$ is a maximal
subuniverse of $\m a^n$, then either
$M$ is induced by a relation of arity less than $k$
or $M=M_U\times A^V$ is induced by a critical maximal
subuniverse $M_U\leq \m a^U$ that has the parallelogram property.
(In the latter case, $M$ itself will also have the parallelogram
property.) Our next step is to investigate the structure
of maximal subuniverses with the parallelogram property.

The paper \cite{parallelogram} analyzes arbitrary
compatible relations with the parallelogram property
in congruence modular varieties. It is shown in
\cite{bimmvw} that any algebra with a $0$-pointed
$k$-cube term generates a congruence modular
variety, so the results of \cite{parallelogram} apply here.
The first step in the analysis is the ``reduction''
of a relation, which we describe next.

Suppose that $R\leq \m a^n$ is a compatible relation with
the parallelogram property; as a special case, suppose
that $M\leq \m a^n$ is a maximal critical subuniverse 
with the parallelogram property.
For the first step in the reduction, 
realize $R$ as a subdirect product
$R\leq_{\text{sd}} \prod_{i=1}^n \m a_i$, 
where $\m a_i:= \pi_i(R)\leq \m a$.
In the special case involving the maximal subuniverse
$M$ we will have 
$\m a_i= \pi_i(M) =  \m a$ unless the projection of $M$
onto one single coordinate is not surjective.
This happens only if $M$ has a support of size one,
which, by criticality, implies that $M$ is a unary relation.
We henceforth consider only $M$
of arity at least two, so that in our special case 
$\pi_i(M)=A$ for all $i$. Thus, in the first step in reduction,
nothing happens if $M$ is maximal and of arity greater than one.

Second, define relations, called \emph{coordinate kernels}
in \cite{parallelogram}, 
\[
\theta_i = \{(a,b)\in A_i^2\;|\;\exists \wec{c}\in \prod_{j\neq i}\m a_j\;
(a\wec{c}\in R\;\&\;b\wec{c}\in R)\}.
\]
It is proved in Lemma~2.3 of \cite{parallelogram} that
(i) each $\theta_i$ is a congruence on $\m a_i$, and
(ii) $R$ is induced by the homomorphism 
$\psi\colon \prod \m a_i\to \prod \m a_i/{\theta_i}$ that is the natural
map in each coordinate. (This means: if $\overline{R} = \psi(R)$,
then $R = \psi^{-1}(\overline{R})$.)
The relation $\overline{R}$ is the \emph{reduction} of $R$.

In our special case $M\leq \m a^n$, where $M$ is critical and
maximal, we observe 
%two things. 
%First, we observe that each coordinate kernel is a proper congruence
%on $\m a$. To see this, assume instead that, say, $\theta_n$ is the total
%binary relation on $A$. Since $M\neq \m a^n$, there
%is a tuple $\wec{a}c\in A^n\setminus M$. Since $M$ is critical
%and $n>1$, $\pi_{[n-1]}(M) = A^{n-1}$, 
%so there is a tuple $\wec{a}d\in M$ whose first $n-1$
%coordinates agree with those of the previous tuple.
%The fact
%that $\theta_n$ is the total binary relation on $A$ implies
%that there exists $\wec{b}\in A^{n-1}$ such that $\wec{b}c, \wec{b}d\in M$.
%Now we have a contradiction to the
%paralellogram property: $\wec{a}d, \wec{b}d, \wec{b}c\in M$
%and $\wec{a}c\notin M$. We conclude that no coordinate
%kernel is the total binary relation.
%Second, we observe 
that $\overline{M} = \psi(M)$ is a maximal
subuniverse of $\prod \m a/{\theta_i}$. For, if $M<M'<\prod \m a/{\theta_i}$,
then 
$M=\psi^{-1}(M)<\psi^{-1}(M') < \m a^n$, contradicting the maximality of $M$.

The next result is a specialization of 
(some parts of) Theorem~2.5 of \cite{parallelogram}
to the case where $M$ is a critical maximal subuniverse of $\m a^n$
and $n$ is greater than one. We maintain the numbering
of \cite{parallelogram}, but delete the unused parts
of the theorem.

\begin{thm}\label{para_main}
Let $M$ be a critical maximal subuniverse of $\m a^n$
that satisfies the parallelogram property,
and let $\overline{M}\leq \prod \m a/{\theta_i}$
be its reduction. If
%With the notation and assumptions of the previous paragraph,
%along with the assumptions that 
$n>1$ and $\m a$ lies in a congruence modular variety, 
then the following hold.
\begin{enumerate}
\item[$(1)\hphantom{^*}$] 
$\overline{M}\leq \prod \m a/{\theta_i}$ is a representation 
of $\overline{M}$ as a subdirect product of subdirectly
irreducible algebras.
%\item[$(3)^*$]
%The projection of $\overline{M}$ onto any $n-1$ coordinates
%is an isomorphism.
\item[$(5)^*$]
If $n>2$, then the monolith of $\m a/{\theta_i}$
is the total relation; i.e. $\m a/{\theta_i}$ is simple.
\item[$(7)^*$] 
If $n>2$, then each simple algebra 
$\m a/{\theta_i}$ is abelian.
\end{enumerate}
\end{thm}

Here, items 
%(3), 
(5) and (7) are marked with asterisks, because
we have altered the statement of (5) from \cite{parallelogram}
in order to take into account that $\overline{M}$
is a maximal subuniverse of $\prod \m a/{\theta_i}$ 
%(this was
%taken into account in the formulation of $(5)^*$)
%and that the projection of $\overline{M}$
%onto any $n-1$ coordinates is surjective (this was
%taken into account in the formulation of $(3)^*$).
and we have altered the statement of (7)
%Item (7) is marked 
in order 
to take into account the
conclusion from $(5)^*$ that 
$\m a/{\theta_i}$ is simple.

We explain what this theorem contributes to our current investigation.
Suppose that $\m a$ has a $0$-pointed
$k$-cube term. Suppose also that
$M\leq \m a^n$ is maximal, $U$
is the minimal support of $M$,
and $M=M_U\times A^V$ is induced by $\pi_U\colon \m a^n\to \m a^U$.
If $|U|$ is at least as large as $\max\{3, k\}$,
then the theorem proves that 
$M_U$ is induced by a homomorphism 
$\psi\colon \m a^U\to \prod_U \m a/{\theta_i}$ where 
each factor $\m a/{\theta_i}$ is a simple abelian algebra.
Thus, $M$ itself is induced by the composition
of the surjective homomorphisms
\[
\m a^n 
\stackrel{\pi_U}{\longrightarrow} 
\m a^U
{\longrightarrow} 
\left(\m a/{[1,1]}\right)^U
\longrightarrow
\prod_U \m a/{\theta_i},
\]
where the last two maps are a factorization
of the map 
$\psi\colon \m a^U\to \prod_U \m a/{\theta_i}$ which induces $M_U$,
and these two maps are defined
coordinatewise by the natural maps
$\m a\to \m a/{[1,1]}\to \m a/\theta_i$.
(We have $\theta_i\geq [1,1]$, since $\m a/{\theta_i}$ is abelian.)
Hence $M$ is induced by the sub-composition
$\m a^n 
\stackrel{\pi_U}{\longrightarrow} 
\m a^U
{\longrightarrow} 
\left(\m a/{[1,1]}\right)^U$, which may be factored another way
as 
$\m a^n 
\stackrel{\eta}{\longrightarrow} 
\left(\m a/{[1,1]}\right)^n
\stackrel{\pi_U}{\longrightarrow} 
\left(\m a/{[1,1]}\right)^U$.
Hence $M$ is induced by the single map
$\eta$, which maps $\m a^n$ onto its abelianization.
Altogether this proves the desired result:

\begin{thm}\label{induced}
Assume that $\m a$ has a $0$-pointed $k$-cube term.
If $M\leq \m a^n$ is a maximal subuniverse, then either
\begin{enumerate}
\item[($\pi$)] 
$M$ is induced by a projection $\pi_U\colon \m a^n\to \m a^U$
for some subset $U\subseteq [n]$ satisfying $|U| < \max\{3,k\}$, or 
\item[($\eta$)]
$M$ is induced by $\eta\colon \m a^n\to (\m a/{[1,1]})^n$.
\end{enumerate}
\end{thm}

\subsection{A solution to a combinatorial problem}\label{solution}

The problem considered here is:
If $A$ is a finite set and $n\geq k>1$ are 
integers, then how small can a set $G\subseteq A^n$ be
if its projection onto any subset of $k$ coordinates
is surjective? 

If $A$ is finite, $G\subseteq A^n$ and $|G|=g$, 
then $G$ can be linearly ordered
and taken to be the sequence of 
rows of a $g\times n$ matrix of elements
of $A$, say $[a_{i,j}]$. If 
\[
\sigma\colon \quad 1 \leq j(1) < \cdots < j(k) \leq n
\]
is a selection of $k$ numbers between $1$ and $n$, then 
the projection of $G$ onto the
coordinates in $\sigma$ 
is the set of row vectors
$(a_{1, j(1)}, \ldots, a_{1, j(k)}), \ldots, 
(a_{g, j(1)}, \ldots, a_{g, j(k)})$ which 
occur as the set of rows of the $g\times k$
minor of $[a_{i,j}]$ whose column indices
are the indices in $\sigma$.
$G$ projects surjectively onto each $k$ coordinates of $A^n$ 
if and only if, for each choice $\sigma$
of $k$ column indices, the set of row vectors 
of the corresponding $g\times k$ minor of $[a_{i,j}]$ exhausts $A^k$.
Therefore, call a $g\times k$ matrix of elements of $A$
a \emph{bad minor} (or \emph{bad matrix})
if its rows fail to exhaust $A^k$.
The desired property of $G$ is that its associated
matrix has no bad minors.

\begin{thm}\label{surj_projections}
Let $A$ be a finite set of size $|A|=a > 1$.
Let $n\geq k > 1$ be natural numbers, and set
%$\displaystyle u = \frac{a^k}{a^k-1}$.
$\displaystyle u = a^k/(a^k-1)$.
If 
$
g\geq k\log_u\left(n\right) + \log_u\left(a^k/k!\right),
$
then there is a matrix in $A^{g\times n}$ with no bad minors.
\end{thm}

\begin{proof}
This is a probabilistic proof.
Our sample space is the set $A^{g\times n}$ of all $g\times n$
matrices of elements of $A$. 
Our probability distribution is the uniform one, 
so each individual matrix $M\in A^{g\times n}$ 
has probability $P(M)=|A^{g\times n}|^{-1} = a^{-gn}$.
For each matrix $M\in A^{g\times n}$ and 
each sequence of $k$ column indices, 
\[
\sigma\colon \quad 1 \leq j(1) < \cdots < j(k) \leq n,
\]
let $M_{\sigma}$ denote the $g\times k$ minor of $M$
whose column indices are those enumerated by $\sigma$
(called the \emph{$\sigma$-minor} of $M$).
Let $X_{\sigma}$ be the random variable whose value 
at the element $M\in A^{g\times n}$ is $1$ if 
$M_{\sigma}$ is a bad minor and $0$ otherwise,
i.e., $X_{\sigma}$ is the indicator variable for bad $\sigma$-minors.

\begin{clm}\label{expectation_clm}
For any $\sigma$, 
the expected value of $X_{\sigma}$ satisfies
\begin{equation}\label{first_estimate}
E(X_{\sigma})\leq a^k\left(a^k-1\right)^ga^{-gk}.
\end{equation}
\end{clm}

The expectation is computed 
\[
\begin{array}{rl}
E(X_{\sigma}) 
&=\; \sum_{M\in A^{g\times n}} (X_{\sigma}(M)\cdot P(M))\\
&=\; \sum_{M\in A^{g\times n}} (X_{\sigma}(M)\cdot a^{-gn})\\
&=\; \left(\sum_{M\in A^{g\times n}} X_{\sigma}(M)\right)a^{-gn},
\end{array}
\]
where the sum $\sum_{M\in A^{g\times n}} X_{\sigma}(M)$
on the last line represents the number matrices
in $A^{g\times n}$ whose $\sigma$-minor is bad.
By definition, a $g\times k$ matrix is bad if some
tuple $\wec{a}\in A^k$ does not appear among its rows.
So, 
for each $\wec{a}\in A^k$, let $\mathcal U_{\wec{a}}$
denote the set of all $g\times k$ matrices where
$\wec{a}$ does not appear among the rows.
$|\mathcal U_{\wec{a}}|$ can be computed by noting that
the $g$ rows of a matrix in $\mathcal U_{\wec{a}}$
may be freely chosen
from the set $A^k-\{\wec{a}\}$, which has
size $a^k-1$, so $|\mathcal U_{\wec{a}}| = (a^k-1)^g$.
The bad $g\times k$ matrices are those from
$\bigcup_{\wec{a}\in A^k} \mathcal U_{\wec{a}}$. Since
the cardinality of the union is no more than the sum of the
individual cardinalities, and these summands have the
same size,  we get that the number of bad
$g\times k$ matrices is no more than
$|A^k|\cdot |\mathcal U_{\wec{a}}| = a^k(a^k-1)^g$.
Each bad $g\times k$ matrix $N$ can be extended in 
$a^{g(n-k)}$ ways to a matrix $M\in A^{g\times n}$ whose
$\sigma$-minor satisfies $M_{\sigma}=N$, so 
the number of matrices in $A^{g\times n}$ with
a bad $\sigma$-minor is no more than 
$a^k(a^k-1)^ga^{g(n-k)}$.
Hence 
\[
E(X_{\sigma}) = 
\left(\sum_{M\in A^{g\times n}} X_{\sigma}(M)\right)a^{-gn}
\leq a^k(a^k-1)^g a^{g(n-k)} a^{-gn} = a^k(a^k-1)^g a^{-gk},
\]
as claimed.

\medskip

If $X:=\sum_{\sigma} X_{\sigma}$ is the sum of all
$X_{\sigma}$ as $\sigma$ ranges
over all $\binom{n}{k}$ choices of $k$ column indices
and $M\in A^{g\times n}$, then $X(M)$
equals the number of bad $g\times k$ minors of $M$. Since 
expectation is linear, and since 
$\binom{n}{k} < n^k/k!$ when $n\geq k > 1$, we get 
from (\ref{first_estimate})
that 
\[
E(X) = \sum_{\sigma} E(X_{\sigma}) \leq \binom{n}{k} a^k(a^k-1)^g a^{-gk} <
n^k(a^k/k!)(a^k-1)^g a^{-gk}.
\]
If it is the case that 
\begin{equation}\label{bound}
n^k(a^k/k!)(a^k-1)^g a^{-gk}\leq 1,
\end{equation}
then we will have $E(X) < 1$, meaning that the expected number of 
bad minors in an element of $A^{g\times n}$
is strictly less than $1$. This can happen only
if matrices without bad minors exist.
Rewriting (\ref{bound}) as
\[
n^k \leq 
\left(\frac{a^{k}}{\left(a^k-1\right)}\right)^g(a^k/k!)^{-1} = 
u^g(a^k/k!)^{-1},
\]
using the definition $u = a^k/(a^k-1)$, we can solve for $g$
to get
\begin{equation}\label{4rth}
g\geq k\log_u(n) + \log_u(a^k/k!).
\end{equation}
When this inequality holds
we get that (\ref{bound}) holds,
so a matrix with no bad minors exists. This is exactly the statement
of the theorem.
\end{proof}

\begin{cor}\label{surj_projections_cor}
Let $A$ be a finite set of size $|A|=a > 1$.
Let $n\geq k > 1$ be natural numbers, and set
$u = a^k/(a^k-1)$.
\begin{enumerate}
\item[(1)]
If 
$
g=\lceil k\log_u\left(n\right) + \log_u\left(a^k/k!\right)\rceil,
$
then there exists a subset $G\subseteq A^n$ of size $g$
whose projection onto any
$k$ coordinates of $A^n$ is surjective.
\item[(2)]
Let $G\subseteq A^n$ be a subset.
\begin{enumerate}
\item[(i)]
If the projections 
${\rm pr}_i\colon G\to A$ and 
${\rm pr}_j\colon G\to A$ 
onto distinct coordinates are different
functions for all $i$ and $j$, then
$
|G| \geq \log_a(n).
$
\item[(ii)]
If no two projections 
${\rm pr}_i\colon G\to A$ and 
${\rm pr}_j\colon G\to A$ 
differ by a permutation of $A$
(i.e., if it is not the case that 
$\pi\circ {\rm pr}_i={\rm pr}_j$
for any $i$ and $j$ or $\pi\in {\textrm{Sym}(A)}$),
then
$
|G| \geq \log_a(n) + \log_a(a!).
$
\item[(iii)]
If all projections of $G$
onto pairs of coordinates are surjective, then
$
|G| \geq \log_a(n) + \log_a(a!).
$
\end{enumerate}
\end{enumerate}
\end{cor}

When reading the statement of
this corollary one should imagine that $a$ and $k$ 
(and hence $u$) are
fixed while $n$ ranges. Item (1) implies that
there is a subset $G\subseteq A^n$ of size $O(\log(n))$
that projects surjectively onto any $k$ coordinates.
Item (2) implies that any set $G\subseteq A^n$ 
that projects surjectively onto each set of $k$ coordinates
must have size $\Omega(\log(n))$ if $k$ is at least $2$. Taken together
these statements show that, as a function of $n$, 
the least size of a set $G\subseteq A^n$ that projects onto
each $k$ coordinates is $\Theta(\log(n))$.

\begin{proof}[Proof of Corollary~\ref{surj_projections_cor}]
Part (1) is an immediate consequence of Theorem~\ref{surj_projections}.

For (2)(i), suppose that 
$G\subseteq A^n$ is a subset 
whose projections 
onto distinct coordinates are different
functions for different coordinates. 
Then the set of projections onto distinct coordinates
constitute $n$ distinct elements of the function 
space $A^G$, which has cardinality $a^g$. 
It follows that $n\leq a^g$, or $|G|=g\geq \log_a(n)$.

The argument for (2)(ii) is essentially the same as for 
(2)(i). The assumptions imply that the
set of all functions of the form $\pi\circ {\rm pr}_i\colon G\to A$,
$\pi\in {\textrm{Sym}(A)}$, are distinct. 
This yields $|{\textrm{Sym}(A)}|\cdot n = a!\cdot n$ functions
from the set $A^G$, which has size $a^g$,
hence $a!\cdot n\leq a^g$. Solving for $g$
yields $|G|=g\geq \log_a(n) + \log_a(a!)$.

For (2)(iii), if two projections differ by a permutation,
say $\pi\circ {\rm pr}_i = {\rm pr}_j$, then the projection
of $G$ onto the $i$-th and $j$ coordinates maps $G$ into the
the set of pairs of the form $(x,\pi(x))$. All such pairs
lie on the graph of $\pi$, which is a proper
subset of $A^2$ when $|A|>1$. Thus, if all projections
of $G$ onto pairs of coordinates are surjective, 
no two single coordinate projections
can differ by a permutation. This shows that 
item (2)(iii) follows from (2)(ii).
\end{proof}

\begin{rem}
It is possible to refine the estimate
in Corollary~\ref{surj_projections_cor}~(1)
when we are dealing with algebras instead of just sets.
Suppose that there is a subset $B\subseteq A$ such
that $B^k$ generates $\m a^k$. Then we can apply
the arguments above to obtain a set $G\subseteq B^n$
whose projection onto any $k$ coordinates
is $B^k$. According to the arguments, we can find such a $G$
of size 
\[
|G| = \lceil k\log_v\left(n\right) + \log_v\left(b^k/k!\right)\rceil,
\]
where $b = |B|$ and $v = b^k/(b^k-1)$. This is a larger
base for the logarithm and a smaller constant on the right,
so an overall smaller estimate for $G$.
The subalgebra of $\m a^n$ generated
by $G$ will project surjectively onto each $k$
coordinates, since its projection contains the generating
set $B^k$. Note here that there is no longer any
need to assume that $\m a$ is finite, only that $B$
is finite.

Another refinement can be made on top of the preceding one.
Suppose that there is a subset $P\subseteq B^k$
of size $|P|=p$
that generates $\m a^k$. (Here we do not assume that $P$
is the $k$-th power of a subset of $A$.)
The argument we have given does not generalize to produce
a subset $G\subseteq A^n$ whose projection onto any $k$ coordinates
is exactly $P$, but it does generalize to produce a subset
$G\subseteq A^n$ whose projection onto any $k$ coordinates will be
contained in $B^k$ and will contain $P$. This enough to ensure
that the subalgebra of $\m a^n$ that
is generated by $G$ projects surjectively
onto any $k$ coordinates. How much of an improvement do
we get in our estimate of $|G|$?

The assumption that $P\;(\subseteq B^k)$
generates $\m a^k$ affects the estimate 
in Claim~\ref{expectation_clm} as follows.
For each $\wec{b}\in P$, let $\mathcal U_{\wec{b}}$
denote the set of all matrices in $B^{g\times k}$
where $\wec{b}$ does not appear among the rows.
Then $|\mathcal U_{\wec{b}}| = (b^k-1)^g$, so 
$|\bigcup_{\wec{b}\in P}\mathcal U_{\wec{b}}| \leq |P|(b^k-1)^g=p(b^k-1)$.
Each of these $g\times k$ matrices can be extended in 
$b^{g(n-k)}$ ways to a matrix in $B^{g\times n}$, so 
we obtain the estimate $E(X_{\sigma})\leq p(b^k-1)b^{-gk}$.
This allows us to choose $G$ so that it
is slightly smaller, namely 
\[
|G|= \lceil k \log_v(n) + \log_v(p/k!) \rceil.
\]

Let's summarize the two refinements.
Let $\m a$ be a (possibly infinite) algebra.
Suppose that $P\subseteq A^k$ is a finite
generating set for $\m a^k$. Let $B$ be the smallest
subset of $A$ for which $P\subseteq B^k$, i.e., let
$B\subseteq A$
be the set of elements that appear in the coordinates
of the tuples in $P$. 
Let $b = |B|$,  $p = |P|$,
and $v = b^k/(b^k-1)$.
Assertion: There is a set $G\subseteq A^n$ of size 
$\lceil k \log_v(n) + \log_v(p/k!) \rceil$ that generates
a subalgebra of $\m a^n$ which projects surjectively onto any 
$k$ coordinates.
\end{rem}

\subsection{The Baker-Pixley Theorem, Part I}\label{bak_pix1}

In \cite{baker_pixley}, Kirby Baker and Alden Pixley
characterized the existence of a $(k+1)$-ary near
unanimity term operation in several ways, one of which
is the following.

\begin{thm}\label{baker_pix}
%{\rm \cite{baker_pixley}}
A variety $\mathcal V$ has a $(k+1)$-ary near unanimity
term operation if and only if, in $\mathcal V$,
whenever $\m b$ is a subalgebra of a direct product 
$\m a_1\times \cdots \times \m a_n$, $(n\geq k)$,
then $\m b$ can be uniquely determined from a knowledge 
of its $k$-fold coordinate projections.
\end{thm}

The statement that $\m b\leq \prod \m a_i$ is uniquely
determined by its $k$-fold projections means that 
if $\m c\leq \prod \m a_i$ has the same $k$-fold
projections as $\m b$, then $\m c=\m b$.
The particular case that interests us is 
the one where $\m a_1 = \cdots = \m a_n = \m a$
and $\m c = \m a^n$. In this case the theorem says
that if $\m a$ has a $(k+1)$-ary near unanimity term
and $\m b\leq \m a^n$ projects 
surjectively onto each subset of $k$ coordinates, then
$\m b = \m a^n$.

These observations prompt the following purely combinatorial
question: If $A$ is a finite set and $n\geq k$ are positive
integers, then how small can a set $G\subseteq A^n$ be
if its projection onto any subset of $k$ coordinates
is surjective? For if $A$ is the universe of an algebra
with a $(k+1)$-ary near unanimity term operation 
and $\m b = \lb G\rb$ is the subalgebra generated
by a set $G$ with the property just described,
then $\m b$ will project surjectively onto 
any subset of $k$ coordinates, hence 
$\m a^n = \m b = \lb G\rb$. That is,
a set $G$ that projects surjectively onto 
any $k$ coordinates will be a generating for $\m a^n$
set if $\m a$ has a $(k+1)$-ary near unanimity term operation.

If $k=1$, then one choice for a smallest subset $G\subseteq A^n$ 
that projects surjectively onto any $k$ coordinates is the
set of constant tuples in $A^n$. This set certainly 
projects onto each single coordinate;
on the other hand, no set of size strictly less than
$|A|$ can project onto any single coordinate.
Hence, when $k=1$, the smallest size for $G$ is $|A|$.
In the following subsection, we consider the problem for
$k>1$ and prove that the least size for such $G\subseteq A^n$
is $\Theta(\log(n))$.

\subsection{The Baker-Pixley Theorem, Part II}\label{bak_pix2}

The combinatorial arguments given in the preceding
subsection 
may be combined with Theorem~\ref{baker_pix} to produce 
a result about growth rates.

\begin{thm}
{\rm (Near unanimity terms
enforce the logarithmic monochotomy.)}
If $\m a$ is a finite algebra with a near
unanimity term operation, then  $d_{\m a}\in \Theta(log(n))$.
\end{thm}

This theorem will be generalized in Theorem~\ref{????}.




\subsection{Growth rates for algebras with a cube term}\label{growth_cube}


%\subsection{Enumerating maximal subuniverses}\label{enum}



Let $\max(n)$ (or $\max_{\m a}(n))$ denote the number
of maximal subuniverses of $\m a^n$, and let 
$\crit(n)$ (or $\crit_{\m a}(n)$) denote the number
of critical maximal subuniverses of $\m a^n$.

\begin{cor}
(Need CP, or at least that maximal subuniverses 
have the parallelogram property,
so that subuniverses induced by critical maximals
are maximal.)
If $\max(n)$ is a finite-valued function, then it is related to 
$\crit(n)$ by the equations
\[
\max(n) = \sum_{k=1}^n \binom{n}{k} \crit(k)
\quad {\textrm and}\quad 
\crit(n) = \sum_{k=1}^n (-1)^k\binom{n}{k} \max(k).
\]
\end{cor}


\section{Discussion, extensions, problems}\label{discussion}


\subsection{Growth rates of infinite algebras}

Any set $\Sigma$ of basic identities that does not entail 
the existence of a pointed cube term is also nonrestrictive
for infinite algebras. This can be shown in essentially
the same way we showed it for finite algebras,
but the situation is clearer for infinite algebras 
because we can say explicitly which growth rates
are possible.
In Example~\ref{big} we show that if
$d\colon \mathbb Z^+\to\mathbb Z^+$ is an arbitrary
monotone function, then there is a countably infinite
algebra 
%$\m a$ 
whose growth rate is $d$.
%such that $d_{\m a}(n) = d(n)$ for all $n$.
In Example~\ref{realize} we show how to modify the
example so that it realizes a given set $\Sigma$
of basic identities, provided $\Sigma$ does not entail
the existence of a pointed cube term. The cardinality
of this second example can be any infinite cardinal
that is at least as large as the number of constant
symbols appearing in $\Sigma$. 
%Thus the study of possible growth rates of infinite algebras
%is not interesting at the universal algebraic level.

\begin{exmp}\label{big}
Let $d\colon \mathbb Z^+\to\mathbb Z^+$ be an arbitrary
monotone function. We construct a 
countably infinite partial algebra $\m a$
such that $d_{\m a}(n) = d(n)$ for all $n$.
The one-point completion 
of $\m a$ will be a total algebra
with the same growth rate.

Let $M^{(1)}, M^{(2)}, \ldots$ 
be a sequence of matrices with the following properties.
\begin{enumerate}
\item[(1)] $M^{(n)} = [a_{i,j}^{(n)}]$ is an $n\times d(n)$ matrix.
\item[(2)] All elements of all matrices are different from one another.
\end{enumerate}
Let $A = \{a_{i,j}^{(n)}\}$ 
be the set of all entries appearing in these matrices.
Our aim is to equip $A$ with partial operations
ensuring that the $d(n)$ columns of $M^{(n)}$ form a smallest size
generating set for $\m a^n$. If we achieve this, then we will
have $d_{\m a}(n) = d(n)$ for all $n$.

For each $n\in \mathbb Z^+$ and each 
$\wec{b}\in A^n$, introduce a $d(n)$-ary partial operation
$F_{\wec{b}}$ for which $F_{\wec{b}}(M^{(n)})=\wec{b}$. This means that
$F_{\wec{b}}$ has domain of size $n$, consisting of the $n$ rows of 
$M^{(n)}$, and that $F_{\wec{b}}(a_{i,1}^{(n)},\ldots,a_{i,d(n)}^{(n)})=b_i$ for each
$i=1,\ldots,n$.

Our partial algebra is $A$ equipped with all partial operations
of the type described in the previous paragraph.
Since $F_{\wec{b}}(M^{(n)}) = \wec{b}$ whenever 
$\wec{b}\in A^n$, the columns of $M^{(n)}$
form a generating set of size $d(n)$ for $\m a^n$. 
The following claim will help us to prove that there is no smaller
generating set for $\m a^n$.

\begin{clm}
If a subset $G\subseteq A^n$ has fewer than $d(n)$ tuples
whose coordinates are all distinct, then the same
is true for $\lb G\rb$.
%The only tuples in $\lb G\rb$ whose coordinates
%are distinct are those already in $G$.
\end{clm}

If the claim is not true, then it must be possible to 
generate in one step
a tuple $\wec{c}\in A^n$ whose coordinates are all distinct
using other tuples, where fewer than $d(n)$ of these other tuples 
have the property that their coordinates are all distinct.
If the partial operation used is some
$F_{\wec{b}}$, $\wec{b}\in A^m$ for some $m$,
and the tuples used to generate are $\wec{x}_1,\ldots, \wec{x}_{d(m)}$,
then the following row equations must be satisfied.
\begin{equation}\label{distinct}
\small{
F_{\wec{b}}(\wec{x}_1,\ldots,\wec{x}_{d(m)})=
F_{\wec{b}}\left( 
\left[
\begin{array}{c}
x_{1,1}\\
\vdots\\
x_{n,1}
\end{array}
\right],
\ldots,
\left[
\begin{array}{c}
x_{1,d(m)}\\
\vdots\\
x_{n,d(m)}\\
\end{array}
\right]
\right) = 
\left[
\begin{array}{c}
c_1\\
\vdots\\
c_n
\end{array}
\right]=\wec{c}.
}
\end{equation}
Considering the definition of $F_{\wec{b}}$, it is clear that
the (distinct!) entries of $\wec{c}$ are among the entries
of $\wec{b}$, so $m=|\wec{b}|\geq |\wec{c}|=n$.
Moreover, the row equations $F_{\wec{b}}(x_{i,1},\ldots,x_{i,d(m)})=c_i$
can be solved in only one way, namely by using the appropriate
row of $M^{(m)}$. This forces all entries of $[x_{i,j}]$ to be distinct.
But this means there are $d(m)$ columns, $\wec{x}_j$,
whose coordinates are distinct,
and we assumed that there were fewer than $d(n)$ such columns.
Altogether this yields that $m\geq n$ and $d(m)<d(n)$, contradicting
the monotonicity of $d(n)$. The claim is proved.
\medskip

The claim completes the argument, since a subset $G\subseteq A^n$
of size less than $d(n)$ must have fewer than $d(n)$
tuples whose coordinates are distinct. Such a set cannot generate
$\m a^n$, since the generated subalgebra $\lb G\rb$ contains fewer than
$d(n)$ tuples whose coordinates are distinct while
$\m a^n$ has infinitely many such tuples.
\end{exmp}


\begin{exmp}\label{realize}
Here we explain how to modify the algebra from 
Example~\ref{big}, without changing its growth rate,
so that it realizes 
a given set $\Sigma$ of basic identities.
The only requirement on $\Sigma$ is that it does
not entail the existence of a pointed cube term.

The construction is like the one in Subsection~\ref{pointed_subsection},
so we only outline it. Recall that we started with an algebra
$\m a$, enlarged it to $\m a_{z_1,\ldots,z_p,0}$
by iterating the one-point completion
construction, and then merged it with the model $\m v$
of $\Sigma$ to create $\m a_{\Sigma}$, which realized
$\Sigma$ and had the same growth rate as $\m a$.
In this construction, we used the one-point
completion construction $p$ times, where
$p$ was the number of equivalence classes of 
constant symbols under $\Sigma$-provable equivalence.
The only thing different here is that we may not have finitely
many constant symbols. However, we may well-order
the equivalence classes of constants (say, by 
stipulating that $[c]<[d]$ if the least constant
in class $[c]$ is smaller than the least constant in $[d]$
under the well-order from the proof of Kelly's Theorem).
Now, rather than using the one-point completion 
construction $p$ times, we use the idea of the construction 
exactly once
to adjoin a well-ordered set $\{0\}\cup Z$ to $\m a$ to create
$\m a_{Z,0}$. Here the well-order is $0 < z_1 < z_2 < \cdots$,
with $0$ the least element, and $\lb Z; <\rb$ is a well-ordered
set for which there is a bijection $\varphi\colon [C]\to Z$
from the set of equivalence classes of constants.
The algebra has universe $A_{Z,0}$ equal to the disjoint
union of $A$, $Z$ and $0$. If $F$ is a function
symbol in the language of $\m a$, then it is defined
on $A_{Z,0}$ by 
\[
F^{\m A_{Z,0}}(\wec{a}) = \begin{cases}
F^{\m A}(\wec{a}) & \text{\rm if $\wec{a}\in A^n$};\\
\min\{\{a_1,\ldots,a_n\}\cap (\{0\}\cup Z)\} & \text{\rm else.}
\end{cases}
\]
We also define binary operations corresponding
to the operation $x\wedge y$ of the one-point
completion, namely $x\wedge_z y$ for $z\in Z\cup \{0\}$.
Here 
\[
x\wedge_z y = \begin{cases}
x & \text{\rm if $x=y$};\\
z & \text{\rm if $x\neq y$ and $x, y\in A\cup [z)$};\\
\min\{\{x,y\}\cap (\{0\}\cup Z)\} & \text{\rm else.}
\end{cases}
\]
Arguments similar to those in Theorem~\ref{partial}
show that $\m a$ and $\m a_{Z,0}$ have the
same minimal
generating sets, so the same growth rate.
We can merge this example with a model $\m v$
from Definition~\ref{V} to obtain a model
$\m a_{\Sigma}$, as we did in Subsection~\ref{pointed_subsection}.
Using the same arguments as before, it can be shown that
this model will have the same growth rate as $\m a$
unless $\Sigma$ entails the existence of a pointed
cube term.
\end{exmp}

In particular, it is possible to find infinite algebras
generating congruence distributive and congruence $3$-permutable
varieties whose growth rate is any prescribed
monotone function.

\medskip

We showed in Subsection~\ref{processing}
that a finite algebra with a $p$-pointed $k$-cube term 
has growth that is bounded above by a polynomial.
When $p\geq 1$, the constant $|A|^{k-1}$ appears 
in the polynomial, so the argument requires $\m a$ to be finite.
However, some version of the argument holds
when $\m a$ is infinite.

Let $P$ be the set of $p$ ($\geq 1$) constant symbols that appear in 
some $k$-cube term. Say that a term operation $t^{\m a}$ 
\emph{centralizes} $P$
if $t^{\m a}(c^{\m a},\ldots,c^{\m a}) = c^{\m a}$ for all $c\in P$.
The argument we gave in Subsection~\ref{processing}
may be extended to prove:

\begin{thm}
Let $\m a$ be an algebra with an $m$-ary, $p$-pointed, $k$-cube term, and
let $P$ be the set of constant symbols of appearing in the cube term.
Let $\widehat{\m a}$ be the reduct of $\m a$
to the term operations that centralize $P$.
If the reduct $\widehat{\m a}^{k-1}$ is finitely generated, 
then the growth rate of $\m a$ is bounded above by a 
polynomial of degree at most $\log_w(mp)$,
for $w = 2k/(2k-1)$.
\end{thm}

\begin{proof}
The proof is the same as the proof of Theorem~\ref{pointed_polynomial},
except that we apply one more step after fully processing
all tuples. Let $F\subseteq A$ be a finite subset with the property that
$F^{k-1}$ generates $\widehat{\m a}^{k-1}$.

Recall that a fully processed tuple
$\wec{a}$ has the structure that all but $k-1$ 
of its coordinates may
be divided into a small number of intervals, and in each interval
the coordinate value is the interpretation of a constant
symbol from $P$. Write such a tuple as $\wec{a} = \wec{p}\wec{u}$,
splitting it into its processed part and its unprocessed part.
If we choose tuples $\wec{u}_i\in F^{k-1}$ so that 
$\wec{u} = t(\wec{u}_1,\ldots,\wec{u}_r)$
for some term $t$ which centralizes $P$, 
then
$\wec{a} = \wec{p}\wec{u}=t(\wec{p}\wec{u}_1,\ldots,\wec{p}\wec{u}_r)$.
Hence our previously fully processed tuple $\wec{p}\wec{u}$
may be processed one more step into fully processed 
$\wec{p}\wec{u}_1, \ldots, \wec{p}\wec{u}_r$,
where the
unprocessed part lies in $F^{k-1}$. 
Now the finite number $|F|^{k-1}$ may replace our
use of the number $|A|^{k-1}$ in the proof of 
Theorem~\ref{pointed_polynomial}, yielding a polynomial
upper bound on growth rate.
\end{proof}

This result motivates the 
following problem.

\begin{prb}
Is it true that, for an arbitrary
infinite algebra $\m a$ with
a pointed $k$-cube term,  if $\m a^{k-1}$ is finitely generated,
then $d_{\m a}(n)$ is bounded above by a polynomial?
Is it even true that 
$\m a^{k-1}$ being finitely generated implies that
$\m a^n$ is finitely generated for all $n$?
\end{prb}

The first statement of the problem has an affirmative answer
for $0$-pointed cube terms, as we explain next.

\medskip

In Section~\ref{comb} we proved that if $\m a$ has a $0$-pointed
$k$-cube term, then the growth rate of $\m a$
is logarithmic if $\m a$ is perfect and linear if
$\m a$ is imperfect. Our lower bounds on growth rates
required $\m a$ to be finite, but our upper bounds did not.

Given an algebra $\m a$, let $e_{{\m a},k}(n)$ denote the 
least of a subset $G\subseteq A^n$ for which the generated
subuniverse $\lb G\rb$ projects 
surjectively onto any $k$ coordinates.
It is easy to see that $e_{{\m a},k}(n)\leq d_{\m a}(n)$
for all $k$ and $n$.











linear maltsev conditions do not need the finiteness condition


\begin{prb}
Is there a finite algebra $\m a$ where
$d_{\m a}(n)\notin O(\log(n))$ and 
$d_{\m a}(n)\notin \Omega(n)$?
\end{prb}


\begin{prb}
Is it true that a finite algebra with a $2$-sided
unit for some binary term has logarithmic or linear growth?
\end{prb}



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