\chapter{Introduction}

This monograph is concerned with the relationships
between Maltsev conditions, commutator theories and the
shapes of congruence lattices in varieties of algebras.

\section{Shapes of Congruence Lattices}

Carl F.~Gauss,\index{Gauss, C. F.} 
in \cite{disquisitiones}, 
introduced the notation 
\begin{equation}\label{gauss}
a\equiv b\pmod{m},
\end{equation}
which is read as ``$a$ is congruent to $b$ modulo $m$'',
to mean that the integers $a$ and $b$ have the
same remainder upon division  by the integer modulus $m$,
equivalently that $a-b\in m\mathbb Z$.
As the notation suggests, congruence modulo $m$
is an equivalence relation on $\mathbb Z$.
It develops that congruence modulo $m$
is compatible with the ring
operations of $\mathbb Z$, and that 
the only equivalence relations on $\mathbb Z$
that are compatible with the ring operations
are congruences modulo $m$ for $m\in {\mathbb Z}$.

Richard Dedekind conceived of
a more general notion of ``integer'',
which nowadays we call an ideal in a
number ring. Dedekind extended the notation
(\ref{gauss}) to 
\begin{equation}\label{rdedekind}
a\equiv b\pmod{\mu}
\end{equation}
where $a, b\in \mathbb C$ 
%are potential elements of a number ring 
and $\mu\subseteq \mathbb C$;
%is a potential ideal. 
(\ref{rdedekind}) is defined to hold if $a-b\in \mu$.
Dedekind called a subset $\mu\subseteq \mathbb C$
a {\bf module}\index{module!(Dedekind's definition)} 
if it could serve as the modulus of a congruence,
i.e., if this relation of 
congruence modulo $\mu$ is an equivalence relation
on $\mathbb C$. This happens precisely when 
$\mu$ is closed under subtraction.
For Dedekind, therefore, a ``module'' 
was an additive subgroup of $\mathbb C$.

The set of Dedekind's modules
is closed under the operations
of intersection and sum. These two operations
make the set of modules into a lattice.
Dedekind proposed and investigated the problem of 
determining the identities of this lattice
(the ``laws of congruence arithmetic'').
%, which may
%be called the 
In 1900,  in \cite{dedekind},\index{Dedekind, R.}
he published the discovery that
if $\alpha, \beta, \gamma\subseteq \mathbb C$ 
are modules, then 
\begin{equation}\label{mod_id}
\alpha\cap (\beta + (\alpha\cap\gamma)) = 
(\alpha\cap\beta)+(\alpha\cap\gamma).
\end{equation}
This $3$-variable law of the lattice of modules
is now called the {\bf modular law}.\index{modular law}
Dedekind went on to prove that any equational 
law of congruence arithmetic
that can be expressed with at most $3$-variables
is a consequence of the modular law and the laws
valid in all lattices.

Dedekind did not write the law
in the form (\ref{mod_id}), which is an identity,
but rather as a quasi-identity: for all modules 
$\alpha, \beta, \gamma\subseteq \mathbb C$ 
\begin{equation}\label{mod_qid}
\alpha\supseteq \gamma \longrightarrow 
\alpha\cap (\beta + \gamma) = 
(\alpha\cap\beta)+\gamma.\footnote{In fact, Dedekind
used the symbols $+$ and $-$ instead of $+$ and $\cap$.}
\end{equation}

Dedekind also discovered a useful
``omitting sublattices'' 
version of the modular law. It is the assertion that
there do not exist modules 
$\alpha, \beta, \gamma\subseteq \mathbb C$ 
which generate a sublattice isomorphic to $\m n_5$\index{$\m n_5$}
(Theorem I.7.12 of \cite{birkhoff}).
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{.9mm}
\begin{picture}(20,40)
\put(0,20){\circle*{1.2}}
\put(10,5){\circle*{1.2}}
\put(10,35){\circle*{1.2}}
\put(20,15){\circle*{1.2}}
\put(20,25){\circle*{1.2}}

\put(10,5){\line(-2,3){10}}
\put(10,5){\line(1,1){10}}
\put(10,35){\line(-2,-3){10}}
\put(10,35){\line(1,-1){10}}
\put(20,25){\line(0,-1){10}}

\put(22,24){$\alpha$}
\put(-5,18){$\beta$}
\put(22,14){$\gamma$}

\put(-3.5,0){$\beta\cap\alpha = \beta\cap\gamma$}
\put(-4,38){$\beta+\alpha = \beta+\gamma$}

%\put(-25,18){$\m n_5$}
\end{picture}
\end{center}
\caption{The lattice $\m n_5$}\label{fig(i)}
\end{figure}

More generally, a 
{\bf congruence}\index{congruence!defined informally}
on an arbitrary algebra $\m a$ is an equivalence
relation on the universe of $\m a$ that 
is compatible with the operations of $\m a$.
Equivalently, it is the kernel of a homomorphism
with domain $\m a$. The set of all congruences
is a sublattice of the lattice of equivalence
relations on $A$. A 
{\bf congruence identity}\index{congruence identity!of an algebra}
of $\m a$ is an identity that holds
in the lattice $\Conb(\m a)$
of all congruences of $\m a$.
Dedekind's observation 
is that the modular law is a congruence
identity satisfied by $\mathbb C$ as an abelian group.
In fact, it is a congruence identity of any group,
ring, vector space, Boolean algebra or lattice.

Dedekind's result initiated many lines of research in the
20th century, but to avoid losing focus we mention
only a few. Garrett Birkhoff\index{Birkhoff, G.} 
observed that the congruences of 
any group {\bf permute},\index{permuting congruences}
meaning that $\alpha\circ \beta = \beta\circ \alpha$,
and that any lattice of permuting equivalence relations 
is modular. This ``explains'' Dedekind's result to some degree,
but in \cite{fun-nak} N. Funayama\index{Funayama, N.}
and T. Nakayama\index{Nakayama, T.}
showed that the lattice of congruences of
a lattice satisfies the {\bf distributive law}:\index{distributive law}
for all $\alpha, \beta, \gamma$
\begin{equation}\label{distributive_law}
\alpha\cap (\beta + \gamma) = (\alpha\cap \beta) + (\alpha \cap \gamma),
\end{equation}
which is stronger than the modular law, yet 
congruences of lattices need not permute.
Thus permutability implies modularity, but not vice versa.
Bjarni J\'{o}nsson\index{J\'{o}nsson, B.} 
then refined the results of Birkhoff
and Dedekind by showing in \cite{jonsson0} that 
any lattice of permuting equivalence relations 
satisfies the arguesian law, which is stronger
than the modular law. Conversely, he showed that 
any complemented lattice 
satisfying the arguesian law is embeddable into
the lattice of congruences of an abelian group.
Here the 
{\bf arguesian law}\index{arguesian law} 
is the $6$-variable law
asserting that for all $\alpha_i, \beta_i$, $i=0,1,2$,
\begin{equation}\label{arguesian_law}
(\alpha_0+\beta_0)\cap(\alpha_1+\beta_1)\cap(\alpha_2+\beta_2) \leq
\alpha_0\cap (\gamma + \alpha_1) + 
\beta_0\cap (\gamma + \beta_1)
\end{equation}
where $\gamma = (\alpha_0+\alpha_1)\cap 
(\beta_0+\beta_1)\cap (\delta_0+\delta_1)$, 
$\delta_0= (\alpha_0+\alpha_2)\cap (\beta_0+\beta_2)$
and $\delta_1=(\alpha_1+\alpha_2)\cap (\beta_1+\beta_2)$.
Thus J\'{o}nsson solved the special instance
of Dedekind's problem 
(of determining the laws of modules) which concerns
only complemented lattices of modules.
The full problem is still open, although 
in the same paper J\'{o}nsson gave an example of a finite
noncomplemented arguesian lattice that is not embeddable
in the congruence lattice of a group, and in \cite{haiman_th} 
Mark Haiman\index{Haiman, M.} 
proved that lattices
of permuting equivalence relations satisfy 
laws stronger than the arguesian law.

\section{Maltsev Conditions}

Let $\vr R$ be the variety of rings and $\vr A$
be the variety of abelian groups.
The fact that rings have a 
term-definable underlying 
abelian group structure may be denoted
$\vr A\leq \vr R$, i.e., rings
have the structure of abelian groups and more.
A.~I.~Maltsev\index{Maltsev, A.~I.} 
proved in \cite{maltsev} that 
a variety $\vr V$ of algebras has the property
that the congruence lattices of its members
consist of permuting equivalence relations if and only
if $\vr P\leq \vr V$ where $\vr P$ is the variety
with one ternary basic operation symbol $p$ that 
is axiomatized by the {\bf Maltsev identities}:\index{Maltsev!identities}
\begin{equation}\label{maltsev_id}
p(x,y,y)\approx x\qquad {\rm and} \qquad 
p(y,y,x)\approx x.
\end{equation}
In other words, $\vr V$ consists of congruence
permutable algebras if and only if there 
is a ternary $\vr V$-term $p$,
called a {\bf Maltsev term},\index{Maltsev!term} 
such that the identities
(\ref{maltsev_id}) hold in $\vr V$.
Thus, one may refine Birkhoff's earlier explanation
of Dedekind's modularity result to:
groups are congruence modular because 
groups have a Maltsev term $p(x,y,z) = xy^{-1}z$,
so groups have permuting congruences, 
and lattices of permuting equivalence relations 
are modular.

Let $\vr U$ be a finitely presented\footnote{I.e., $\vr U$ is a 
finitely axiomatized variety with finitely many basic 
operation symbols.}
variety.
The condition on a variety $\vr V$
that $\vr U\leq \vr V$ may be expressed
as ``there is a finite set of $\vr V$-terms 
corresponding to the basic operation
symbols of $\vr U$ such that the finite set 
of identities corresponding to the axioms of $\vr U$
hold in $\vr V$.'' A condition of this type
is called a {\bf strong Maltsev condition}%
\index{Maltsev condition!strong --- defined informally},
and given a finitely presented variety $\vr U$ 
the class $\{\vr V\;|\;\vr U\leq \vr V\}$
is the {\bf strong Maltsev class}\index{Maltsev class!strong}
that is {\bf defined}\index{Maltsev condition!defined by a (strong) ---}
by this condition. 
Given a descending sequence $\cdots \leq {\vr U}_2\leq 
{\vr U}_1\leq {\vr U}_0$ of finitely presented varieties,
the class $\{\vr V\;|\;\exists n ({\vr U}_n\leq \vr V)\}$
is the {\bf Maltsev class}\index{Maltsev class}
that is defined by this sequence, and the
{\bf Maltsev condition}\index{Maltsev condition!defined informally} 
associated to this sequence is the assertion that for some $n$ 
there is a finite set of $\vr V$-terms 
corresponding to the basic operation
symbols of ${\vr U}_n$ such that the finite set 
of identities corresponding to the axioms of ${\vr U}_n$
hold in $\vr V$.

After Maltsev published the condition
from (\ref{maltsev_id})
defining the class of varieties with permuting
congruences, Alden Pixley\index{Pixley, A.} 
found in \cite{pixley1} 
a strong Maltsev condition defining the class of varieties
with distributive and permuting congruences,
B.~J\'{o}nsson found in \cite{jonsson1}
a Maltsev condition defining the class of varieties
with distributive congruence lattices, and 
Alan Day\index{Day, A.} 
found in \cite{day1} 
a Maltsev condition defining the class of varieties
with modular congruence lattices.
These results (and many others
like them) culminated in the theorem,
obtained independently by Pixley \cite{pixley2}
and Rudolf Wille\index{Wille, R.} 
\cite{wille}, that if $\varepsilon$
is any lattice identity, then the class of varieties
whose congruence lattices satisfy $\varepsilon$
is the intersection of countably many Maltsev classes.
This proves, in particular, that if two varieties
satisfy the same Maltsev conditions, then they
satisfy the same congruence identities.
This result of Pixley and Wille includes an algorithm
for generating Maltsev conditions associated with
congruence identities, paving the way for a deeper
study of congruence identities. 

J.~B.~Nation\index{Nation, J.~B.} 
discovered in \cite{nation}
that there exist lattice identities that are 
strictly weaker than the modular law when
considered as lattice identities, but equivalent to 
the modular law when considered as 
congruence identities. That is, there exists a variety 
$\vr W$ of
lattices that strictly contains all modular lattices, yet
any variety $\vr V$ of algebras with the property that the
congruence lattice of every algebra from 
$\vr V$ is found in $\vr W$ is
actually congruence modular.
Nation's
Theorem brought attention to
the possibility that there might be ``few'' 
different {\bf congruence varieties},\index{congruence variety}
which are varieties of lattices of the form 
\begin{equation}\label{congruence_var}
\CON({\vr V}) = 
\Ho\Su\Pd\left(\{\Conb(\m a)\;|\;\m a\in \vr V\}\right).
\end{equation}
For example, it suddenly became plausible
that the only proper nondistributive congruence
varieties might be those of the form
$\Con({\vr M})$ where $\vr M$ is a 
variety of modules.
Nation's result led Ralph McKenzie\index{McKenzie, R.}
to conjecture in \cite{mckenzie2} 
that if $\varepsilon$ is any nontrivial 
lattice identity, then $\varepsilon$
implies modularity when considered as a congruence identity.
Studies of identities 
of small complexity yielded positive 
evidence for McKenzie's Conjecture in 
\cite{day2,day3,freese-nation2,jonsson2,mederly}.
The most general of these results
is the theorem of Ralph Freese\index{Freese, R.} 
and J.~B.~Nation 
that any lattice identity $\varepsilon$
that can be written as an inclusion of the form 
\[\bigwedge\bigvee\bigwedge\mbox{\rm variables}\leq
\bigvee\bigwedge\bigvee\mbox{\rm variables}\]
implies modularity as a congruence identity.
Similar results were proved at the same time
about congruence identities implying 
congruence distributivity, and in 
\cite{freese-jonsson} one finds the result
that any congruence identity implying congruence
modularity must also imply the stronger
arguesian identity as a congruence identity.

McKenzie's Conjecture was refuted 
by S.~V.~Polin\index{Polin, S.~V.} 
in his famous
paper \cite{polin}. Polin constructed a 
locally finite variety $\vr P$ 
that is not congruence
modular, but satisfies a nontrivial 
congruence identity. To describe
a congruence identity that holds in Polin's 
variety, let $x,y$ and $z$ be lattice variables
and let $y_1 = y, z_1 = z, 
y_{n+1} = y+(x\cap z_n)$, and 
$z_{n+1} = z+(x\cap y_n)$. 
Let $\delta_n$ be the weakened distributive law:
\[x\cap(y+z) \approx (x\cap y_n) + (x\cap z_n).\]
The identity $\delta_1$ is the 
usual distributive law. 
Although $\delta_2$ is strictly weaker
than distributivity as a lattice identity,
it implies distributivity as a congruence
identity.\footnote{We know this because Ralph Freese
showed us that $\delta_2$ fails in 
$\Con(\m f_{\vr P}(1))$. Using this observation 
and the main result of \cite{day-freese} it is easy
to see that $\delta_2$ implies
distributivity as a congruence identity.}
Polin's variety satisfies $\delta_3$ as a congruence 
identity.\footnote{Polin showed that $\vr P$ satisfies
a different congruence identity. Alan
Day was the one to emphasize the importance 
of the identities $\delta_n$.}
Since it does not satisfy $\delta_2$, 
and since any lattice variety that
satisfies $\delta_3$ and not $\delta_2$ is nonmodular,
$\Con({\vr P})$ is an example of a 
proper, nondistributive congruence
variety that differs
from $\Con({\vr M})$ for any variety of
modules $\vr M$.\footnote{We now know that there are
continuumly many different nonmodular congruence varieties
(see \cite{day-freese}), 
and even some congruence varieties of groups
that differ from any congruence variety
of modules (see \cite{p3-szabo}).}
Polin's variety was thoroughly investigated
by Day and Freese in \cite{day-freese}
with the result that
we now have an efficient algorithm 
for determining if an identity implies
modularity as a congruence identity.
In an unpublished manuscript, Day 
showed that the Polin construction
can be iterated\footnote{This and other
modifications of Polin's construction 
were rediscovered and appear in \cite{k-valeriote}
and \cite{newrly-valeriote}.}
to produce varieties satisfying weaker and weaker 
nontrivial congruence identities.

Not all of the preceding results were 
originally proved via Maltsev 
conditions, but they could have been,
and the analysis of Maltsev conditions
is a powerful method for obtaining further 
results like these. Moreover, there are results
on congruence identities 
that have been proved by a careful analysis of 
the Maltsev conditions defining a given class of varieties
that seem unreachable by any other method, such as 
Paolo Lipparini's\index{Lipparini, P.} 
results on congruence identities satisfied by congruence
$n$-permutable varieties\footnote{Congruences $\alpha$ and $\beta$
{\bf $n$-permute}\index{permuting congruences!$n$- } 
if the alternating composition
$\alpha\circ_n\beta = \alpha\circ\beta\circ \alpha\cdots$,
with $n-1$ occurrences of $\circ$,
equals $\beta\circ_n\alpha$. A variety is 
{\bf congruence $n$-permutable}\index{congruence $n$-permutable variety}
if the congruences on all members $n$-permute.}
in \cite{lip1, lipparini1, lipparini4, lip3, lip4}.

\section{Commutator Theories}

Birkhoff's $\Ho\Su\Pd$ Theorem\index{HSP Theorem} asserts that
if $\vr K$ is a class of similar algebras, then 
any model of the identities true in $\vr K$ 
may be constructed as a homomorphic image of 
a subalgebra of a product of algebras in $\vr K$,
that is as $\m b/\theta$ where $\m b\leq \prod \m a_i$,
$\m a_i\in \vr K$. Replacing each $\m a_i$ by the
projection of $\m b$ onto the $i$-th factor, 
we find that a typical model has the form 
$\m b/\theta$ where $\m b$ is a subdirect product 
of subalgebras of members of $\vr K$. 
It is therefore important to know how to construct
congruences on subdirect products.
The case of two factors is already difficult.
If $\m b\leq_{\rm sd} \m a_1\times \m a_2$
and $\alpha_i\in \Con(\m a_i)$, then the restriction
of $\alpha_1\times \alpha_2$ to $\m b$ is 
a {\bf product congruence}\index{congruence!product}
on $\m b$. All other congruences are 
{\bf skew}.\index{congruence!skew}
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{.85mm}
\begin{picture}(30,40)
\put(0,20){\circle*{1.2}}
\put(15,5){\circle*{1.2}}
\put(15,20){\circle*{1.2}}
\put(15,35){\circle*{1.2}}
\put(30,20){\circle*{1.2}}

\put(15,5){\line(-1,1){15}}
\put(15,5){\line(0,1){15}}
\put(15,5){\line(1,1){15}}
\put(15,35){\line(-1,-1){15}}
\put(15,35){\line(0,-1){15}}
\put(15,35){\line(1,-1){15}}

\put(10,0){$0\times 0$}
\put(-14,19.5){$0\times 1$}
\put(33,19.5){$1\times 0$}
\put(10,37){$1\times 1$}
\put(9,19.5){$\Delta$}
\end{picture}
\end{center}
\caption{$\Conb({\mathbb Z}_2\times {\mathbb Z}_2)$}\label{fig(ii)}
\end{figure}
For example, the congruence lattice of the group
$\m b = {\mathbb Z}_2\times {\mathbb Z}_2$
is pictured in 
Figure~\ref{fig(ii)}. 
There is one skew congruence, $\Delta$,
which is the congruence that has 
the diagonal subgroup 
of  ${\mathbb Z}_2\times {\mathbb Z}_2$ as a class.

To understand skew congruences like this one,
let us attempt to describe a typical ``diagonal 
congruence'' on a typical ``diagonal subalgebra''
$\m b$ of $\m a^2$ for some algebra $\m a$
in a congruence permutable variety $\vr V$. 
Here a ``diagonal subalgebra'' is one containing
the diagonal of $\m a^2$, and a ``diagonal skew congruence''
is a congruence generated by a set of pairs of 
elements from the diagonal. More formally, if
$\delta\colon \m a\to \m a^2\colon a\mapsto (a,a)$ 
is the canonical diagonal
embedding, then $\m b\leq \m a^2$ is a diagonal subalgebra
if $\delta$ factors through the inclusion of $\m b$
into $\m a^2$.
A diagonal congruence on $\m b$ is the extension to
$\m b$ of a congruence on $\delta(\m a)$.

Since we have assumed that $\m a$ lies in a congruence
permutable variety, it has a Maltsev term. 
Hence the only diagonal subalgebras $\m b$ of $\m a^2$ are 
those whose universe $B$ is a congruence on $\m a$,
say $B = \beta$. Since the diagonal of 
$\m a^2$ supports a subalgebra 
isomorphic to $\m a$, the only diagonal
congruences on $\m b$ are the extensions to $\m b$ of 
congruences $\delta(\alpha)$ for $\alpha\in\Con(\m a)$.
Thus, the construction of a typical diagonal congruence
on a typical diagonal subalgebra $\m b\leq \m a^2$
involves a pair of congruences $\alpha, \beta\in\Con(\m a)$.
Specifically, we are considering $\m b$ to be the subalgebra
supported by $\beta$, and a diagonal congruence
$\Delta=\Delta_{\alpha,\beta}$ on $\m b$ that is generated  by 
$
\{\lb (u,u), (v,v)\rb\;|\;(u,v)\in \alpha\}.
$
We next try to understand how $\Delta$ is related to the
product congruences of $\m b$.

Since we have assumed that $\m a$ has a Maltsev term,
and $\m b$ is in the variety generated by $\m a$,
$\Conb(\m b)$ is modular. It is a consequence of 
Dedekind's analysis of $3$-variable consequences
of the modular law that the $3$-generated
free modular lattice has only $28$ elements, hence any
$3$-generated modular lattice can be easily drawn.
\begin{figure}[ht]
\begin{center}
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\put(45,20){\circle*{1.2}}
\put(45,40){\circle*{1.2}}
\put(45,60){\circle*{1.2}}
\put(60,10){\circle*{1.2}}
\put(60,30){\circle*{1.2}}
\put(60,50){\circle*{1.2}}
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\put(75,40){\circle*{1.2}}
\put(90,30){\circle*{1.2}}

\put(30,60){\circle*{1.2}}
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\put(45,70){\circle*{1.2}}
\put(60,60){\circle*{1.2}}

\put(30,40){\circle*{1.2}}
\put(45,30){\circle*{1.2}}

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\put(0,30){\line(3,2){45}}
\put(45,0){\line(-3,2){45}}
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\put(75,20){\line(-3,2){45}}
\put(90,30){\line(-3,2){45}}
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\put(60,60){\line(-3,2){15}}

\put(30,60){\line(0,-1){10}}
\put(45,70){\line(0,-1){10}}
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\thicklines
\put(30,40){\line(3,2){15}}
\put(60,10){\line(3,2){15}}
\put(15,40){\line(3,2){15}}

\put(92,29){$\eta_2=1\times 0$}
\put(-26,29){$\eta_1=0\times 1$}
\put(15,59){$\alpha\times 1$}
\put(62,59){$1\times \alpha$}
\put(41,60){$\theta$}
\put(46,47){$\psi$}
\put(24,39){$\Delta$}
\put(90,60){\small $\theta=\beta\times 1 = 1\times \beta$}
\put(100,47){\small $\psi=\alpha\times\alpha$}
\end{picture}
\end{center}
\caption{$\lb\eta_1,\eta_2,\Delta\;|\;\eta_1\cap\eta_2=0\rb$}\label{fig(iii)}
\end{figure}
In particular, the modular lattice with the presentation
$\lb\eta_1,\eta_2,\Delta\;|\;\eta_1\cap\eta_2=0\rb$
is the one in Figure~\ref{fig(iii)}.
Hence the sublattice of $\Conb(\m b)$ that is generated
by the projection kernels
$\eta_1=0\times 1$ and $\eta_2=1\times 0$ together
with $\Delta=\Delta_{\alpha,\beta}$
is a homomorphic image of this lattice.
The smallest product congruence containing $\Delta$, 
the ``product cover'' of $\Delta$,
is 
$$
(\Delta + (0\times 1))\cap (\Delta + (1\times 0)) = \alpha\times \alpha = \psi.
$$
We take the interval $I[\Delta,\psi]$, from
$\Delta$ to its product cover, to be a measure of the skewness
of $\Delta$. Meeting with $\eta_2$ and joining
with $\eta_1$ projects this interval in two steps
to the isomorphic interval 
\begin{equation}\label{latter_int}
I[\eta_1+(\Delta\cap \eta_2),\eta_1+(\psi\cap \eta_2)]=
I[\eta_1+(\Delta\cap \eta_2),(\alpha\cap \beta)\times 1],
\end{equation}
so the interval in 
(\ref{latter_int}) can also be taken to be a measure
of the skewness of $\Delta$. But 
the interval in (\ref{latter_int})
lies entirely in the interval above the first coordinate 
projection kernel $\eta_1$, which is naturally
isomorphic to $\Conb(\m a)$, so we can measure
the skewness of $\Delta$ by considering the
corresponding congruence interval of $\m a$.
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{.75mm}
\begin{picture}(90,45)
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\put(30,30){\circle*{1.2}}

\put(15,20){\line(0,-1){20}}
\put(15,20){\line(3,2){15}}
\put(15,20){\line(-3,2){15}}
\put(15,40){\line(3,-2){15}}
\put(15,40){\line(-3,-2){15}}

\put(70,30){\circle*{1.2}}
\put(85,0){\circle*{1.2}}
\put(85,10){\circle*{1.2}}
\put(85,20){\circle*{1.2}}
\put(85,40){\circle*{1.2}}
\put(100,30){\circle*{1.2}}

\put(85,20){\line(0,-1){20}}
\put(85,20){\line(3,2){15}}
\put(85,20){\line(-3,2){15}}
\put(85,40){\line(3,-2){15}}
\put(85,40){\line(-3,-2){15}}

\put(-3,-5){$\eta_1=0\times 1$}
\put(-15,28.5){$\alpha\times 1$}
\put(32,28.5){$\beta\times 1$}
\put(-20,8.5){$\eta_1 + (\Delta\cap \eta_2)$}
\put(-18,18){$(\alpha\cap\beta)\times 1$}
\put(-3,42){$(\alpha+\beta)\times 1$}

\put(84,-5.5){$0$}
\put(65,28.5){$\alpha$}
\put(102,28.5){$\beta$}
%\put(87,8.5){$[\alpha,\beta]$}
\put(87,8.5){?}
\put(87,18.5){$\alpha\cap\beta$}
\put(78,42){$\alpha+\beta$}
\end{picture}
\end{center}
\caption{$I[0\times 1, 1\times 1]$ vs.\ $\Conb(\m a)$}\label{fig(iiia)}
\end{figure}
The left lattice in Figure~\ref{fig(iiia)}
indicates the interval in $\Conb(\m b)$ above
$\eta_1$, while the right lattice shows
the corresponding congruences on $\m a$.
Note especially the question mark by the
right lattice, which labels the congruence on $\m a$
induced by $\Delta$. This induced congruence
is a function of $\alpha$ and $\beta$,
and will be written $[\alpha,\beta]$
and called the 
{\bf commutator}\index{commutator!modular}
of $\alpha$ and $\beta$. For groups
it is the ordinary group commutator.\footnote{This means
that if $N_{\alpha}$ and $N_{\beta}$ are the normal
subgroups corresponding to the subscripted congruences,
then $[N_{\alpha},N_{\beta}] = N_{[\alpha,\beta]}$.}
The interval from the commutator $[\alpha,\beta]$
to the intersection $\alpha\cap\beta$ is
the interval in $\Conb(\m a)$ that measures the
skewness of $\Delta$.

In analogy with group theory, we call a congruence
interval $I[\sigma,\tau]$ 
{\bf abelian}\index{abelian!congruence interval}
if $[\tau,\tau]\leq \sigma$, 
and call $\m a$ abelian
if the entire lattice $\Conb(\m a)=[0,1]$ is an abelian congruence
interval. If $\tau$ is defined to equal
$\alpha\cap\beta$ in the last paragraph, then
$\tau\leq \alpha$ and $\tau\leq \beta$ so 
$[\tau,\tau] \leq [\alpha,\beta]$.
This shows that the interval 
$I\left[ [\alpha,\beta],\alpha\cap\beta \right]$,
which measures the skewness of $\Delta$, is abelian.
Conversely, every abelian interval $I[\sigma,\tau]$
is contained in an interval that measures the
skewness of some diagonal congruence, namely $\Delta_{\tau,\tau}$.
Hence diagonal congruences fail to be skew in $\vr V$
if and only if $\vr V$ omits abelian congruence intervals
if and only if the commutator trivializes throughout 
$\vr V$ in the sense that
\begin{equation}\label{comm_id}
\forall \alpha,\beta([\alpha,\beta] = \alpha\cap\beta)
\end{equation}
holds throughout $\vr V$.
It can be shown that for varieties with
a Maltsev term the commutator identity (\ref{comm_id})
is equivalent to congruence distributivity, and from
congruence distributivity one can derive that there
are no skew congruences of any type 
in finite subdirect products of algebras in $\vr V$.
Thus, although it seems that we examined only a very special
instance of a diagonal skew congruence, which then resulted
in the definition of the commutator, this commutator
controls all skew congruences in finite subdirect products
in varieties with a Maltsev term.

In fact, the assumption that $\m a$ has a Maltsev term 
was used only to deduce that every diagonal
subalgebra of $\m a^2$ has a congruence as its universe.
This fact turns out to be inessential;
the theory can be developed under the weaker
hypothesis that $\m a$ lies in a congruence
modular variety.
The theory of this commutator was developed
by Jonathan~D.~H. Smith\index{Smith, J.~D.~H.} 
(when $\vr V$ has a Maltsev term) 
in \cite{smith},
Joachim Hagemann\index{Hagemann, J.}
and Christian Herrmann\index{Herrmann, C.}
in \cite{hagemann-herrmann}, 
Heinz-Peter Gumm\index{Gumm, H.~P.} 
in \cite{gumm},
and Freese\index{Freese, R.} 
and McKenzie\index{McKenzie, R.} 
in \cite{freese-mckenzie}.
The strength of the theory lies in 
the representation theorem for abelian
algebras, proved by Herrmann in \cite{herrmann},
which states that an abelian algebra
in a congruence modular variety 
is {\bf affine}.\index{affine}
This theorem associates to an abelian algebra
or congruence interval a
module in the modern sense, i.e., a module over a ring.
The fact that the commutator links every skew
congruence with a module is the key to dealing with 
these congruences.

To emphasize the link with what we have said earlier,
the commutator in congruence modular varieties encodes 
the existence of skew congruences. 
When the commutator trivializes,
skew congruences are omitted, and this restricts
the shape of the congruence lattices in some way.
In the case of the modular commutator, this restriction
on shapes is expressible by a congruence identity 
stronger than modularity, namely the distributive law.

When $\vr V$ is not congruence modular the types
of skew congruences multiply. Different commutator
theories have been invented which deal with this.
The {\bf TC-commutator}\index{commutator!TC-}
is a commutator based on the {\bf term condition}
(Section~\ref{the_term_condition}).
It was invented 
%by Ralph McKenzie\index{McKenzie, R.}
to generalize the modular commutator.
As such it encodes the existence of diagonal
skew congruences. A useful and 
completely general representation
theorem for abelian algebras and congruences
is not likely to exist, but Keith Kearnes 
and \'{A}gnes Szendrei\index{Szendrei, \'{A}.}
extended Herrmann's representation theorem
for congruence modular varieties 
to any variety satisfying some nontrivial
idempotent Maltsev condition in \cite{k-szendrei2}.

McKenzie next introduced the {\bf strong term condition},
\cite{mckenzie3, hobby-mckenzie},
a concept similar in spirit to the 
ordinary term condition. He proved its
usefulness in many ways, but he did not 
develop a commutator theory for it.
Later McKenzie defined the concept
of {\bf rectangulation}, \cite{kiss-prohle},
and established its usefulness.
His definition was slightly short of
a term condition for rectangulation, and he
did not develop a commutator theory for it.

In the early 1990's, we (= Kearnes and Kiss)
worked on the problem of 
developing a commutator theory for McKenzie's
strong term condition. We found that certain 
``cross-diagonal'' skew congruences occur 
naturally in algebras. These are congruences on 
symmetric diagonal subalgebras
$\m b\leq \m a^2$ generated by pairs of the form 
$\lb (u,v), (v,u)\rb$.
The behavior of this type of skew congruence
is encoded in a commutator we call
the {\bf strong rectangular commutator}.
It is so named because it has an associated
term condition related to McKenzie's description
for rectangulation.
We developed the theory of this commutator in \cite{k-kiss}
and proved a representation for its abelian
algebras and congruences. We found that McKenzie's
strong term condition was exactly the conjunction of his
original term condition (TC) and our term condition for 
strong rectangulation, therefore a commutator theory
for the strong term condition follows from the
theories of the TC-commutator and the strong rectangular
commutator without further effort.

In this monograph we introduce a term condition
for rectangulation, develop the theory of the 
rectangular commutator,
and prove a representation theorem for its abelian
algebras and congruences (Chapter~\ref{rectangulation1}).

There are actually many more commutators than the four
described above, but the others that are known seem to be
approximations to these four.
Each of these commutators has an ideal model
of an abelian algebra. For the TC commutator,
the ideal model is an abelian group
expanded by unary endomorphisms, i.e., a module over a ring.
For the strong commutator it is a set 
expanded by unary endomorphisms, i.e., a unary algebra.
For the rectangular commutator
it is a semilattice expanded by unary endomorphisms,
which is a natural type of semimodule.
For the strong rectangular commutator it is a reduct
to an antichain of a 
semilattice expanded by unary endomorphisms.
The strangeness of this fourth notion of `abelianness'
is a reminder that these commutator theories do not begin by 
postulating the structure of an ideal abelian algebra, but rather
by identifying a natural type of skew congruence.

\section{The Results of This Monograph}

The theorem which prompted us to write this monograph
was our 1999 discovery that a variety satisfies a nontrivial
congruence identity if and only if it satisfies
an idempotent Maltsev condition that fails in the variety 
of semilattices (Theorem~\ref{main1}).
David Hobby\index{Hobby, D.}
and Ralph McKenzie\index{McKenzie, R.}
proved in Chapter~9 of \cite{hobby-mckenzie}
that a certain Maltsev condition, 
that we will call in this introduction 
``HM'', defines the class of all varieties satisfying
an idempotent Maltsev condition that fails in the variety 
of semilattices. {\rm HM} is a disjunction
 $\bigvee {\rm HM}_n$ of strong Maltsev conditions.
The proof of the 1999 theorem is based on a careful analysis
of various Maltsev conditions equivalent to HM.
A novel feature of the proof is that it 
deals only with the local effects of HM
on nonsolvable intervals in congruence lattices
of algebras in a variety $\vr V$ satisfying
some ${\rm HM}_n$.
Namely, we prove that every nonmodular interval
in a congruence lattice contains what we call
a ``solvability obstruction''.
We use ${\rm HM}_n$ to introduce a rank function 
on solvability obstructions that appear in congruence 
lattices of algebras in $\vr V$.
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{.8mm}
\begin{picture}(20,35)
\put(0,20){\circle*{1.2}}
\put(10,5){\circle*{1.2}}
\put(10,35){\circle*{1.2}}
\put(20,15){\circle*{1.2}}
\put(20,25){\circle*{1.2}}
\put(0,20){\line(2,3){10}}
\put(20,25){\line(-1,1){10}}
\put(10,5){\line(1,1){10}}
\bezier{200}(20,15)(16,20)(20,25)
\bezier{200}(20,15)(24,20)(20,25)
\bezier{200}(10,5)(0,9)(0,20)
\bezier{200}(10,5)(10,16)(0,20)
\put(22,14){$\beta$}
\put(22,24){$\alpha$}
\put(-5,19){$\gamma$}
\put(9,0){$\sigma$}
\end{picture}
\end{center}
\caption{ \ }\label{fig(iv)}
\end{figure}
Then we prove that for every solvability
obstruction that appears in the interval 
$I[\beta,\alpha]$ of some copy of $\m n_5$ 
in some congruence lattice 
there is associated
a solvability obstruction of strictly smaller rank
in the interval $I[\sigma,\gamma]$ (cf.\ Figure~\ref{fig(iv)}).
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{.6mm}
\begin{picture}(120,94)
\put(0,45){\circle*{1.6}}
\put(15,45){\circle*{1.6}}
\put(30,45){\circle*{1.6}}
\put(60,5){\circle*{1.6}}
\put(60,85){\circle*{1.6}}
\put(67.5,10){\circle*{1.6}}
\put(67.5,80){\circle*{1.6}}
\put(75,15){\circle*{1.6}}
\put(75,75){\circle*{1.6}}
\put(90,45){\circle*{1.6}}
\put(105,35){\circle*{1.6}}
\put(105,55){\circle*{1.6}}
\put(112.5,40){\circle*{1.6}}
\put(112.5,50){\circle*{1.6}}
\put(0,45){\line(3,2){60}}
\put(0,45){\line(3,-2){60}}
\put(15,45){\line(3,2){52.5}}
\put(15,45){\line(3,-2){52.5}}
\put(30,45){\line(3,2){45}}
\put(30,45){\line(3,-2){45}}
\put(90,45){\line(3,2){15}}
\put(90,45){\line(3,-2){15}}
\put(60,5){\line(3,2){22.5}}
\put(60,85){\line(3,-2){22.5}}
\put(112.5,40){\line(-3,-2){15}}
\put(112.5,50){\line(-3,2){15}}
\put(112.5,50){\line(0,-1){10}}
\put(-9,44){$\gamma_{0}$}
\put(7,44){$\gamma_{1}$}
\put(82,44){$\gamma_{\ell}$}
\put(57,-1){$\sigma_{0}$}
\put(57,89){$\tau_{0}$}
\put(67,5){$\sigma_{1}$}
\put(67,83){$\tau_{1}$}
\put(107,32){$\sigma_{\ell}$}
\put(107,55){$\tau_{\ell}$}
\put(115.5,38.5){$\beta$}
\put(115.5,48.5){$\alpha$}
\put(90,25){\circle*{.9}}
\put(90,65){\circle*{.9}}
\put(93,27){\circle*{.9}}
\put(93,63){\circle*{.9}}
\put(87,23){\circle*{.9}}
\put(87,67){\circle*{.9}}
\put(60,45){\circle*{.9}}
\put(67.5,45){\circle*{.9}}
\put(52.5,45){\circle*{.9}}
\end{picture}
\end{center}
\caption{The lattice $\N {\ell+5}$}\label{fig(v)}
\end{figure}
By iterating this observation,
one obtains that deeply nested copies of $\m n_5$,
as depicted in Figure~\ref{fig(v)},
can appear as a sublattice of a congruence
lattice of an algebra in $\vr V$ only if solvability obstructions of 
large rank exist. We complete the proof by 
proving that the complexity of ${\rm HM}_n$
induces a uniform (finite) bound on the rank
of solvability obstructions throughout $\vr V$, hence 
for some $\ell$ it is not possible to embed
$\m n_{\ell+5}$ into the congruence lattice
of any member of $\vr V$.
This fact is converted into a nontrivial 
congruence identity satisfied by $\vr V$.\footnote{It 
is interesting that Dedekind's investigation
into the arithmetic of congruences of $\lb \mathbb C; +, -, 0\rb$ 
led to the law postulating the omission of 
sublattices isomorphic to $\m n_5$,
while the natural extension of
that investigation to arbitrary algebras
leads to ``generalized modular laws'' 
postulating the omission of 
sublattices isomorphic to $\m n_{\ell+5}$ for some $\ell$.}

This theorem gives a simple answer to the question
``Which varieties satisfy nontrivial congruence identities?''
But more importantly, the effort to prove
the theorem answers the questions
``Which varieties have trivial rectangular commutator?''
and ``What kinds of restrictions on congruence
lattice shapes are linked to having trivial
rectangular commutator?'',
since the combination
of Theorems~\ref{rect_Malt_cond2} and \ref{main1}
prove that a variety $\vr V$
satisfies a nontrivial congruence identity if and
only if the rectangular commutator is 
trivial throughout $\vr V$.
%In fact, $\vr V$ satisfies a congruence identity
%if and only if 
%all commutators that we know about trivialize
%throughout $\vr V$, except
%for the one based on the original term condition,
%and this one behaves on solvable intervals 
%exactly as it does in congruence modular varieties
%(Chapter~\ref{solvability}).

This monograph contains many characterizations of the class
of varieties whose rectangular commutator is trivial:
by the shapes of congruence lattices in these varieties
%(expressed in many ways)
%terms of the 
%congruence identities they satisfy, 
%by a certain congruence quasi-identity they satisfy,
%by omitted sublattices, or by factorizations
%of congruence lattices), 
and by the Maltsev conditions they satisfy
(see, for example,  Sections~\ref{omit_rect_tol}, 
\ref{sec_cong_id} and \ref{8.2}).
But it also contains characterizations
of other classes of varieties with parallel descriptions.
Namely, we consider the classes
of varieties (i) whose strong commutator
is trivial (Sections~\ref{str_ab}, \ref{4.2} and \ref{4.3}), 
(ii) whose strong rectangular commutator
is trivial 
(Sections~\ref{join_term}, \ref{4.2} and \ref{4.3}), 
(iii) whose TC commutator is trivial (Section~\ref{sdm_sect}), 
and (iv) whose rectangular and TC commutators
are both trivial (Section~\ref{8.3}). 
These characterizations of classes
of varieties are also in terms
of the shapes of congruence lattices in these varieties
and the Maltsev conditions they satisfy.
One byproduct of this investigation is our
first real understanding of congruence
join semidistributivity: a variety $\vr V$ is congruence
join semidistributive if and only if 
it is congruence meet semidistributive and satisfies a nontrivial
congruence identity (equivalently, if all 
commutators trivialize throughout $\vr V$).

We end the monograph with a chapter on residually small varieties satisfying
congruence identities (Chapter~\ref{RS_chap}).  The main result of this
chapter is that  a residually small variety satisfies a congruence identity if
and only if it is congruence modular, hence McKenzie's Conjecture is true for
residually small varieties.  We apply the main result of the chapter  to show
that there is no  ``almost congruence distributive'' variety that  satisfies a
nontrivial idempotent Maltsev condition.

A number of problems are posed in this monograph.
Those posed in the text are restated after Chapter 9,
together with some new problems.

%Tame congruence theory is not used in this monograph,
%but the influence of the tame congruence theory manual,
%\cite{hobby-mckenzie} by Hobby and McKenzie, will be evident
%to anyone familiar with that book. The term conditions,
%Maltsev conditions and congruence lattice shapes
%that we consider are those related to some of the 
%``omitting types theorems''
%of tame congruence theory. 


