\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 $a, b\in \mathbb C$ and $\mu\subseteq
\mathbb C$; (\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'').  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 \big(\beta +
(\alpha\cap\gamma)\big) =
(\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
\mathrel{\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){$\beta+\alpha = \beta+\gamma$}
\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 \big(\alpha_0\cap (\gamma + \alpha_1)\big) +
\big(\beta_0\cap (\gamma + \beta_1)\big)
\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{That is, $\vr
U$ 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 $\{\vr V\mid \vr U\leq \vr V\}$ 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 $\{\vr
V\mid \exists n ({\vr U}_n\leq \vr V)\}$ 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\big(\{\Conb(\m a)\mid \m a\in \vr V\}\big)\,.
\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 continuum many
  different nonmodular congruence varieties (see
  \cite{hutchinson-czedli}), 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
$\beta$ are said to 
{\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~$\m b\leq_\textrm{sd} \m a_1\times
\m a_2$ 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 $\{\lb (u,u), (v,v)\rb\mid (u,v)\in \alpha\}$. 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. Dedekind's analysis of $3$-variable consequences
of the modular law shows that the $3$-generated free modular
lattice has only $28$ elements, hence any $3$-generated
modular lattice can be easily drawn.
\begin{figure}[ht]
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\begin{picture}(90,75) \put(0,30){\circle*{1.2}}
\put(15,20){\circle*{1.2}} \put(15,40){\circle*{1.2}}
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\put(30,50){\circle*{1.2}} \put(45,0){\circle*{1.2}}
\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}}
\put(75,20){\circle*{1.2}} \put(75,40){\circle*{1.2}}
\put(90,30){\circle*{1.2}}

\put(30,60){\circle*{1.2}} \put(45,50){\circle*{1.2}}
\put(45,70){\circle*{1.2}} \put(60,60){\circle*{1.2}}

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

\thinlines \put(45,0){\line(3,2){45}}
\put(30,10){\line(3,2){45}} \put(15,20){\line(3,2){45}}
\put(0,30){\line(3,2){45}} \put(45,0){\line(-3,2){45}}
\put(60,10){\line(-3,2){45}} \put(75,20){\line(-3,2){45}}
\put(90,30){\line(-3,2){45}} \put(30,40){\line(3,2){30}}
\put(30,60){\line(3,2){15}} \put(45,30){\line(-3,2){15}}
\put(45,50){\line(-3,2){15}} \put(60,60){\line(-3,2){15}}

\put(30,60){\line(0,-1){10}} \put(45,70){\line(0,-1){10}}
\put(60,60){\line(0,-1){10}} \put(45,50){\line(0,-1){30}}

\thicklines \put(30,40){\line(3,2){15}}
\put(60,10){\line(3,2){15}} \put(15,40){\line(3,2){15}}

\put(93,29){$\eta_2=1\times 0$} \put(-27,29){$\eta_1=0\times
1$} \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(85,60){\small $\theta=\beta\times
1 = 1\times \beta$} \put(100,47){\small
$\psi=\alpha\times\alpha$}
\end{picture}
\end{center}
\caption{$\lb\eta_1,\eta_2,\Delta\mid
  \eta_1\cap\eta_2=0\rb$}\label{fig(iii)}
\end{figure}
In particular, the modular lattice with the presentation
$\lb\eta_1,\eta_2,\Delta\mid \eta_1\cap\eta_2=0\rb$ 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
\[
\big(\Delta + (0\times 1)\big)\cap \big(\Delta +
(1\times 0)\big) = \alpha\times \alpha = \psi\,.
\]
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\big[\eta_1+(\Delta\cap \eta_2),\;\eta_1+(\psi\cap
\eta_2)\big]= I\big[\eta_1+(\Delta\cap
\eta_2),\;(\alpha\cap \beta)\times 1\big],
\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]
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\put(15,0){\circle*{1.2}} \put(15,10){\circle*{1.2}}
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\put(30,30){\circle*{1.2}}

\put(15,20){\line(0,-1){20}} \put(15,20){\line(3,2){15}}
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\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(-12,-1.5){$\eta_1=0\times 1$}
\put(-15,28.5){$\alpha\times 1$} \put(32,28.5){$\beta\times
1$} \put(-19,8.5){$\eta_1 + (\Delta\cap \eta_2)$}
\put(-18,18.5){$(\alpha\cap\beta)\times 1$}
\put(-3,42){$(\alpha+\beta)\times 1$}

\put(87,-1.5){$0$} \put(64,28.5){$\alpha$}
\put(102,28.5){$\beta$} \put(87,8.5){?}
\put(89,18.5){$\alpha\cap\beta$}
\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 the algebra~$\m a$
abelian if the entire lattice $\Conb(\m a)=[0,1]$ is an
abelian congruence interval. If $\tau$ is defined to be
equal to~$\alpha\cap\beta$ in the last paragraph, then
$\tau\leq \alpha$ and $\tau\leq \beta$ so \hbox{$[\tau,\tau]
\leq [\alpha,\beta]$.}  This shows that the interval $I\big[
[\alpha,\beta],\alpha\cap\beta \big]$, 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\big([\alpha,\beta] = \alpha\cap\beta\big)
\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 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} in
\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} (see~\cite{kiss-prohle}), and established
its usefulness.  His definition was slightly short of a term
condition for rectangulation, and he did not develop a
corresponding commutator theory.

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 $\vr V$.

This monograph contains many characterizations of the class
of varieties whose rectangular commutator is trivial: by the
shapes of congruence lattices in these varieties 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 (see page
\pageref{problems}), together with some new problems.

We attach an appendix which collects
the many characterizations from the monograph of the classes 
of varieties that have Taylor terms or Hobby-McKenzie terms.

\section{Thanks!}

We are happy to express our gratitude to those
who supplied comments, corrections and encouragement
during the preparation of this monograph,
including:
Joel Berman,
Ralph Freese,
Pawe{\l} Idziak,
Bill Lampe,
Paolo Lipparini,
Ralph McKenzie,
George McNulty,
J.~B.~Nation, 
Peter Ouwehand,
James Raftery,
\'Agnes Szendrei,
Walter Taylor,
Jir\'i Tuma,
Matthew Valeriote,
Fred Wehrung, and
Ross Willard.
