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% The Shape of Congruence Lattices
% Kearnes, Kiss, 2011.
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\begin{document}
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\frontmatter
\title{The Shape of Congruence Lattices}
\author{Keith A. Kearnes}
\address{Department of Mathematics\\
University of Colorado\\
Boulder, CO 80309-0395\\ 
USA}
\email{kearnes@euclid.colorado.edu}
\urladdr{http://spot.colorado.edu/\~{}kearnes}
\thanks{The first author was supported in part
by NSF Grant \#9802922.}
\author{Emil W. Kiss}
\address[Emil W. Kiss]{Lor{\'a}nd E{\"o}tv{\"o}s University\\
Department of Algebra and Number Theory\\
1117 Budapest, P{\'a}zm{\'a}ny P{\'e}ter
s{\'e}t{\'a}ny 1/c\\
Hungary}
\email{ewkiss@cs.elte.hu}
\urladdr{http://www.cs.elte.hu/\~{}ewkiss}
\thanks{The second author was supported in part 
by OTKA Grants \#T043671 and \#T043034.}
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\subjclass[2010]{Primary 08B05;\\ Secondary 08B10}
\keywords{Abelian,
almost congruence distributivity,
commutator theory,
compatible semilattice operation,
congruence identity, 
congruence modularity,
congruence semidistributivity,
Maltsev condition, 
meet continuous lattice,
rectangulation, 
residual smallness,
solvable,
tame congruence theory,
term condition,
variety,
weak difference term.}
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\begin{abstract}
  We develop the theories of the strong commutator, the
  rectangular commutator, the strong rectangular commutator,
  as well as a solvability theory for the nonmodular TC
  commutator. These theories are used to show that each of
  the following sets of statements are equivalent for a
  variety~$\vr V$ of algebras.
  \begin{enumerate}
  \item[(I)]
    \begin{enumerate}
    \item $\vr V$ satisfies a nontrivial congruence
      identity.
    \item $\vr V$ satisfies an idempotent Maltsev condition
      that fails in the variety of semilattices.
    \item The rectangular commutator is trivial
      throughout~$\vr V$.
    \end{enumerate}
  \item[(II)]
    \begin{enumerate}
    \item $\vr V$ satisfies a nontrivial meet continuous
      congruence identity.
    \item $\vr V$ satisfies an idempotent Maltsev condition
      that fails in the variety of sets.
    \item The strong commutator is trivial throughout~$\vr
      V$.
    \item The strong rectangular commutator is trivial
      throughout~$\vr V$.
    \end{enumerate}
  \item[(III)]
    \begin{enumerate}
    \item $\vr V$ is congruence semidistributive.
    \item $\vr V$ satisfies an idempotent Maltsev condition
      that fails in the variety of semilattices and in any
      nontrivial variety of modules.
    \item The rectangular and TC commutators are both
      trivial throughout~$\vr V$.
    \end{enumerate}
  \end{enumerate}
  We prove that a residually small variety that satisfies a
  congruence identity is congruence modular.
\end{abstract}
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\vspace*{3cm}

\begin{center}
\textsl{\large Dedicated to Bjarni J\'{o}nsson,}\\[2pt]
\textsl{\large For his work on congruence varieties,}
\end{center}

\vspace{2cm}

\begin{center}
\textsl{\large To Ralph McKenzie,}\\[2pt]
\textsl{\large For his work on commutator theories,}
\end{center}

\vspace{2cm}

\begin{center}
\textsl{\large And to Walter Taylor,}\\[2pt]
\textsl{\large For his work on Maltsev conditions.}
\end{center}

\vfill
\cleardoublepage

\setcounter{page}{7}

\tableofcontents

\mainmatter
\include{cong_1}
\include{cong_2}
\include{cong_3}
\include{cong_4}
\include{cong_5}
\include{cong_6}
\include{cong_7}
\include{cong_8}
\include{cong_9}
\include{cong_problems}
\appendix
\include{cong_appendix}
\backmatter
\include{cong_bib}
\printindex
\end{document}
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