%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  A Characterization of Varieties That Satisfy A
%  Nontrivial Congruence Identity
%  Feb 1999
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Conventions
%
%  definitions are in bold face, \em is used for emphasis
%  identities expressed with \approx rather than =
%  spelling is: Maltsev
%  \vr for varieties
%  \lneqq for (\leq but not =)
%  \subsetneqq
%  congruences: \alpha, \beta, \gamma, \delta, \theta, \mu, \nu
%  homomorphisms: \varphi, \psi
%  algebra: \m a = \lb A; \{f_i\wrel{|}i\in I\rb
%  Con(A) as a lattice: \Conb(\m a)
%  Eq(A) as a lattice: \Eqb(A)
%
%  (1), (2), ...  Independent statements in Thm.,TFAE     ***FIX!
%  (i), (ii), ... Properties or conditions in Df or Thm.
%                 Also: sublists within a (1), (2), ...--list
%  (a), (b), ...  items in a paragraph
%  no $'s on these, as in $(1)$ or $(1)'$. Write (1) and (1)' instead
%
%  \medskip after the proof of a claim.
%  pairs of pairs: \lb (a,b), (c,d) \rb
%  \rightarrow used for implication in formulas, \Rightarrow(\mimplies) 
%     used as meta-implication
%  (W) is defined by a macro throughout EXCEPT in the definition
%      of (W) and of a quasi-id satisfying (W). This is to allow
%      for bold face in the definition. If the macro is changed,
%      then change the defns too.  
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt,openany]{memo-l}
\usepackage{amsmath,amsthm,amsfonts,latexsym,amssymb} 
\usepackage[all]{xy}
\makeindex

\includeonly{%preface,
cong_1,
cong_2,
cong_3,
cong_4,
cong_5,
cong_6,
cong_7,
cong_8,
cong_9,
%cong_10,
%cong_11,
cong_b,
index}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%   ENVIRONMENTS
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \theoremstyle{plain} %% This is the default
\newtheorem{thm}{Theorem}[chapter]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lm}[thm]{Lemma}
\newtheorem{prp}[thm]{Proposition}
\newtheorem{clm}[thm]{Claim}
\newtheorem{subclm}[thm]{Subclaim}
%
\theoremstyle{definition}
\newtheorem{df}[thm]{Definition}
\newtheorem{notat}[thm]{Notation}
\newtheorem{exmp}[thm]{Example}
\newtheorem{exmps}[thm]{Examples}
\newtheorem{exer}[thm]{Exercise}
\newtheorem{prb}[thm]{Problem}
\newtheorem{xca}[thm]{Exercise}
%
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
\newtheorem{rems}[thm]{Remarks}
%
\numberwithin{section}{chapter}
\numberwithin{figure}{chapter}
\numberwithin{equation}{chapter}
%
\newcommand{\thref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lmref}[1]{Lemma~\ref{#1}}
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Math macros
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Logical symbols
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\DeclareMathOperator{\satf}{\models}
\DeclareMathOperator{\miff}{\Longleftrightarrow} 
                                           % math `if and only if'
\DeclareMathOperator{\mimplies}{\Longrightarrow} 
                                           % math `implies'
\DeclareMathOperator{\mimplied}{\Longleftarrow}  
                                           % math `implied by'
\newcommand{\set}[2]{\{#1\mid\nobreak\text{#2}\}} % { x | ... }
\DeclareMathOperator{\id}{\textsl{id}}     % The identity map
\DeclareMathOperator{\miffdef}{\stackrel{def}{\miff}}
                                           % defining iff
\DeclareMathOperator{\eqdef}{\stackrel{def}{=}}
                                           % defining equation

\DeclareMathOperator{\neareq}{\stackrel{\ae}{=}}
\DeclareMathOperator{\aeqv}{\stackrel{ae}{\equiv}}
\DeclareMathOperator{\aeq}{\stackrel{ae}{=}}
\newcommand{\labelledarrow}[1]{%
\stackrel{#1}{\longrightarrow}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%  Classical math
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\DeclareMathOperator{\BN}{\mathbb{N}} % nonnegative integers
\DeclareMathOperator{\BZ}{\mathbb{Z}} % integers
\DeclareMathOperator{\BQ}{\mathbb{Q}} % rationals
\DeclareMathOperator{\BR}{\mathbb{R}} % reals
\DeclareMathOperator{\BC}{\mathbb{C}} % complex numbers
\DeclareMathOperator{\im}{{\rm im}}         % imaginary part
%
\DeclareMathOperator{\meet}{\wedge}
\DeclareMathOperator{\join}{\vee}
\DeclareMathOperator{\persu}{\nearrow}
\DeclareMathOperator{\persd}{\searrow}
\DeclareMathOperator{\union}{\cup}
\DeclareMathOperator{\inter}{\cap}  % intersection
\DeclareMathOperator{\iso}{\cong}   % isomorphic
\DeclareMathOperator{\piso}{\simeq} % polynomially isomorphic
\newcommand{\Star}{^ *}
\DeclareMathOperator{\compose}{{\scriptstyle \circ}} % f o g
%
\newcommand{\wec}[1]{{\mathbf{#1}}}  % notation for vectors in algebras
%\newcommand{\wec}[1]{{\bar{#1}}} % which one do you prefer?
\newcommand{\wrel}[1]{\;#1\;}     % a\wrel{\alpha}b = a \alpha b
%              % this provides a flexible way to change the spacing
%
%\newcommand{\wrel}[1]{{\def\wrelarg{\qopname\relax{no}{#1}}\wrelarg}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%  Operators on algebras
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%%  \m is used to produce models (algebras) as follows:
%%%  typing `\m a'  produces in bold face, A.
%%%  \vr does the same for varieties
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\m}[1]{{\mathbf{\uppercase{#1}}}}
\newcommand{\vr}[1]{{\mathcal{\uppercase {#1}}}}
\newcommand{\C}[1]{{\mathbf{\uppercase{#1}}}}
%
\DeclareMathOperator{\Var}{\mathsf{V}}
\DeclareMathOperator{\Ho}{\mathsf{H}}
\DeclareMathOperator{\Su}{\mathsf{S}}
\DeclareMathOperator{\Pd}{\mathsf{P}}
\DeclareMathOperator{\Iso}{\mathsf{I}}
\DeclareMathOperator{\Ps}{\Pd_s}
\DeclareMathOperator{\Kluk}{\mathsf{K}}
\DeclareMathOperator{\twin}{Tw}
\newcommand{\fin}[1]{{\mathsf{#1}}_{\text{\upshape fin}}}
%
\DeclareMathOperator{\Pol}{Pol}
\DeclareMathOperator{\Clo}{Clo}
\DeclareMathOperator{\Con}{Con}
\DeclareMathOperator{\CON}{\rm{CON}} % CON(V)
\DeclareMathOperator{\Conb}{\mathbf{Con}} % Con(A) as a lattice
\DeclareMathOperator{\Eqb}{\mathbf{Eq}} % Eq(A) as a lattice
\DeclareMathOperator{\cg}{Cg}
\DeclareMathOperator{\tg}{Tg}
\DeclareMathOperator{\sg}{Sg}
\DeclareMathOperator{\eq}{Eq}
\DeclareMathOperator{\cs}{\#} % Congruence size
\DeclareMathOperator{\chs}{c} % Chief factor size
\newcommand{\Cg}[1]{{\cg}^{\m{#1}}}
\newcommand{\Tg}[1]{{\tg}^{\m{#1}}}
\newcommand{\Sg}[1]{{\sg}^{\m{#1}}}
\newcommand{\Eq}[1]{{\eq}^{{#1}}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  macros for tame congruence theory.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\M}[3]{\text{\upshape M}_{\m{#1}} (#2, #3)}
\newcommand{\minl}[1]{\text{\upshape Min}_{\m{#1}}}
%
\DeclareMathOperator{\typ}{typ}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\Sep}{Sep}
\DeclareMathOperator{\ssb}{ssb}
\DeclareMathOperator{\sst}{sst}
%
\DeclareMathOperator{\utyp}{\mathbf{1}}% unary      
\DeclareMathOperator{\atyp}{\mathbf{2}}% affine     
\DeclareMathOperator{\btyp}{\mathbf{3}}% boolean    
\DeclareMathOperator{\ltyp}{\mathbf{4}}% lattice    
\DeclareMathOperator{\styp}{\mathbf{5}}% semilattice
\DeclareMathOperator{\ityp}{\mathbf{i}}
\DeclareMathOperator{\jtyp}{\mathbf{j}}
%
\DeclareMathOperator{\solv}{\stackrel{s}{\sim}}
\DeclareMathOperator{\ssolv}{\stackrel{ss}{\sim}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Abbreviations
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}
%
\newcommand{\lb}{\langle} % <...>
\newcommand{\rb}{\rangle}
\newcommand{\lv}{{[\![}}  % [[...]]
\newcommand{\rv}{{]\!]}}
%
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
%
\newcommand{\vep}{\varepsilon}
\newcommand{\di}{\diamond}
\newcommand{\mult}{\diamond}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Hyphenation
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\hyphenation{Mal-tsev}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% New Keith-macros
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\tw}{Tw}
\DeclareMathOperator{\rad}{{\rm rad}}    
\newcommand{\s}[1]{\underset{\mbox{\scriptsize\sl #1}}{\textstyle\bold \sim}}
\newcommand{\spec}[1]{\Spec_{#1}}
\newcommand{\N}[1]{{\mathbf{N}}_{#1}}
\newcommand{\XXX}{{\infty}}
\newcommand{\obst}{{\blacktriangleleft}}
\newcommand{\lhdlhd}{{\lhd\!\!\lhd}}
\newcommand{\W}{\mbox{\rm (W)}}
\newcommand{\sdj}{{\rm SD}_{\vee}}
\newcommand{\sdm}{{\rm SD}_{\wedge}}
\newcommand{\Mod}{{\rm Modular}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% End of math definitions
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

\frontmatter
\title{The Shape of Congruence Lattices}
%{Term Conditions, Maltsev Conditions, and the
%Shape of Congruence Lattices}
%\title[Varieties That Satisfy A Nontrivial
%Congruence Identity]
%{A Characterization Of Varieties That Satisfy A Nontrivial
%Congruence Identity}
\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/$^\sim$kearnes}
\thanks{The first author was supported in part
by NSF Grant \#9802922.}
\author{Emil W. Kiss}
\address[Emil W. Kiss]{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}
%\address{Department of Algebra and Number Theory,
%E\"{o}tv\"{o}s Lor\'{a}nd University, 
%M{\'u}zeum krt. 6--8, 
%1088 Budapest, Hungary}
\email{ewkiss@cs.elte.hu}
\urladdr{http://www.cs.elte.hu/$^\sim$ewkiss}
\thanks{The second author was supported in part 
by OTKA Grants \#T043671 and \#T043034.}
%%%\#T029525 and \#T023186.}

\subjclass{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,
%projective lattice,
rectangulation, 
residual smallness,
solvable,
term condition,
variety,
weak difference term.}

\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}

\maketitle


\newpage

\null

\vskip 3cm

\centerline{\sl\large \hfill Dedicated to Bjarni J\'{o}nsson, \hfill\hfill}
\centerline{\sl\large \hfill For his work on congruence varieties, \hfill\hfill}

\vskip 2cm

\centerline{\sl\large \hfill To Ralph McKenzie, \hfill\hfill}
\centerline{\sl\large \hfill For his work on commutator theories, \hfill\hfill}

\vskip 2cm

\centerline{\sl\large \hfill And to Walter Taylor, \hfill\hfill}
\centerline{\sl\large \hfill For his work on Maltsev conditions. \hfill\hfill}

\vfill\newpage


\setcounter{page}{5}


\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_10}
%\include{cong_11}
\backmatter
\include{cong_b}
\include{index}
\printindex
\end{document}






