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\usepackage{graphics,multimedia,mathrsfs,color}

\setbeamertemplate{navigation symbols}{}

\theoremstyle{theorem-KL}
\newtheorem{theorem-KL}[theorem]{Theorem (Kearnes, Li)}

%\theoremstyle{Problem}
%\newtheorem{problem}[theorem]{Problem}

\theoremstyle{Conjecture}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{Question}
\newtheorem{question}[theorem]{Question}

\theoremstyle{Example}
\newtheorem{exmp}[theorem]{Example}

\theoremstyle{Conjecture2}
\newtheorem{conjecture2}[theorem]{Conjecture 2}

\theoremstyle{Theorems}
\newtheorem{theorems}[theorem]{Theorems}

\theoremstyle{Lemmas}
\newtheorem{lemmas}[theorem]{Lemmas}


% This file is a solution template for:

% - Giving a talk on some subject.
% - The talk is between 15min and 45min long.
% - Style is ornate.

% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice. 

\mode<presentation>
{


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%  \usetheme{Berkeley}
%%%%%%  \usetheme{Madrid}
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%  \usetheme{Rochester} %[ works best as \usetheme[height=7mm]{Rochester} ]
%  \usetheme{Copenhagen}
%  \usetheme{Singapore}
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%\usecolortheme[named=PineGreen]{structure} 

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\usefonttheme{serif}

%  \setbeamercovered{transparent}
  % or whatever (possibly just delete it)
}

\usepackage[english]{babel}
% or whatever

\usepackage[latin1]{inputenc}
% or whatever

\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever.  Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.


%%%\title[Short Paper Title] % (optional, use only with long paper titles)
%%%{Presentation Title}
\title[Finitely based algebras]{
Finitely based finite algebras
}

%%%\subtitle
%%%{Presentation Subtitle} % (optional)

%%%\author[Author, Another] % (optional, use only with lots of authors)
%%%{F.~Author\inst{1} \and S.~Another\inst{2}}
%%%% - Use the \inst{?} command only if the authors have different
%%%%   affiliation.
\author{Kearnes, Szendrei, Willard}
%\author{Kearnes\inst{U Colorado}, Szendrei\inst{U Colorado}, Willard\inst{U Waterloo}}
%\author{Kearnes}


%%%\institute[Universities of Somewhere and Elsewhere] % (optional, but mostly needed)
%%%{
%%%  \inst{1}%
%%%  Department of Computer Science\\
%%%  University of Somewhere
%%%  \and
%%%  \inst{2}%
%%%  Department of Theoretical Philosophy\\
%%%  University of Elsewhere}
%%%% - Use the \inst command only if there are several affiliations.
%%%% - Keep it simple, no one is interested in your street address.
%\institute{University of Colorado}

%%%\date[Short Occasion] % (optional)
%%%{Date / Occasion}
%\date{April 28, 2006\\ University of Hawaii}
\date[]{}
%76. Arbeitstagung Allgemeine Algebra\\May 22-25, 2008\\Linz, Austria}

%%%\subject{Talks}
%%%% This is only inserted into the PDF information catalog. Can be left
%%%% out. 


%%%% If you have a file called "university-logo-filename.xxx", where xxx
%%%% is a graphic format that can be processed by latex or pdflatex,
%%%% resp., then you can add a logo as follows:

%\pgfdeclareimage[height=2cm]{university-logo}{university-logo-NSAC2013.jpg}
%\logo{\pgfuseimage{university-logo}}


%%%% Delete this, if you do not want the table of contents to pop up at
%%%% the beginning of each subsection:
%%%\AtBeginSubsection[]
%%%{
%%%  \begin{frame}<beamer>
%%%    \frametitle{Outline}
%%%    \tableofcontents[currentsection,currentsubsection]
%%%  \end{frame}
%%%}


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command: 

%\beamerdefaultoverlayspecification{<+->}


\usepackage{amsmath,amsthm,amsfonts,latexsym,amssymb,amscd,mathrsfs}

\begin{document}

\begin{frame}
  \titlepage

\vspace{-2cm}
\begin{center}
\includegraphics[height=2cm]{NSAC2013.jpg}
\end{center}


\end{frame}

%%%\begin{frame}
%%%  \frametitle{Outline}
%%%  \tableofcontents
%%%  % You might wish to add the option [pausesections]
%%%\end{frame}


% Since this a solution template for a generic talk, very little can
% be said about how it should be structured. However, the talk length
% of between 15min and 45min and the theme suggest that you stick to
% the following rules:  

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.



\beamerdefaultoverlayspecification{<+->}
%%%\subsection[Short First Subsection Name]{First Subsection Name}
%%\subsection{Apr 3--7, 2006}


\def\m#1{\mathbf{\uppercase{#1}}}
\def\df{{\mathrm{Def}}}
\def\thm{{\mathrm{Thm}}}
\def\sd{{\mathrm{sd}}}
\def\EE{{\sf E}}
\def\UU{{\sf U}}
\def\MM{{\sf M}}
\def\PP{{\sf P}}
\def\Clo{{\mathrm{Clo}}}
\def\Pol{{\mathrm{Pol}}}
\def\Sub{{\mathrm{Sub}}}
\def\Con{{\mathrm{Con}}}
\def\Aut{{\mathrm{Aut}}}
\def\pr{{\mathrm{pr}}}
\def\var#1{{\mathcal #1}}
\def\wec#1{{\mathbf #1}}

\def\FF{{\mathfrak F}}

\begin{frame}[t]
\frametitle{Finite bases}
\pause
\noindent
The laws (or identities) defining groups are
\begin{enumerate}
\item[(1)]
$x(yz)=(xy)z$, $x1=x$, and $xx^{-1}=1$.
\end{enumerate}
\medskip

\pause
\noindent
The symmetric group $S_3$ satisfies the additional identities
\begin{enumerate}
\item[(2)]
$x^6=1$ and $x^2y^2=y^2x^2$.
\end{enumerate}
\pause

\medskip

\noindent
All identities true in $S_3$ are consequences 
of those in (1) and (2), so $S_3$ is \emph{finitely based}.

\medskip

\pause
In fact, every finite group, ring, Lie algebra, or lattice
is finitely based.
\pause
Every algebra with only 2 elements is finitely based. 
\pause
${\raisebox{-.15in} {\includegraphics[height=1cm]{liar.jpeg}}}$
%$\left({\raisebox{-.15in} {\includegraphics[height=1cm]{liar.jpeg}}}\right)$

\medskip
\pause
\noindent
An algebra ${\bf A}$ is finitely based iff the variety
it generates, ${\mathcal V}({\bf A}) = {\mathsf {HSP}}({\bf A}) = 
{\mathsf {Mod}}({\mathsf {Th_{Eq}}}({\bf A}))$ 
is finitely first-order axiomatizable.

\smallskip

\pause
\noindent
Words to know: 
\pause
algebra,
\pause
identity,
\pause
basis,
\pause
variety.
\end{frame}

\begin{frame}[t]
\frametitle{Lyndon's groupoid, Murskii's groupoid}
\pause
\noindent
Here is an groupoid that is not finitely based.
\[
{\small
\begin{array}{|c||c|c|c|c|c|c|c|}
\hline
 & 0 & 1 & 2 & 3 & 4 & 5 & 6\\
\hline
\hline
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\hline
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\hline
2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\hline
3 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
\hline
4 & 0 & 4 & 5 & 6 & 0 & 0 & 0\\
\hline
5 & 0 & 5 & 5 & 5 & 0 & 0 & 0\\
\hline
6 & 0 & 6 & 6 & 6 & 0 & 0 & 0\\
\hline
\end{array}
\hspace{.5in}
\text{Lyndon, 1954}
}
\]
\medskip

\pause
\noindent
Here is another.
\[
{\small
\begin{array}{|c||c|c|c|}
\hline
 & 0 & 1 & 2 \\
\hline
\hline
0 & 0 & 0 & 0\\
\hline
1 & 0 & 0 & 1\\
\hline
2 & 0 & 2 & 2\\
\hline
\end{array}
\hspace{.5in}
\text{Murskii, 1965}
}
\]
\end{frame}

\begin{frame}[t]
\frametitle{INFB}
\pause
\noindent
Murskii's groupoid is \emph{{\color{red}inherently} 
\color{black}nonfinitely based},
which means that it belongs to no finitely based
locally finite variety.

\medskip

\pause
\noindent
Lyndon's groupoid is not finitely based, but not INFB.
\medskip

\pause
\noindent
There exist finite abelian algebras that are not
finitely based, but there do not exist
exist finite abelian algebras that are INFB.
\medskip

\pause
\noindent
Although nonfinitely based/INFB algebras are fairly rare
in a probabilistic sense, some occur naturally:

\medskip

\pause
\noindent
{\bf Thm.} (Sapir) 
The multiplicative semigroup $M_n(\mathbb F_q)$ is INFB if $n>1$.

\medskip

\pause
\noindent
{\bf Thm.} (Dolinka) 
The semiring of all binary relations on a set satisfying 
$1<|X|<\infty$ is INFB.
\end{frame}


\begin{frame}[t]
\frametitle{Tarski's question}
\pause
\noindent
Tarski asked whether there is an algorithm 
to determine if a finite algebra is finitely based.

\medskip

\pause
\noindent
McKenzie answered the question negatively by 
constructing, from a Turing machine $\mathcal T$,
a finite algebra ${\bf A}({\mathcal T})$,
such that 
\pause
\begin{enumerate}
\item[(1)] ${\bf A}({\mathcal T})$ is finitely based if $\mathcal T$
halts, while
\pause
\item[(2)] ${\bf A}({\mathcal T})$ is INFB if $\mathcal T$
does not halt.
\end{enumerate}

\medskip
\pause
\noindent
Hence there is no algorithm to determine whether a
finite algebra is finitely based or is INFB.

\medskip

\pause
\noindent
Interestingly, 
the variety generated by
${\bf A}({\mathcal T})$ is finitely based iff
it has a finite \emph{residual bound},
which means: a finite bound on the size
of its subdirectly irreducible members.

\medskip
\pause
\noindent
Words to know: 
\pause
subdirectly irreducible algebra,
\pause
congruence,
\pause
covering pair/atom/monolith.
\end{frame}

\begin{frame}[t]
\frametitle{Park's Conjecture}
\pause
\noindent
{\bf Park's Conjecture.}
Every variety with a finite residual bound
is finitely based.
\medskip

\pause
Robert Park was a student of Kirby Baker.

\pause
He made this conjecture in his 1976 PhD thesis.

\pause
It is likely that Park was motivated by Baker's Theorem,
which proves that the statement of Park's Conjecture
is true for any variety whose members have distributive
congruence lattices.
\end{frame}

\begin{frame}[t]
\frametitle{Baker's Theorem+extensions, in the language of TCT\\
Part 1: TCT}
\pause
\noindent
{\color{red}T}\color{black}ame 
{\color{red}c}\color{black}ongruence 
{\color{red}t}\color{black}heory 
was invented by David Hobby and Ralph
McKenzie in the mid-1980's.

\medskip

\pause
\noindent
The theory explains how to assign an algebra invariant
to each congruence covering of a finite algebra, which reflects
its ``local polynomial behavior''.

\medskip

\pause
\noindent
The invariant algebras have been assigned
numbers, which have nearly no significance:
\begin{enumerate}
\item[{\bf 1}] = a simple $G$-set
\item[{\bf 2}] = a 1-dimensional vector space
\item[{\bf 3}] = 2-element Boolean algebra (or a 2-element field)
\item[{\bf 4}] = 2-element lattice
\item[{\bf 5}] = 2-element semilattice
\end{enumerate}
\end{frame}

\begin{frame}[t]
\frametitle{Baker's Theorem+extensions, in the language of TCT\\
Part 2: the theorem}

\pause
\parbox{3.5in}{
{\bf Thm.}  (Baker, 1977)
{\color{purple}Park's Conjecture} is true for varieties 
whose types are in the set
{\color{orange} $\{{\bf 3}, {\bf 4}\}$}.
}
\pause
{\raisebox{-.25in} {\includegraphics[height=1.5cm]{shock.jpeg}}}
\pause
\parbox{3.5in}{
Or, any congruence distributive variety with a finite
residual bound is finitely axiomatizable.
}

\pause
\parbox{3.75in}{
{\bf Thm.}  (McKenzie, 1987)
Park's Conjecture is true for varieties 
whose types are in the set
$\{{\bf 2}, {\bf 3}, {\bf 4}\}$.
}
\pause
{\raisebox{-.25in} {\includegraphics[height=1.5cm]{ClaudeRains.jpg}}}
\pause
\parbox{3.5in}{
Or, any congruence modular variety with a finite
residual bound is finitely axiomatizable.
}

\pause
\parbox{3.75in}{
{\bf Thm.}  (Willard, 2000)
Park's Conjecture is true for varieties 
whose types are in the set
$\{{\bf 3}, {\bf 4}, {\bf 5}\}$.
}
\pause
{\raisebox{-.25in} {\includegraphics[height=1.5cm]{rosshat.jpg}}}
\pause
\parbox{3.75in}{
Or, any variety omitting nontrivial abelian
congruences, which has a finite
residual bound, is finitely axiomatizable.
}
\end{frame}

\begin{frame}[t]
\frametitle{Proof sketches, I:}
\pause
\noindent
Let's start with Baker's Theorem that a 
congruence distributive (CD) variety $\mathcal V$
with a finite residual bound is finitely axiomatizable.

\pause
\begin{enumerate}
\item[1] Find a Maltsev characterization of the
class of varieties being considered. %(J\'onsson)
\pause
\item[2] Use it 
to show that the relation 
${\rm Cg}(a,b)\cap {\rm Cg}(c,d)=0$ is uniformly 
definable throughout $\mathcal V$, say by 
$\Omega(a,b,c,d)$. 
\pause
\item[3] Show that there is an ``enforcing'' 
%first-order 
sentence
%, called ``Clint'', 
that asserts ``$\Omega$ works''.
\pause
{\raisebox{-.25in} {\includegraphics[height=1.5cm]{clint.jpeg}}}
\pause
\item[End] Choose finitely many identities $\Sigma_1$
of $\mathcal V$ that prove 
%``Clint'' 
%that 
the enforcing sentence.
% is true.
%(Do we really need any identities at all to prove this?)
%\pause
\pause
Axiomatize $\mathcal K$ with 
$\Sigma_1$ and
$\Omega(a,b,c,d)\to (a=b\;\text{or}\;
c=d)$.
\pause
$\mathcal K\cap {\mathcal V}={\mathcal V}_{FSI}
\pause
={\mathcal V}_{SI}$.
\pause
Choose finitely many identities $\Sigma_2$
of $\mathcal V$ that prove 
$\mathcal K\cap {\mathcal V}$ is the class
of SI's of $\mathcal V$. \pause 
$\Sigma_1\cup \Sigma_2$
axiomatizes $\mathcal V$.

\end{enumerate}
\end{frame}

\begin{frame}[t]
\frametitle{Proof sketches, II:}
\pause
\noindent
Willard's Theorem: proved the same way.
The key innovations are:

\pause
\begin{enumerate}
\item[1] Willard reframed the known Maltsev condition
for the 
class of varieties with types in $\{{\bf 3}, {\bf 4}, {\bf 5}\}$
in a really new and useful way.
\pause
\item[2] The rest of his argument is like Baker's, but
proving that the relation 
${\rm Cg}(a,b)\cap {\rm Cg}(c,d)=0$ is uniformly 
definable throughout $\mathcal V$ is harder.
\end{enumerate}
\medskip

\pause
\noindent
McKenzie's Theorem: starts out in a similar way,
but the bulk of the proof is based on 
an entirely new and rich set of ideas.
\pause
The problem that forces a change of strategy is that
${\rm Cg}(a,b)\cap {\rm Cg}(c,d)=0$ is 
not uniformly 
definable throughout $\mathcal V$. 
\pause
Rather, 
an analogous relation 
$[{\rm Cg}(a,b), {\rm Cg}(c,d)]=0$ is uniformly
definable,
but this fact is not strong enough on its
own to follow the template set out by Baker.
\pause
(For example, if $\mathcal V$ is generated
by a finite solvable group, then the class
$\mathcal K$ axiomatized by
$\Sigma_1$ and
$[{\rm Cg}(a,b),{\rm Cg}(c,d)]=0 \to (a=b\;\text{or}\;
c=d)$ is trivial, so can't help
prove the finite axiomatizability of $\mathcal V$.)


\end{frame}

\begin{frame}[t]
\frametitle{Reasonable contexts for the next step}
\pause
\noindent
Of course, one should try to prove Park's 
conjecture for varieties whose types are in the set
$\{{\bf 2}, {\bf 3}, {\bf 4}, {\bf 5}\}$ next.
\pause
These varieties 
are exactly those that have a \emph{weak difference term},
which is a term $d(x,y,z)$ such that, whenever
${\bf A}\in {\mathcal V}$ and ${\rm Cg}(a,b)$
is abelian, then $d(a,b,b)=a$ and $d(a,a,b)=b$ hold.
\pause
Necessarily $d(x,y,z)=x-y+z$ on 
the classes of any abelian congruence of a member of $\mathcal V$.

\medskip
\pause
\noindent
There is a slightly more restrictive (but natural) condition:
a term $d(x,y,z)$ is 
a \emph{difference term}
for $\mathcal V$ if $d(x,x,y)=y$ holds throughout $\mathcal V$
and $d(a,b,b)=a$ holds when ${\rm Cg}(a,b)$ is abelian.
\pause
\noindent
In the language of TCT, $\mathcal V$ has a difference term iff
its types are in $\{{\bf 2}, {\bf 3}, {\bf 4}, {\bf 5}\}$ 
\emph{\color{red} and} type-{\bf 2} minimal sets have empty tail.
\pause
This class of varieties includes all of the varieties
covered by McKenzie's and Willard's Theorems, but
is nicer than the class of varieties with a weak
difference term 
because $[\alpha,\beta]=[\beta,\alpha]$
for any congruences $\alpha$ and $\beta$
on an algebra in this kind of variety.
\end{frame}

\begin{frame}[t]
\frametitle{Toward Park's Conjecture
for varieties with a difference term}
\pause
\noindent
{\bf Thm.} (K, Szendrei, Willard)
If $\mathcal V$ is a variety with a difference term
and $\mathcal V$ has a finite residual bound, then
$\mathcal V$ is not INFB.

\medskip

\pause
\noindent
The structure of the proof is similar to the proof
of McKenzie's Theorem.
The main new ingredient is a proof
that $[{\rm Cg}(a,b), {\rm Cg}(c,d)]=0$ is uniformly
definable in varieties with a difference term
and a finite residual bound.

\medskip
\pause
\noindent
We actually prove something stronger
than the  above theorem, but
we do not prove that $\mathcal V$ is finitely
based. \pause
We have found an obstacle to proving this,
which does not appear if type {\bf 2} is missing
(as in Willard's Theorem) or if
atomic congruences are uniformly definable
(as in McKenzie's Theorem).
\end{frame}

\begin{frame}[t]
\frametitle{}
\pause
\noindent
I now think Park's Conjecture is false
(even for varieties with a difference term),
but the following weakening might still be true:

\medskip
\pause
\noindent
\emph{A variety with a finite residual bound is not INFB.}

\medskip
\pause
\noindent
A recent result of Kate Owens lends support to this statement:

\medskip
\pause
\noindent
{\bf Thm.}
A variety that can be shown to be INFB using the
shift automorphism method must contain an infinite
subdirectly irreducible algebra.

\end{frame}

\end{document}

