%
% file: aaa-brno.tex
% purpose: talk at AAA91, Brno, Feb 2016
% created: pasha jan 28,31-feb 3 2016
% modified:
% modification:
%
\documentclass{beamer}
\usepackage{mathrsfs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\set}[2]{\ensuremath{\{ #1 \>|\> #2 \}}}
\newcommand{\modulus}[1]{\ensuremath{|\, #1 \,|}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\setcounter{framenumber}{-1}
\setbeamertemplate{navigation symbols}{}
\title{Robinson--Amitsur ultrafilters}
\author{Pasha Zusmanovich}
\institute{University of Ostrava}
\date{February 6, 2016}
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\titlepage
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% from now on, display counters
\setbeamertemplate{headline}
{
\vskip2pt\hskip1pt\insertframenumber / \inserttotalframenumber
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{A theorem from 1960s}
\begin{block}{Theorem (S. Amitsur, A. Robinson)}
If a prime associative ring $R$ embeds in a direct product of associative
division rings, then $R$ embeds in an associative division ring.
\end{block}
\begin{block}{Proof}
\smallskip
Given embedding: $R \subseteq \prod_{i\in \mathbb I} A_i$.
\smallskip
$\mathcal S = \set{\set{i\in \mathbb I}{f_i \ne 0}}{f\in R, f\ne 0}$.
\smallskip
Primeness of $R$ $\Rightarrow$
finite intersection property of $\mathcal S$ $\Rightarrow$
$\mathcal S$ extends to an ultrafilter $\mathcal U$.
\smallskip
$R \subseteq \prod_{\mathcal U} A_i$ \hskip 15pt + {\L}o\'s' theorem.
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{A generalization from 2010s}
\begin{block}{Theorem (``Robinson--Amitsur for algebraic systems'')}
For any algebraic system $A$ the following are equivalent:
\begin{enumerate}[(i)]
\item
$A$ is finitely subdirectly irreducible;
\item
For any set $\{ B_i \}_{i\in \mathbb I}$ of algebraic systems,
$A \subseteq \prod_{i\in \mathbb I} B_i \>\Rightarrow\>
\exists \text{ ultrafilter } \mathscr U \text{ on } \mathbb I:
A \subseteq \prod_{\mathscr U} B_i$.
\end{enumerate}
\end{block}
\begin{block}{Remark}
For rings and algebras, primeness $\Rightarrow$ finite subdirect irreducibility.
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Birkhoff meets Robinson--Amitsur}
\begin{block}{Criterion for absence of nontrivial identities}
Let $\mathfrak V$ be a variety of algebraic systems such that any free system in
$\mathfrak V$ is finitely subdirectly irreducible. Then for an algebraic system
$A \in \mathfrak V$ the following are equivalent:
\begin{enumerate}[(i)]
\item
$A$ does not satisfy nontrivial identities within $\mathfrak V$;
\item any free system of $\mathfrak V$ embeds in an ultrapower of $A$;
\item any free system of $\mathfrak V$ embeds in a system elementarily equivalent to $A$.
\end{enumerate}
\end{block}
\begin{block}{Proof}
(i) $\Rightarrow$ (ii) follows from Birkhoff's theorem + Robinson--Amitsur.
(ii) $\Rightarrow$ (iii) follows from {\L}o\'s' theorem.
(iii) $\Rightarrow$ (i) is trivial.
\end{block}
\begin{block}{Applicable to:}
All groups, Burnside varieties of groups,
all algebras, associative algebras, Lie algebras.
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Semigroups?}
\begin{block}{Question}
What about semigroups? Inverse semigroups? Burnside varieties of semigroups?
etc...
\end{block}
\begin{block}{An obstacle}
Free semigroups are not finitely subdirectly irreducible.
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Applications}
\begin{block}{``Baby'' Regev's theorem}
If $A$ is a finite-dimensional associative algebra, and $B$ is PI, then
$A\otimes B$ is PI.
\end{block}
\begin{block}{Algebras with the same identities (Kushkulei, Razmyslov, et al.)}
If $\mathfrak g_1, \mathfrak g_2$ are finite-dimensional simple objects
in some classes of algebras (Lie, Jordan, etc.), then
$Var(\mathfrak g_1) = Var(\mathfrak g_2) \>\Leftrightarrow\>
\mathfrak g_1 \simeq \mathfrak g_2$.
\end{block}
\begin{block}{Growth sequence of Tarski's monsters}
Under some additional assumptions, the growth sequence (number of generators of
$\underbrace{G \times \dots \times G}_{n \text{ times}}$) of Tarski's monster
$G$ is constant, equal to $2$.
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Another generalization}
\begin{block}{Theorem (``Robinson--Amitsur: from $\omega$ to $\kappa$'')}
For any algebraic system $A$, and any cardinal $\kappa > 2$ such that any
$\kappa$-complete filter can be extended to a $\kappa$-complete ultrafilter,
the following are equivalent:
\begin{enumerate}[(i)]
\item
$A$ is $\kappa$-subdirectly irreducible;
\item
For any set $\{ B_i \}_{i\in \mathbb I}$ of algebraic systems,
$A \subseteq \prod_{i\in \mathbb I} B_i \>\Rightarrow\>$
\newline
$\exists$ $\kappa$-complete ultrafilter $\mathscr U \text{ on } \mathbb I:
A \subseteq \prod_{\mathscr U} B_i$.
\end{enumerate}
\end{block}
\begin{block}{A disappointment}
No corollary similar to criterion for absence of nontrivial identities
(second-order logic, big cardinals, ...)
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Dual situation}
\begin{block}{Theorem (Bergman--Nahlus)}
For any algebraic system $A$, and any cardinal $\kappa > 2$, the following are
equivalent:
\begin{enumerate}[(i)]
\item
For any surjective homomorphism $f: \prod_{i\in \mathbb I} B_i \to A$,
$\modulus{\mathbb I} < \kappa$, there is $i_0 \in \mathbb I$ such that
$f$ factors through the canonical projection
$\prod_{i\in \mathbb I} B_i \to B_{i_0}$.
\item
For any surjective homomorphism $f: \prod_{i\in \mathbb I} B_i \to A$,
there is a $\kappa$-complete ultrafilter
$\mathscr U$ on $\mathbb I$ such that $f$ factors through the
canonical homomorphism $\prod_{i\in \mathbb I} B_i \to \prod_{\mathscr U} B_i$.
\end{enumerate}
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{More questions}
\begin{block}{Question (Zilber)}
Whether an ultraproduct of finite groups can be mapped surjectively on $SO(3)$?
\end{block}
\begin{block}{Remark}
By Bergman--Nahlus, ``ultraproduct'' can be replaced by ``direct product''.
\end{block}
\begin{block}{Another question}
Robinson--Amitsur for metric ultraproducts?
\end{block}
(Related to sofic groups, continuous first-order logic, etc.)
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}
Based on:
\begin{itemize}
\item
On the utility of Robinson-Amitsur ultrafilters,
J. Algebra \textbf{388} (2013), 268--286; \textsf{arXiv:0911.5414}
\item
On the utility of Robinson-Amitsur ultrafilters. II, \textsf{arXiv:1508.07496}
\end{itemize}
Slides at \url{http://www1.osu.cz/~zusmanovich/math.html}
\vskip 1.5cm
\begin{center}
{\Huge That's all. Thank you.}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
% eof