%%%% David Pierce's abstract for LC 2005, Athens
%%%% prepared  2005.04.15
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\title{Model-theory of Lie-rings}

\author{David Pierce}
%\revauthor{Pierce, David}

\address{Mathematics Department\\
Middle East Technical University\\
Ankara 06531, Turkey}

\email{dpierce@math.metu.edu.tr}
\urladdr{http://www.math.metu.edu.tr/\textasciitilde dpierce/}

\newcommand{\End}{\operatorname{End}}

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The prototypical Lie-ring is
$(\End(E),\mathord{\circ}-\mathord{\circ}')$, where $E$ is an abelian
group, $\End(E)$ is the abelian group of its endomorphisms, $\circ$ is
composition, and $x\circ'y=y\circ x$.  Then an arbitrary ring
$(E,\cdot)$ is a Lie-ring if
\begin{gather*}
  x\cdot y+y\cdot x=0,\\
(x\cdot y)\cdot z=x\cdot(y\cdot z)-y\cdot(x\cdot z).
\end{gather*}
The latter identity means that the map $u\mapsto(z\mapsto u\cdot z)$
is a homomorphism from $(E,\cdot)$ to
$(\End(E),\mathord{\circ}-\mathord{\circ}')$.  Commutative rings then
have a parallel definition: $x\cdot y-y\cdot x=0$, and
$u\mapsto(z\mapsto u\cdot z)$ is a homomorphism from $(E,\cdot)$ to
$(\End(E),\circ)$ .  Other rings can be defined in terms of
different linear combinations $*$ of $\circ$ and $\circ'$, but they don't
behave so nicely: we do not in general have that 
$u\mapsto(z\mapsto u*z)$ is a homomorphism from $(\End(E),*)$ to
$(\End(\End(E)),*)$ unless $*$ is $\circ$ or
$\mathord{\circ}-\mathord{\circ}'$ or trivial.

To what extent will the model-theory of Lie-rings parallel that of
commutative rings (integral domains, fields)?
The set of derivations of a commutative ring is naturally a Lie-ring.
Hence, for example, the model-theory of differential fields gives rise
to model-complete and $\omega$-stable theories of (expansions of)
Lie-rings. 


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