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\begin{center}{\huge Relative Categoricity in Modules}\end{center}
\begin{center}{\large Wilfrid Hodges}\end{center}
\begin{center}{\large Queen Mary and Westfield College, 
London}\end{center}

\bigskip

This talk is an attempt to clear the decks of some work begun in the 1980s
on spectra of relative categoricity between abelian groups;  Anatolii
Yakovlev helped me plug the chief gap last year.  We consider a first-order
theory $T$ with a distinguished 1-ary relation symbol $P$, such that if $M$ is
any model of $T$ then its restriction to $P$ is a substructure $M_P$.  
For a pair
of cardinals $\kappa, \lambda$ we consider the property:  
if $M$, $N$ are models of
$T$ of cardinality $\kappa$ and $M_P$, $N_P$ are isomorphic and of cardinality
$\lambda$, then the isomorphism extends to an isomorphism from $M$ to $N$.  
If $T$
is the theory of an abelian group with subgroup defined by $P$, the property
is (for uncountable cardinals) independent of the choice of $\kappa$ and
$\lambda$.  This is not so for general $T$.

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