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This paper treats a small but significant error in the model-theoretic
literature on unimodularity.
A unimodular strongly minimal set is one in which for any interalgebraic
independent tuples $\bar{a}$ and $\bar{b}$, we have equality of multiplicities
$mult(\bar{a}/\bar{b})=mult(\bar{b}/\bar{a})$. Unimodularity was defined and
studied in \cite{HrUnimod}, wherein it is shown that unimodularity implies
local modularity, generalising Zilber's proof that locally finite (e.g.
$\omega$-categorical) strongly minimal sets are locally modular.
In the abstract and introduction of \cite{HrUnimod} it is indicated that
we can take as an equivalent definition of unimodularity: if $f_1$ and $f_2$
are definable surjective functions $U \rightarrow V$ which are everywhere
$k_1$-to-1 and $k_2$-to-1 respectively, then $k_1 = k_2$.
The authors of the reviewed paper term this ``weak unimodularity'', and
demonstrate with a simple example that it is in fact strictly weaker than
unimodularity. They further show that if ``definable'' is replaced with
``type-definable'' in the definition of weak unimodularity, it does become
equivalent to unimodularity (one might deduce from the discussion at the start
of section 2 of \cite{HrUnimod} that this is how the definition in that paper
was in fact meant to read).
Meanwhile, the error of assuming equivalence of unimodularity and weak
unimodularity has propogated through the literature on measurable structures
in the sense of Macpherson and Steinhorn \cite{MacphSteinAsymp}.
The papers \cite{MacphSteinAsymp} and \cite{ElwesAsymp} each conflate weak
unimodularity with unimodularity, and the latter gives an erroneous proof of
equivalence.
This paper deals with this confusion. The authors show that the Zilber
functions defined in \cite{HrUnimod} correspond to
Macpherson-Steinhorn-measures in the strongly minimal case, hence that for
strongly minimal sets measurablity is equivalent to unimodularity. They
further give a correct proof of the result in \cite{ElwesAsymp} that
Macpherson-Steinhorn-measurable stable theories are 1-based - using Buechler's
dichotomy to reduce to strongly minimal sets, then applying Hrushovski's
theorem that unimodular strongly minimal sets are locally modular.
The paper is clear and precise. It has no prerequisites beyond a familiarity
with the basic notions of stability theory.
\begin{thebibliography}{Hru92}
\bibitem[Elw07]{ElwesAsymp}
Richard Elwes.
\newblock Asymptotic classes of finite structures.
\newblock {\em J. Symbolic Logic}, 72(2):418--438, 2007.
\bibitem[Hru92]{HrUnimod}
Ehud Hrushovski.
\newblock Unimodular minimal structures.
\newblock {\em J. London Math. Soc. (2)}, 46(3):385--396, 1992.
\bibitem[MS08]{MacphSteinAsymp}
Dugald Macpherson and Charles Steinhorn.
\newblock One-dimensional asymptotic classes of finite structures.
\newblock {\em Trans. Amer. Math. Soc.}, 360(1):411--448 (electronic), 2008.
\end{thebibliography}
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