Logic Seminar
John Baldwin
UIC
Strongly minimal sets that do not admit elimination of imaginaries
Abstract: Let $M$ be strongly minimal and constructed by a `Hrushovski
construction' with a single ternary relation. If the Hrushovski
algebraization function $\mu$ is in a certain class $\Tscr$ ($\mu$ triples)
we show that for independent $I$ with $|I| >1$, $\dcl^*(I)= \emptyset$ (*
means not in $\dcl$ of a proper subset). This implies the only definable
truly $n$-ary functions $f$ ($f$ `depends' on each argument), occur when
$n=1$.
We prove
% , indicating the dependence on $\mu$,
for Hrushovski's original construction and for the strongly minimal
$k$-Steiner systems of Baldwin and Paolini that the symmetric definable
closure, $\sdcl^*(I) =\emptyset$ (Definition~\ref{defsdcl}). Thus, no such
theory admits elimination of imaginaries. As, we show that in an arbitrary
strongly minimal theory, elimination of imaginaries implies $\sdcl^*(I)
\neq \emptyset$.
We usually meet in room 300 at 3:30
Tuesday September 26, 2023 at 4:00 PM in 636 SEO