John Baldwin : Strongly minimal sets that do not admit elimination of imaginaries

Posted by John Baldwin , part of the Logic Seminar.

At

Sept. 26, 2023, 4 p.m.

In

636 SEO

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 $\sdcl^*(I) \neq \emptyset$.

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