Logic Seminar
John Baldwin
UIC
Strongly minimal Steiner systems
Abstract: With Gianluca Paolini, we constructed families of strongly minimal Steiner
$(\infty,2,k)$ systems for every $k \geq 3$. Here we show that the
$2^{\aleph_0}$ Steiner $(2,3)$-systems are coordinatized by strongly
minimal Steiner quasigroups and the Steiner $(2,4)$-systems are
coordinatized by strongly minimal $SQS$-Skeins. Further the Steiner
$(2,4)$-systems admit Steiner quasigroups but it is open whether their
theory is strongly minimal. We exhibit strongly minimal uniform Steiner
triple systems (with respect to the associated graphs $G(a,b)$ (Cameron and
Webb) with varying numbers of finite cycles. This work inaugurates a
program of differentiating the many strongly minimal sets, whose geometries
of algebraically closed sets are (locally isomorphic) to the original
Hrushovski example, but with varying properties in the object language.
Tuesday August 28, 2018 at 3:30 PM in 427 SEO