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On strongly minimal Steiner systems Baldwin, John


With Gianluca Paolini (in preparation), we constructed families of strongly minimal Steiner $( systems for every $k 3$. A quasigroup is a structure with a binary operation such that for each equation $xy=z$ the values of two of the variables determines a unique value for the third. Here we show that the $2^{ Steiner $(2,3)$-systems are definably coordinatized by strongly minimal Steiner quasigroups and the Steiner $(2,4)$-systems are definably coordinatized by strongly minimal $SQS$-Skeins. Further the Steiner $(2,4)$-systems admit Stein quasigroups but depending on the choice of theory may or may not admit a definable binary function and be definably coordinatized by an $ Stein quasigroup. 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. We show how to vary the theory to obtain $2$ or $3$-transitivity. This work inaugurates a program of differentiating the many strongly minimal sets, whose geometries of algebraically closed sets may be (locally) isomorphic to the original Hrushovski example, but with varying properties in the object language. In particular, can one organize these geometries by studying the associated algebra. This work differs from traditional work in the infinite combinatorics of Steiner systems by considering the relationship among different models of the same first order theory.

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