Strongly Minimal Steiner Systems II: Coordinatization and Quasigroups

TL;DR

This paper explores the coordinatization of strongly minimal Steiner systems using quasigroups, showing that for prime-power parameters, such systems can be strongly minimal and definably coordinatized within a specific quasigroup variety.

Contribution

It demonstrates that strongly minimal Steiner systems with prime-power parameters can be coordinatized by a strongly minimal quasigroup variety, refining previous constructions.

Findings

Coordinatization by quasigroups is possible for prime-power Steiner systems.

Refined construction yields a strongly minimal, definably coordinatized Steiner system.

Existence of a (2,k)-variety of quasigroups with these properties.

Abstract

We note that a strongly minimal Steiner $k$-Steiner system $(M,R)$ from (Baldwin-Paolini 2020) can be `coordinatized' in the sense of (Gantner-Werner 1975) by a quasigroup if $k$ is a prime-power. But for the basic construction this coordinatization is never definable in $(M,R)$. Nevertheless, by refining the construction, if $k$ is a prime power there is a $(2,k)$-variety of quasigroups which is strongly minimal and definably coordinatizes a Steiner $k$-system.