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We note that a strongly minimal Steiner \(k\)-Steiner system \((M,R)\) from [1] can be `coordinatized' in the sense of [2] 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. We present a number of further questions about the connections with varieties of quasigroups.

References:

1. John T. Baldwin and G. Paolini, Strongly Minimal Steiner Systems I, Journal of Symbolic Logic (2021)
   arXiv
2. Bernhard Ganter and Heinrich Werner, Equational classes of Steiner systems, Algebra Universalis 5:125–140, (1975).