Strongly minimal Steiner systems II: coordinatization and quasigroups

Abstract

Each strongly minimal Steiner k-system (M, R) (where is R is a ternary collinearity relation) can be ‘coordinatized’ in the sense of (Ganter–Werner 1975) by a quasigroup if k is a prime-power. We show this coordinatization is never definable in (M, R) and the strongly minimal Steiner k-systems constructed in (Baldwin–Paolini 2020) never interpret a quasigroup. Nevertheless, by refining the construction, if k is a prime power, in each (2, k)-variety of quasigroups (Definition 3.10) there is a strongly minimal quasigroup that interprets a Steiner k-system.