A Steiner triple system becomes oriented when each of its three-point blocks is given a cyclic order, recorded by an orientation function on ordered pairs of distinct points. An oriented Steiner quasigroup is built as a Schreier-type f-extension of a small cyclic group (order 2, or order 3 for the canonical version) by the Steiner quasigroup on those points, with the extension's factor system chosen to restrict to that orientation function - which is what turns a combinatorial orientation into an algebraic object. Studying it means asking which weak-associativity and inversion laws survive: these quasigroups are flexible, semi-symmetric and cross-inverse in the order-2 case, inverse-property but not flexible in the canonical case, never idempotent, and never Bol or Moufang.
An oriented Steiner triple system is a Steiner triple system - a point set in which every pair of distinct points lies in exactly one three-point block, existing on n points precisely when n is 1 or …