On nondistributive Steiner quasigroups

A well known result of R. Dedekind states that a lattice is nonmodular if and only if it has a sublattice isomorphic to ${\displaystyle N_5}$. Similarly a lattice is nondistributive if and only if it has a sublattice isomorphic to ${\displaystyle N_5}$ or ${\displaystyle M_3}$ (see [11]). Recently a few results in this spirit were obtained involving the number of polynomials of an algebra (see e.g. [1], [3], [5], [6]). In this paper we prove that a nondistributive Steiner quasigroup (G,·) has at least 21 essentially ternary polynomials (which improves the recent result obtained in [7]) and this bound is achieved if and only if (G,·) satisfies the identity (xz·yz)·(xy)z = (xz)y·x. Moreover we prove that a Steiner quasigroup (G,·) with 21 essentially ternary polynomials contains isomorphically a certain Steiner quasigroup (M,·), which we describe in Section 1.