An Example of a Quasigroup with a Distributive Subquasigroup Lattice

We first show (using ideas from Pióro, in Quasigroups Relat Syst 15:309–316, 2007) that if a finitely generated quasigroup or more general, an algebra satisfying some quasigroup-like condition, A has a distributive subalgebra lattice and satisfies the descending chain condition for finitely generated subalgebras, then A is cyclic (i.e. has one generator). But the main aim of the paper is to construct a two-generated non-cyclic quasigroup with a distributive subquasigroup lattice (as a part of this construction we will obtain an analogous example for groupoids). Thus distributivity of subquasigroup lattice is an essentially weaker condition than in the case of groups (it is well-know that such groups are locally cyclic).