Structure Theory of CI-Groupoids of Bol-Moufang Type
A groupoid identity $p \approx q$ is of Bol-Moufang type if (i) the same 3 variables appear in $p$ and $q$, (ii) one of the variables appears twice in $p$ and $q$, (iii) the remaining two variables appear once in $p$ and $q$, and (iv) the variables appear in the same order in $p$ and $q$. Phillips and Vojtechovsky studied quasigroup and loop varieties defined by one additional identity of Bol-Moufang type, showing that there are exactly 26 and 14 such varieties, respectively.
Aided by Prover9/Mace4 and the Universal Algebra Calculator, we show that there are exactly eight varieties of commutative, idempotent groupoids defined by one additional identity of Bol-Moufang type. Five of them are congruence meet-semidistributive. We investigate the structure of the variety of commutative, idempotent groupoids satisfying $x(x(yz)) \approx (x(xy))z$, and show that it is the regularization of the variety of squags. We also show that every finite commutative, idempotent, distributive groupoid is a Plonka sum of quasigroups.