Distributive groupoids and symmetry-by-mediality (Q801421)

It is proved that every distributive idempotent groupoid \(G\) is symmetric-by-medial, i.e. there exists a congruence \(\rho\) of \(G\) such that \(G/\rho\) is symmetric and every block of \(\rho\) is a medial subgroupoid of \(G\). This result has several important consequences. One of them: every free distributive groupoid is cancellative.