M. Polonijo
On medial-like identities

The description of the quasigroups that satisfy the identities of the form (ab)(cd) = (p(a)p(b))(p(c)p(d)), where p is a certain permutation on a,b,c,d, is given. Those quasigroups include internally medial (ab)(cd) = (ac)(bd), externally medial (ab)(cd) = (db)(ca) and palindromic (ab)(cd) = (dc)(ba) quasigroups. There are six identities that are the equivalents of commutativity, and fourteen identities are the equivalents of commutative mediality.