Pentagonal quasigroups, their translatability and parastrophes

Any pentagonal quasigroup Q is proved to have the product xy = phi(x) + y - phi(y), where (Q, +) is an Abelian group, q) is its regular automorphism satisfying phi(4) - phi(3) + phi(2) - phi + epsilon = 0 and c is the identity mapping. All Abelian groups of order n < 100 inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity (xy . x) y . x = y is proved to be the variety of commutative, pentagonal quasigroups, whose spectrum is {11(n) : n = 0, 1, 2,...}. We prove that the only translatable commutative pentagonal quasigroup is xy = (6x + 6y) (mod 11). The parastrophes of a pentagonal quasigroup are classified according to well-known types of idempotent translatable quasigroups. The translatability of a pentagonal quasigroup induced by the group Z n and its automorphism phi(x) = ax is proved to determine the value of a and the range of values of n.