Pentagonal quasigroups, their translatability and parastrophes

Abstract

Any pentagonal quasigroup QQ is proved to have the product xy=φ(x)+y−φ(y)xy=\varphi \left(x)+y-\varphi (y), where (Q,+)\left(Q,+) is an Abelian group, φ\varphi is its regular automorphism satisfying φ4−φ3+φ2−φ+ε=0{\varphi }^{4}-{\varphi }^{3}+{\varphi }^{2}-\varphi +\varepsilon =0 and ε\varepsilon is the identity mapping. All Abelian groups of order n<100n\lt 100 inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity (xy⋅x)y⋅x=y\left(xy\cdot x)y\cdot x=y is proved to be the variety of commutative, pentagonal quasigroups, whose spectrum is {11n:n=0,1,2,…}\left\{1{1}^{n}:n=0,1,2,\ldots \right\}. We prove that the only translatable commutative pentagonal quasigroup is xy=(6x+6y)(mod11)xy=\left(6x+6y)\left({\rm{mod}}\hspace{0.33em}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 Zn{{\mathbb{Z}}}_{n} and its automorphism φ(x)=ax\varphi \left(x)=ax is proved to determine the value of aa and the range of values of nn.

Keywords

• quasigroup
• pentagonal quasigroup
• translatability
• idempotent
• 20n02
• 20n05