# Pentagonal quasigroups, their translatability and parastrophes

## TL;DR

This paper characterizes pentagonal quasigroups using automorphisms of abelian groups, classifies their parastrophes, and determines conditions for their translatability, including explicit formulas for certain cases.

## Contribution

It provides a complete algebraic description of pentagonal quasigroups, including their automorphism-based structure, classification of parastrophes, and translatability conditions.

## Key findings

- Explicit form of pentagonal quasigroups using automorphisms
- Classification of parastrophes for these quasigroups
- Conditions for translatability and specific formulas

## Abstract

Any pentagonal quasigroup is proved to have the product xy = R(x)+y-R(y) where (Q,+) is an Abelian group, R is its regular automorphism satisfying R^4-R^3+R^2-R+1 = 0 and 1 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+6x)(mod11). 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 additive group Zn of integers modulo n and its automorphism R(x) = ax is proved to determine the value of a and the possible values of n.