# medial quasigroup

A *medial quasigroup* is a quasigroup^{} such that, for any choice of four elements $a,b,c,d$, one has

$$(a\cdot b)\cdot (c\cdot d)=(a\cdot c)\cdot (b\cdot d).$$ |

Any commutative^{} quasigroup is trivially a medial quasigroup. A nontrivial class of examples may be constructed as follows. Take a commutative group $(G,+)$ and two automorphisms^{} $f,g:G\to G$ which commute with each other, and an element $c$ of $G$. Then, if we define an operation^{} $\cdot :G\times G\to G$ as

$$x\cdot y=f(a)+g(b)+c,$$ |

$(G,\cdot )$ is a medial quasigroup.

Reference:

V. D. Belousov, Fundamentals of the theory of quasigroups and loops (in Russian)

Title | medial quasigroup |
---|---|

Canonical name | MedialQuasigroup |

Date of creation | 2013-03-22 16:27:33 |

Last modified on | 2013-03-22 16:27:33 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 5 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 20N05 |