Reference : Reducibility of n-ary semigroups: from quasitriviality towards idempotency |

E-prints/Working papers : Already available on another site | |||

Physical, chemical, mathematical & earth Sciences : Mathematics Engineering, computing & technology : Computer science | |||

Computational Sciences | |||

http://hdl.handle.net/10993/40481 | |||

Reducibility of n-ary semigroups: from quasitriviality towards idempotency | |

English | |

Couceiro, Miguel [] | |

Devillet, Jimmy [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] | |

Marichal, Jean-Luc [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] | |

Mathonet, Pierre [] | |

23-Sep-2019 | |

13 | |

No | |

[en] Semigroup ; polyadic semigroup ; Abelian group ; reducibility ; quasitriviality ; idempotency | |

[en] Let $X$ be a nonempty set. Denote by $\mathcal{F}^n_k$ the class of associative operations $F\colon X^n\to X$ satisfying the condition $F(x_1,\ldots,x_n)\in\{x_1,\ldots,x_n\}$ whenever at least $k$ of the elements $x_1,\ldots,x_n$ are equal to each other. The elements of $\mathcal{F}^n_1$ are said to be quasitrivial and those of $\mathcal{F}^n_n$ are said to be idempotent. We show that $\mathcal{F}^n_1=\cdots =\mathcal{F}^n_{n-2}\varsubsetneq\mathcal{F}^n_{n-1}\varsubsetneq\mathcal{F}^n_n$.
The class $\mathcal{F}^n_1$ was recently characterized by Couceiro and Devillet \cite{CouDev}, who showed that its elements are reducible to binary associative operations. However, some elements of $\mathcal{F}^n_n$ are not reducible. In this paper, we characterize the class $\mathcal{F}^n_{n-1}\setminus\mathcal{F}^n_1$ and show that its elements are reducible. In particular, we show that each of these elements is an extension of an $n$-ary Abelian group operation whose exponent divides $n-1$. | |

University of Luxembourg - UL | |

Researchers ; Professionals ; Students | |

http://hdl.handle.net/10993/40481 | |

https://hal.inria.fr/hal-02293908/document |

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