2019/09/23 by Miguel Couceiro, Jimmy Devillet, Jean‐Luc Marichal +2
Computer Science · Mathematics · #Abelian group #Advanced Algebra and Logic #Advanced Graph Theory Research #Advanced Topology and Set Theory #Class (philosophy) #Combinatorics #Computer science #Discrete mathematics #Exponent #Geometric and Algebraic Topology #Idempotence #Mathematics #Pure mathematics #Rings, Modules, and Algebras #Semigroup #math.GR #math.RA #msc:16B99 #msc:20K25 #msc:20M10 #msc:20N15 #semigroups and automata theory
paper · pdf · open access · doi:10.1007/s13366-020-00551-2
published in Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 63(1), 149-166 (Springer Science+Business Media)
openalex publication_date 2021/01/09 · arxiv created 2022/03/14 · arxiv updated 2022/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
Let X be a nonempty set. Denote by Fnk the class of associative operations F\colon Xn→ X satisfying the condition F(x1,…,xn)∈\x1,…,xn\ whenever at least k of the elements x1,…,xn are equal to each other. The elements of Fn1 are said to be quasitrivial and those of Fnn are said to be idempotent. We show that Fn1=⋯ =Fnn-2\subseteqFnn-1\subseteqFnn and we give conditions on the set X for the last inclusions to be strict. The class Fn1 was recently characterized by Couceiro and Devillet, who showed that its elements are reducible to binary associative operations. However, some elements of Fnn are not reducible. In this paper, we characterize the class Fnn-1\setminusFn1 and show that its elements are reducible. We give a full description of the corresponding reductions and show how each of them is built from a quasitrivial semigroup and an Abelian group whose exponent divides n-1.