On the structure of symmetric n-ary bands

2020/04/26 by Devillet, Jimmy, Mathonet, Pierre · 1 citation
#20K25 #20M14 #20N15 #FOS: Mathematics #Group Theory (math.GR) #Primary 20M10 #Rings and Algebras (math.RA) #Secondary 08A30

paper · doi:10.48550/arxiv.2004.12423

Abstract

We study the class of symmetric n-ary bands. These are n-ary semigroups (X,F) such that F is invariant under the action of permutations and idempotent, i.e., satisfies F(x,…,x)=x for all x∈ X. We first provide a structure theorem for these symmetric n-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong n-ary semilattice of n-ary semigroups and we show that the symmetric n-ary bands are exactly the strong n-ary semilattices of n-ary extensions of Abelian groups whose exponents divide n-1. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric n-ary band to be reducible to a semigroup.

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