On Sets of Lengths in Monoids of plus-minus weighted Zero-Sum Sequences

2024/04/26 by Geroldinger, Alfred, Kainrath, Florian · 2 citations
#11B30 #13A05 #20M13 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2404.17258

Abstract

Let G be an additive abelian group. A sequence S = g₁ ⋅ … ⋅ g_ℓ of terms from G is a plus-minus weighted zero-sum sequence if there are ε₁, …, ε_ℓ ∈ \-1, 1\ such that ε₁ g₁ + … + ε_ℓ g_ℓ=0. We study sets of lengths in the monoid \mathcal B_± (G) of plus-minus weighted zero-sum sequences over G. If G is finite, then sets of lengths are highly structured. If G is infinite, then every finite, nonempty subset of \mathbb N_{≥ 2} is the set of lengths of some sequence S ∈ \mathcal B_± (G).

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• On the system of length sets of power monoids
• The set of minimal distances of the monoid of plus-minus weighted zero-sum sequences and applications to the characterization problem

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