2013/09/22 by Sukumar Das Adhikari, Weidong Gao, Adhikari, Sukumar Das +3
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1309.5588
openalex publication_date 2013/09/22 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
Let S be a finite commutative semigroup written additively, and let exp(S) be its exponent which is defined as the least common multiple of all periods of the elements in S. For every sequence T of elements in S (repetition allowed), let σ(T) ∈ S denote the sum of all terms of T. Define the Davenport constant D(S) of S to be the least positive integer d such that every sequence T over S of length at least d contains a proper subsequence T' with σ(T')=σ(T), and define the Erdős-Ginzburg-Ziv Theorem constant E(S) to be the least positive integer ℓ such that every sequence T over S of length at least ℓ contains a subsequence T' with |T|-|T'|=\lceil(|S|)/(exp(S))\rceilexp(S) and σ(T')=σ(T). When S is a finite abelian group, it is well known that \lceil(|S|)/(exp(S))\rceilexp(S)=|S| and E(S)=D(S)+|S|-1. In this paper we investigate whether E(S)≤ D(S)+\lceil(|S|)/(exp(S))\rceil exp(S)-1 holds true for all finite commutative semigroups S. We provide a positive answer to the question above for some classes of finite commutative semigroups, including group-free semigroups, elementary semigroups, and archimedean semigroups with certain constraints.