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Erdős-Ginzburg-Ziv theorem for finite commutative semigroups
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TL;DR
The paper studies an Erdős–Ginzburg–Ziv-type inequality for finite commutative semigroups, asking whether \(E(S) \le D(S)+\kappa(S)-1\), where \(\kappa(S)=\lceil |S|/\exp(S)\rceil \exp(S)\). It proves this bound for group-free semigroups, subdirectly irreducible/elementary semigroups, and gives related results for archimedean semigroups under a nilpotency-index constraint.
Key findings
- For any group-free finite commutative semigroup \(S\), Theorem 2.2 proves \(E(S) \le D(S)+\kappa(S)-1\); Corollary 2.4 gives the same bound for finite commutative nilsemigroups.
- For any subdirectly irreducible finite commutative semigroup, Theorem 3.1 proves the conjectured bound; for elementary semigroups \(S=G\cup N\), the paper notes equality \(E(S)=D(S)+\kappa(S)-1\) because \(S\) has an identity element.
- For any finite archimedean semigroup \(S\), Theorem 4.6 proves \(E(S) \le D(S)+\kappa(S)\); if the nilsemigroup \(S/K\) has nilpotency index at most three, then \(E(S) \le D(S)+\kappa(S)-1\), where \(K\) is the kernel.
- Proposition 1.2 records that for a finite commutative semigroup, \(d(S)<\infty\) and \(D(S)=d(S)+1\).
Methods
- Theoretical work in zero-sum theory and commutative semigroup theory; no empirical data or sample are used.
- The paper defines the small Davenport constant \(d(S)\), large Davenport constant \(D(S)\), the invariant \(E(S)\), and \(\kappa(S)=\lceil |S|/\exp(S)\rceil \exp(S)\), then investigates Conjecture 1.4: \(E(S)\le D(S)+\kappa(S)-1\) for all finite commutative semigroups.
- Proofs use Green’s preorder/congruence, group-free and nilsemigroup lemmas, subdirect decomposition, semilattice decomposition into archimedean components, Rees quotient semigroups, kernels, and nilpotency index.
Limitations
- The paper states that it gives a positive answer only for some classes of finite commutative semigroups: group-free semigroups, elementary semigroups, and archimedean semigroups with certain constraints; the general conjecture is not resolved.
- For archimedean semigroups, the stronger \(-1\) inequality is proved only when \(S/K\) has nilpotency index at most three; otherwise only \(E(S)\le D(S)+\kappa(S)\) is shown.
- The results are purely theoretical; no computational or empirical validation is reported.
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0 Valid · 1 Not valid · 8 Not checked · 11 Own results and interpretations
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In the paper
“2. D ( S ) = d ( S ) + 1 .”
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A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Lemma 3.7. Let N be a finite commutative nilsemigroup, and let a , b be two elements of N with a < H b. Then ∞ : b ⫋ ∞ : a.”
plainstates a cause- Asserted: passes — The authors put it forward as true.
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In the paper
“Theorem 4.6. For any finite archimedean semigroup S , E ( S ) ≤ D ( S ) + κ ( S ) .”
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Theorem 2.2. For any group-free f.c.s. S ,”
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In the paper
“Lemma 3.6. Let N be a finite commutative nilsemigroup, and let a , b be two elements in N. If a + b = a then a = ∞ N.”
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In the paper
§ Proposition 1.2. Let S be a finite commutative semigroup. Then, · p. 5
“Note that if S = G is a finite abelian group, the invariants E ( S ) and D ( S ) are consistent with the classical invariants D ( G ) and E ( G ), respectively.”
§ Proposition 1.2. Let S be a finite commutative semigroup. Then, · p. 5
“Definition 1.3. Define E ( S ) of any f.c.s. S as the smallest positive integer ℓ such that, every sequence A ∈ F ( S ) of length ℓ contains a subsequence B with σ ( B ) = σ ( A ) and | A |-| B | = κ ( S ) , where”
Citations
Cites
Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory
Geroldinger, A., & Halter-Koch, F. (2006) · doi:10.1201/9781420003208 · Cited by 366
Zero-sum problems in finite abelian groups: A survey
Gao, W., & Geroldinger, A. (2006) · Expositiones Mathematicae · doi:10.1016/j.exmath.2006.07.002 · Cited by 262
Self-citationRees, D. (1940) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100017436 · Cited by 231
Grynkiewicz, D. J. (2013) · Developments in mathematics · doi:10.1007/978-3-319-00416-7 · Cited by 170
A theorem in additive number theory
Crocker, R. (1969) · Colloquium Mathematicum · doi:10.4064/cm-20-1-53-56 · Cited by 169
Grillet, P. A. (2001) · doi:10.1007/978-1-4757-3389-1 · Cited by 96
A Combinatorial Problem on Finite Abelian Groups
Gao, W. D. (1996) · Journal of Number Theory · doi:10.1006/jnth.1996.0067 · Cited by 74
Self-citationOn decompositions of a commutative semigroup
Tamura, T., & Kimura, N. (1954) · Kodai Mathematical Journal · doi:10.2996/kmj/1138843534 · Cited by 55
A combinatorial problem in Abelian groups
Rogers, K. (1963) · Mathematical Proceedings of the Cambridge Philosophical Society · doi:10.1017/s0305004100037221 · Cited by 47
Contributions to zero-sum problems
Adhikari, S. D., Chen, Y. G., Friedlander, J. B., Konyagin, S. V., & Pappalardi, F. (2006) · Discrete Mathematics · doi:10.1016/j.disc.2005.11.002 · Cited by 47
Research works printed but not linked: 1
- Thierrin, G.: Sur quelques properiétés de certaines classes de demi-groupes. C.R. Acad. Sci. Paris 239, 1335–1337 (1954)
Cited by
On the algebraic and arithmetic structure of the monoid of product-one sequences
Oh, J. S. (2020) · Journal of Commutative Algebra · doi:10.1216/jca.2020.12.409 · Cited by 17
Davenport constant for semigroups II
Wang, G. (2015) · Journal of Number Theory · doi:10.1016/j.jnt.2015.01.007 · Cited by 16
Self-citationStructure of the largest idempotent-product free sequences in semigroups
Wang, G. (2018) · Journal of Number Theory · doi:10.1016/j.jnt.2018.05.020 · Cited by 15
Self-citationAdditively irreducible sequences in commutative semigroups
Wang, G. (2017) · Journal of Combinatorial Theory Series A · doi:10.1016/j.jcta.2017.07.001 · Cited by 14
Self-citationNote on the Davenport constant of the multiplicative semigroup of the quotient ring 𝔽p[x] 〈f(x)〉
Wang, H., Zhang, L., Wang, Q., & Qu, Y. (2015) · International Journal of Number Theory · doi:10.1142/s1793042116500433 · Cited by 6
Metadata
- Authors
- Das Adhikari, S., Gao, W., & Wang, G.
- Affiliations
- Nankai University (CN)
- Tiangong University (CN)
- Corresponding author
- Wang, Guoqing
- ORCID
- Wang, Guoqing
- Venue
- Semigroup Forum
- Published
- 2013-11-14
- Volume
- 88
- Issue
- 3
- Pages
- 555-568
- ISSN
- 0037-1912 (ISSN-L)
- 1432-2137
- OpenAlex
- W2165097875
- Semantic Scholar
- 47ab255943c252bdf79abd52f0d25f4db6b27174
- Cited by
- 7
- References
- 28
- Citations by year
- 2026: 1 · 2020: 1 · 2018: 1 · 2017: 1 · 2015: 3
- FWCI
- 1.31
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Semigroup Forum (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Natural Science Foundation of China (11301381), 11001035, 11271207)
- Science and Technology Development Fund
- Nankai University
- Topics
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Advanced Combinatorial Mathematics
- Physical Sciences › Mathematics › Algebra and Number Theory › Analytic Number Theory Research
- Physical Sciences › Mathematics › Algebra and Number Theory › Rings, Modules, and Algebras
- Keywords
- Erdős-Ginzburg-Ziv theorem, Davenport constant, subsequence sums
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- Das Adhikari, S., Gao, W., & Wang, G. (2014). Erdős-Ginzburg-Ziv theorem for finite commutative semigroups. Semigroup Forum. https://doi.org/10.1007/s00233-013-9544-1
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