AIMS Mathematics, 2020, 5(4): 2992-3001. doi: 10.3934/math.2020193.

Research article

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

A generalization of Kruyswijk-Olson theorem on Davenport constant in commutative semigroups

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, P. R. China

## Abstract    Full Text(HTML)    Figure/Table    Related pages

Let $\mathcal{S}$ be a finite commutative semigroup written additively. An element $e$ of $\mathcal{S}$ is said to be idempotent if $e+e=e$. The Erdős-Burgess constant of the semigroup $\mathcal{S}$ is defined as the smallest positive integer $\ell$ such that any $\mathcal{S}$-valued sequence $T$ of length $\ell$ must contain one or more terms with the sum being an idempotent of $\mathcal{S}$. If the semigroup $\mathcal{S}$ is a finite abelian group, the Erdős-Burgess constant reduces to the well-known Davenport constant in Combinatorial Number Theory. In this paper, we determine the value of the Erdős-Burgess constant for a direct sum of two finite cyclic semigroups in some cases, which generalizes the classical Kruyswijk-Olson Theorem on Davenport constant of finite abelian groups in the setting of commutative semigroups.
Figure/Table
Supplementary
Article Metrics

Citation: Guoqing Wang. A generalization of Kruyswijk-Olson theorem on Davenport constant in commutative semigroups. AIMS Mathematics, 2020, 5(4): 2992-3001. doi: 10.3934/math.2020193

References

• 1. W. R. Alford, A. Granville, C. Pomerance, There are infinitely many Carmichael numbers, Ann. Math., 140 (1994), 703-722.
• 2. N. Alon, S. Friedland, G. Kalai, Regular subgraphs of almost regular graphs, J. Combin. Theory Ser. B, 37 (1984), 79-91.
• 3. G. Bhowmik, J.-C. Schlage-Puchta, Davenport's constant for groups of the form $\mathbb{Z}_3\oplus \mathbb{Z}_3 \oplus \mathbb{Z}_{3d}$, In: Granville A., Nathanson M.B., Solymosi J. (eds) Additive Combinatorics, CRM Proc. Lecture Notes, 43, pp. 307-326, Am. Math. Soc., 2007.
• 4. D. A. Burgess, A problem on semi-groups, Studia Sci. Math. Hungar., 4 (1969), 9-11.
• 5. K. Cziszter, M. Domokos, A. Geroldinger, The Interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics. In: Chapman S., Fontana M., Geroldinger A., Olberding B. (eds) Multiplicative Ideal Theory and Factorization Theory, Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2016.
• 6. C. Deng, Davenport constant for commutative rings, J. Number Theory, 172 (2017), 321-342.
• 7. P. van Emde Boas, A combinatorial problem on finite abelian groups 2, Report ZW-1969-007, Mathematical Centre, Amsterdam, 1969.
• 8. P. van Emde Boas, D. Kruyswijk, A combinatorial problem on finite abelian groups 3, Report ZW 1969-008, Stichting Math. Centrum, Amsterdam, 1969.
• 9. P. Erdős, A. Ginzburg, A. Ziv, Theorem in the additive number theory, Bull. Res. Council Israel F, 10 (1961), 41-43.
• 10. W. Gao, On Davenport's constant of finite abelian groups with rank three, Discrete Math., 222 (2000), 111-124.
• 11. W. Gao, A. Geroldinger, Zero-sum problems in finite abelian groups: a survey, Expo. Math., 24 (2006), 337-369.
• 12. A. Geroldinger, Additive Group Theory and Non-unique Factorizations. In: A. Geroldinger and I. Ruzsa (Eds.), Combinatorial Number Theory and Additive Group Theory (Advanced Courses in Mathematics-CRM Barcelona), Birkhäuser, Basel, 2009.
• 13. A. Geroldinger, F. Halter-Koch, Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Chapman & Hall/CRC, 2006.
• 14. A. Geroldinger, M. Liebmann, A. Philipp, On the Davenport constant and on the structure of extremal sequences, Period. Math. Hungar., 64 (2012), 213-225.
• 15. D. W. H. Gillam, T. E. Hall, N. H. Williams, On finite semigroups and idempotents, Bull. London Math. Soc., 4 (1972), 143-144.
• 16. P. A. Grillet, Commutative Semigroups, Kluwer Academic Publishers, 2001.
• 17. D. J. Grynkiewicz, Structural Additive Theory, Developments in Mathematics, Springer, Cham, 2013.
• 18. C. Liu, On the lower bounds of Davenport constant, J. Combin. Theory Ser. A, 171 (2020).
• 19. L. E. Marchan, O. Ordaz, I. Santos, et al. Multi-wise and constrained fully weighted Davenport constants and interactions, J. Combin. Theory Ser. A, 135 (2015), 237-267.
• 20. R. Meshulam, An uncertainty inequality and zero subsums, Discrete Math., 84 (1990), 197-200.
• 21. J. E. Olson, A Combinatorial Problem on Finite Abelian Groups, I, J. Number Theory, 1 (1969), 8-10.
• 22. J. E. Olson, A combinatorial problem on finite abelian groups II, J. Number Theory, 1 (1969), 195-199.
• 23. A. Plagne, W. A. Schmid, An application of coding theory to estimating Davenport constants, Des. Codes Cryptogr., 61 (2011), 105-118.
• 24. W. A. Schmid, The inverse problem associated to the Davenport constant for $C_2\oplus C_2\oplus C_{2n}$ and applications to the arithmetical characterization of class groups, Electron. J. Comb., 18 (2011).
• 25. G. Wang, Davenport constant for semigroups II, J. Number Theory, 153 (2015), 124-134.
• 26. G. Wang, Additively irreducible sequences in commutative semigroups, J. Combin. Theory Ser. A, 152 (2017), 380-397.
• 27. G. Wang, Structure of the largest idempotent-product free sequences in semigroups, J. Number Theory, 195 (2019), 84-95.
• 28. H. Wang, L. Zhang, Q. Wang, et al. Davenport constant of the multiplicative semigroup of the quotient ring $\frac{{\rm F}_p[x]}{\langle f(x)\rangle}$, Int. J. Number Theory, 12 (2016), 663-669.
• 29. L. Zhang, H. Wang, Y. Qu, A problem of Wang on Davenport constant for the multiplicative semigroup of the quotient ring of F2[x], Colloq. Math., 148 (2017), 123-130.