Abstract
We prove Sperner-type theorems for the partially ordered set \(\mathcal {P}_M\) of all submodules of a non-free finitely generated module \({}_RM\) over a finite chain ring R. We demonstrate that the partially ordered set \(\mathcal {P}_M\) is not necessarily of Sperner type and solve the problem for modules of shape \(2^11^n\). This result is further generalized for modules of shape \(m^11^n\) over a chain ring of length m.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The authors declare that no datasets were generated or analysed during the current study.
References
Birkhoff G.: Subgroups of abelian groups. Proc. London Math. Soc. 38(2), 385–401 (1934).
Dilworth R.P.: A decomposition theorem for partially ordered sets. Ann. Math. 51(2), 161–166 (1950).
Engel K.: Sperner theory. Cambridge University Press, Cambridge (1997).
Feng C., Nóberga R.W., Kschischang F.R., Silva D.: Communication over finte-chain-ring matrix channels. IEEE Trans. Inf. Theory 60(10), 5899–5917 (2014).
Georgieva N.: Basic algorithms for manipulation of modules over finite chain rings. Serdika J. Comp. 10(3–4), 285–297 (2016).
Hall P.: On representatives of subsets. J. London Math. Soc. 10, 26–30 (1935).
Honold T., Landjev I.: Linear codes over finite chain rings and projective Hjelmslev geometries. In P. Solé, editor, Codes Over Rings, volume 6 of Series on Coding Theory and Cryptology. World Scientific, July (2009).
Honold T., Landjev I.: Codes over rings and ring geometries, Chapter 7. In: De Beule J., Storme L. (eds.) Current research topics in Galois geometries, pp. 161–186. Nova Science Publishers, New York (2012).
Lubell D.: A short proof of Sperner’s lemma. J. Combin. Theory 1, 299 (1966).
McDonald B.R.: Finite rings with Identity. Marcel Dekker, New York (1974).
Meshalkin L.D.: A generalization of Sperner’s theorem on the number of subsets of a finite set (in Russian). Teor. Veroyatnost. i Primenen. 8, 219–220 (1963).
Nechaev A.A.: Finite principal ideal rings. Russian Acad. Sci. Sbornik Math. 20, 364–382 (1973).
Nechaev A.A.: Finite Rings with Applications, in: Handbook of Algebra, Vol. 5 (ed. Hazewinkel, M.:), Elsevier North-Holland, Amsterdam, 213–320 (2008).
Rota G.-C., Harper L. H.: Matching theory, an introduction, In: Advances in Probability Vol. 1, Marcel Dekker, Inc., New York (1971).
Sperner E.: Ein Satz über Untermengen einer endlichen Menge. Mathematische Zeitschrift 27, 544–548 (1928).
Stanley R.: Some applications of algebra to combinatorics. Discret. Appl. Math. 34, 241–277 (1991).
van Lint J.H., Wilson R.M.: A course in combinatorics. Cambridge University Press, Cambridge (1992).
Wang J.: Proof of a conjecture on the Sperner property of the subgroup lattice of an Abelian \(p\)-group. Ann. Combin. 2, 85–101 (1998).
Yamamoto K.: Logarithmic order of free distributive lattices. J. Math. Soc. Japan 6, 343–353 (1954).
Acknowledgements
The first author was supported by the Strategic Development Fund of the New Bulgarian University and by the Bulgarian National Science Research Fund under Contract KP-06-Russia/33 and by the Research Fund of Sofia University under Contract 80-10-177/27.05.2022. The second author was supported by the Research Fund of Sofia University under Contract 80-10-52/27.05.2022.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have influenced the work reported in this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: Finite Geometries 2022”.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Landjev, I., Rogachev, E. Sperner’s theorem for non-free modules over finite chain rings. Des. Codes Cryptogr. (2024). https://doi.org/10.1007/s10623-023-01352-z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10623-023-01352-z