Studying some semisimple modules via hypergraphs

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.

Abstract

Recent studies have shown that hypergraphs are useful in solving real-life problems. Hypergraphs have been successfully applied in various fields. Inspiring by the importance, we shall introduce a new hypergraph assigned to a given module. By the way, vertices of this hypergraph (which we call sum hypergraph) are all nontrivial submodules of a module $P$ and a subset $E$ of the vertices is a hyperedge in case the sum of each two elements of $E$ is equal to $P$ and $E$ is maximal with respect to this condition. Some general properties of such hypergraphs are discussed. Semisimple modules with length $2$ are characterized by their corresponding sum hypergraphs. It is shown that the sum hypergraph assigned to a finite module $P$ is connected if and only if $P$ is semisimple.

Keywords

References

[1] S. Akbari, H. A. Tavallaee and S. Khalashi Ghezelahmad, Intersection graph of submodules of a module, J. Algebra Appl., 11 (2012) 1250019.
[2] S. Akbari, H. A. Tavallaee and S. Khalashi Ghezelahmad, On the complement of the intersection graph of submodules of a module, J. Algebra Appl., 14 (2015) 1550116.
[3] A. Amini, N. Firouzkouhi, A. Gholami, A. R. Gupta, C. Cheng and B. Davvaz, Soft hypergraph for modeling global interactions via social media networks, Expert Syst. Appl., 203 (2022) 117466.
[4] B. Balasundaram and F. M. Pajouh, Graph Theoretic Clique Relaxations and Applications, In: P. Pardalos, DZ. Du, R. Graham (eds), Handbook of Combinatorial Optimization, Springer, New York, NY, 2013.
[5] C. Berge and C. Berge, Graphes and Hypergraphes, 1970. Dunod, Paris, 1967.
[6] C. Berge, Graphs and Hypergraphs, Vol. 7, North-Holland publishing company, Amsterdam, 1973.
[7] S. R. Bulo and M. Pelillo, A game-theoretic approach to hypergraph clustering, In proceedings of the NIPS, (2009) 1571-1579.
[8] A. Bretto, Introduction to hypergraph theory and its use in engineering and image processing, ADV. IMAG. ELECT. PHYS., 131 (2004) 3-64.
[9] A. Bretto, Hypergraph Theory, Mathematical Engineering, Springer International Publishing, 2013.
[10] A. Bretto and L. Gillibert, Hypergraph-based image representation, In Graph-Based Representations in Pattern Recognition: 5th IAPR International Workshop, GbRPR 2005, Poitiers, France, April 11-13, 2005. Proceedings 5, 1-11, Springer Berlin Heidelberg.
[11] A. Ducournau, A. Bretto, S. Rital and B. Laget, A reductive approach to hypergraph clustering: An application to image segmentation, Pattern Recogn., 45 No. 7 (2012) 2788-2803.
[12] R. Fagin, Degrees of acyclicity for hypergraphs and relational database systems, J. Assoc. Comput. Mach., 30 (1983) 514-550.
[13] C. Hebert, A. Bretto and B. Cremilleux, A data mining formalization to improve hypergraph minimal transversal computation, Fundam. Inform., 80 No. 4 (2007) 415-433.
[14] M. Hellmuth, L. Ostermeier and P. F. Stadler, A survey on hypergraph products, Math. Comput. Sci., 6 No. 1 (2012) 1-32.
[15] S. Klamt, U. U. Haus and F. Theis, Hypergraphs and cellular networks, PLoS Comput. Biol., 5 No. 5 (2009) e1000385.
[16] E. V. Konstantinova and V. A. Skorobogatov, Application of hypergraph theory in chemistry, Discrete Math., 235 (2001) 365-383.
[17] L. A. Mahdavi and Y. Talebi, On the small intersection graph of submodules of a module, Algebraic Struc. Appl., 8 No. 1 (2021) 117-130.
[18] A. R. Moniri Hamzekolaee and M. Norouzi, On characterization of finite modules by hypergraphs, Analele Stiint. ale Univ. Ovidius Constanta Ser. Mat., 30 No. 1 (2022) 231-246.
[19] A. R. Moniri Hamzekolaee and M. Norouzi, Applying a hypergraph to determine the structure of some finite modules, J. Appl. Math. Comput., 69 No. 2 (2023)675-687.
[20] X. Ouvrard, J. Le Goff and S. Marchand-Maillet, Networks of collaborations: hypergraph modeling and visualisation, (2017) arXiv preprint arXiv:1707.00115.
[21] D. Shi, Zh. Chen, X. Sun, Q. Chen, Ch. Ma, Y. Lou and G. Chen, Computing cliques and cavities in networks, Commun. Phys., 4 No. 1 (2021).
[22] P. J. Slater, A characterization of soft hypergraphs, Can. Math. Bull., 21 (1978) 335-337.
[23] S. Smorodinsky, On the chromatic number of geometric hypergraphs, SIAM J. Discret. Math., 21 No. 3 (2007) 676-687.
[24] V. Voloshin, Introduction to Graph and Hypergraph Theory, Nova Science Publication, 2009.
[25] D. B. West, Introduction to Graph Theory, 2nd ed, Prentice Hall, 2001.
[26] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991.
[27] A. A. Zykov, Hypergraphs, Uspekhi Mat. Nauk., 29 (1974) 89-154.
Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 18 July 2024

Files

Share

How to cite

Statistics