Construction of Zero-Divisor Graph of a Hyperlattice with Respect to Hyperideals

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎Manipal Institute of Technology‎, ‎Manipal Academy of Higher Education‎, ‎Manipal‎, ‎Karnataka‎, ‎India

2 Department of Chemistry‎, ‎Manipal Institute of Technology‎, ‎Manipal Academy of Higher Education‎, ‎Manipal‎, ‎Karnataka‎, ‎India

Abstract

‎In this paper‎, ‎we define the zero-divisor graph of a meet-hyperlattice‎ with respect to a hyperideal‎. ‎We prove the diameter of a $P-$hyperlattice and Nakano‎ hyperlattice are at most 3 and 4 respectively‎. ‎We obtain that the zero-divisor‎ graph with respect to the intersection of two prime hyperideals is complete bipartite‎. ‎We‎ prove certain properties of these zero-divisor graphs with suitable examples.

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References

[1] D. F. Anderson, T. Asir, A. Badawi and T. T. Chelvam, Graphs from Rings, Springer, 2021.
[2] D. F. Anderson, M. C. Axtell and J. A. Stickles, Zero-divisor graphs in commutative rings. In: M. Fontana, S. E. Kabbaj, B. Olberding, I. Swanson, (eds) Commutative Algebra, Springer, New York, (2011) 23–45, https://doi.org/10.1007/978-1-4419-6990-3_2.
[3] D. F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Commun. Algebra 36 (2008) 3073–3092, https://doi.org/10.1080/00927870802110888.
[4] E. Estaji and K. Khashyarmanesh, The zero-divisor graph of a lattice, Results. Math. 61 (2012) 1–11, https://doi.org/10.1007/s00025-010-0067-8.
[5] V. Joshi, B. N. Waphare and H. Y. Pourali, Zero divisor graphs of lattices and primal ideals., Asian-Eur. J. Math. 5 (2012) p. 1250037, https://doi.org/10.1142/S1793557112500374.
[6] V. Joshi and A. Khiste, Complement of the zero divisor graph of a lattice, Bull. Aust. Math. Soc. 89 (2014) 177–190, https://doi.org/10.1017/S0004972713000300.
[7] T. Tamizh Chelvam and S. Nithya, A note on the zero-divisor graph of a lattice, Trans. Comb. 3 (2014) 51–59, https://doi.org/ 10.22108/TOC.2014.5626.
[8] T. Sahoo, H. Panackal, K. B. Srinivas and S. P. Kuncham, Graph with respect to superfluous elements in a lattice, Miskolc Math. Notes 23 (2022) 929–945, https://doi.org/10.18514/MMN.2022.3620.
[9] T. Sahoo, B. Srinivas Kedukodi, K. Ping Shum, H. Panackal and S. Prasad Kuncham, On essential elements in a lattice and goldie analogue theorem, Asian-Eur. J. Math. 15 (2022) p. 2250091, https://doi.org/10.1142/S1793557122500917.
[10] V. Joshi, B. N. Waphare and H. Y. Pourali, On generalized zero divisor graph of a post, Discrete Appl. Math. 161 (2013) 1490–1495, https://doi.org/10.1016/j.dam.2012.12.019.
[11] M. Konstantinidou, On P-hyperlattices and their distributivity, Rend. Circ. Mat. Palermo 42 (1993) 391–403, https://doi.org/10.1007/BF02844630.
[12] A. Soltani Lashkenari and B. Davvaz, Complete join hyperlattices, Indian J. Pure Appl. Math. 46 (2015) 633–645, https://doi.org/10.1007/s13226-015-0130-y.
[13] R. Ameri, M. Amiri-Bideshki, A. B. Saeid and S. Hoskova-Mayerova, Prime filters of hyperlattices, An. Stiint. Univ. Ovidius Constanta Ser. Mat. 24 (2016) 15–26, https://doi.org/10.1515/auom-2016-0025.
[14] M. Amiri Bideshki, R. Ameri and A. Broomand Saeid, On prime hyperfilters (hyperideals) in$\bigwedge$-hyperlattices, Eur. J. Pure Appl. Math. 11 (2018) 169–188, https://doi.org/10.29020/nybg.ejpam.v11i1.2660.
[15] P. Pallavi, S. P. Kuncham, G. R. B. Vadiraja and P. Harikrishnan, Computation of prime hyperideals in meet-hyperlattices, Bull. Comput. Appl. Math. 10 (2022) 33–58.
[16] S. Rasouli and B. Davvaz, Lattices derived from hyperlattices, Commun. Algebra 38 (2010) 2720–2737, https://doi.org/10.1080/00927870903055230.
[17] B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, USA, 2007.
[18] B. Satyanarayana and K. S. Prasad, Discrete Mathematics and Graph Theory, PHI Learning Pvt. Ltd., 2014.
[19] M. Al-Tahan and B. Davvaz, Algebraic hyperstructures associated to biological inheritance, Math. Biosci. 285 (2017) 112–118, https://doi.org/10.1016/j.mbs.2017.01.002.
[20] B. Davvaz, A. Dehghan Nezhad and M. M. Heidari, Inheritance examples of algebraic hyperstructures, Inform. Sci. 224 (2013) 180–187, https://doi.org/10.1016/j.ins.2012.10.023.
[21] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungsz. Graz 103 (1978) 1–22.
[22] I. Gutman and B. Zhou, Laplacian energy of a graph, Linear Algebra Appl. 414 (2006) 29–37, https://doi.org/10.1016/j.laa.2005.09.008.
Iranian Journal of Mathematical Chemistry
Volume 15, Issue 3
September 2024
Pages 123-135

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