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Intervals of posets of a zero-divisor graph

  • John D. LaGrange
Veröffentlicht/Copyright: 14. August 2024
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Abstract

This article is concerned with bounded partially ordered sets P such that for every pP ∖ {1} there exists qP ∖ {0} such that 0 is the only lower bound of {p, q}. The posets P such that PQ if and only if P and Q have isomorphic zero-divisor graphs are completely characterized. In the case of finite posets, this result is generalized by proving that posets with isomorphic zero-divisor graphs form an interval under the partial order given by PQ if and only if there exists a bijective poset-homomorphism PQ. In particular, the singleton intervals correspond to the posets that are completely determined by their zero-divisor graphs. These results are obtained by exploring universal and couniversal orderings with respect to posets that have isomorphic zero-divisor graphs.

Acknowledgement

The author is grateful for the referee’s suggestions for improving the structure of the article, which benefited the readability of the work.

  1. Communicated by Anatolij Dvurečenskij

References

[1] Akbari, S.—Mohammadian, A.: On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004), 847–855.10.1016/S0021-8693(03)00435-6Suche in Google Scholar

[2] Anderson, D. F.—Lagrange, J. D.: Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph, J. Pure Appl. Algebra 216 (2012), 1626–1636.10.1016/j.jpaa.2011.12.002Suche in Google Scholar

[3] Anderson, D. F.—Frazier, A.—Lauve, A.—Livingston, P. S.: The Zero-Divisor Graph of a Commutative Ring, II. In: Lecture Notes in Pure and Applied Mathematics, Vol. 220, Marcel Dekker, New York, Basel, 2001, pp. 61–72.10.1201/9780429187902-5Suche in Google Scholar

[4] Anderson, D. F.—Livingston, P. S.: The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434–447.10.1006/jabr.1998.7840Suche in Google Scholar

[5] Beck, I.: Coloring of commutative rings, J. Algebra 116 (1988), 208–226.10.1016/0021-8693(88)90202-5Suche in Google Scholar

[6] Devhare, S.—Joshi, V.—Lagrange, J. D.: On the connectedness of the complement of the zero-divisor graph of a poset, Quaest. Math. 42 (2019), 939–951.10.2989/16073606.2018.1502212Suche in Google Scholar

[7] Halaš, R.—Jukl, M.: On Beck’s coloring of posets, Discrete Math. 309 (2009), 4584–4589.10.1016/j.disc.2009.02.024Suche in Google Scholar

[8] Joshi, V.—Khiste, A.: The zero divisor graphs of Boolean posets, Math. Slovaca 64 (2014), 511–519.10.2478/s12175-014-0221-ySuche in Google Scholar

[9] Joshi, V.—Waphare, B. N.—Pourali, H. Y.: The graph of equivalence classes of zero divisors, ISRN Discrete Math. (2014), Art. ID 896270.10.1155/2014/896270Suche in Google Scholar

[10] Kukiela, M.: Reversible and bijectively related posets, Order 26 (2009), 119–124.10.1007/s11083-009-9111-2Suche in Google Scholar

[11] Kukiela, M.: Characterization of hereditarily reversible posets, Math. Slovaca 66 (2016), 539–544.10.1515/ms-2015-0155Suche in Google Scholar

[12] Lagrange, J. D.: Annihilators in zero-divisor graphs of semilattices and reduced commutative semigroups, J. Pure Appl. Algebra 220 (2016), 2955–2968.10.1016/j.jpaa.2016.01.012Suche in Google Scholar

[13] Lagrange, J. D.: Complemented zero-divisor graphs and Boolean rings, J. Algebra 315 (2007), 600–611.10.1016/j.jalgebra.2006.12.030Suche in Google Scholar

[14] Lagrange, J. D.: Order-couniverality of the complete infinitary tree: An application of zero-divisor graphs, J. Pure Appl. Algebra 226 (2022), Art. ID 107114.10.1016/j.jpaa.2022.107114Suche in Google Scholar

[15] Lagrange, J. D.—Roy, K. A.: Poset graphs and the lattice of graph annihilators, Discrete Math. 313(10) (2013), 1053–1062.10.1016/j.disc.2013.02.004Suche in Google Scholar

[16] Lu, D.—Wu, T.: The zero-divisor graphs which are uniquely determined by neighborhoods, Comm. Algebra 35 (2007), 3855–3864.10.1080/00927870701509156Suche in Google Scholar

[17] Lu, D.—Wu, T.: The zero-divisor graphs of posets and an application to semigroups, Graphs Combin. 26 (2010), 793–804.10.1007/s00373-010-0955-4Suche in Google Scholar

[18] Marczewski, E.: Sur l’extension de l’ordre partiel, Fund. Math. 16 (1930), 386–389.10.4064/fm-16-1-386-389Suche in Google Scholar

[19] Mohammadian, A.: On zero-divisor graphs of Boolean rings, Pacific J. Math. 251 (2011), 375–383.10.2140/pjm.2011.251.375Suche in Google Scholar

[20] Mulay, S. B.: Cycles and symmetries of zero-divisors, Comm. Algebra 30 (2002), 3533–3558.10.1081/AGB-120004502Suche in Google Scholar

[21] Patil, A.—Waphare, B. N.—V. Joshi, V.: Zero-divisor graphs of lower dismantlable lattices-II, Math. Slovaca 68 (2018), 225–238.10.1515/ms-2017-0095Suche in Google Scholar

[22] Stanley, R. P.: Enumerative Combinatorics, Vol. 1, 2nd ed., Cambridge University Press, 2012.10.1017/CBO9781139058520Suche in Google Scholar

[23] Tavakkoli, M.—Saeid, A. B.—Poursalavati, N. S.: Classification of posets using Zero-divisor graphs, Math. Slovaca 68 (2018), 21–32.10.1515/ms-2017-0076Suche in Google Scholar

Received: 2022-06-20
Accepted: 2023-12-08
Published Online: 2024-08-14
Published in Print: 2024-08-27

© 2024 Mathematical Institute Slovak Academy of Sciences

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