# Distributive lattices and some related topologies in comparison with zero-divisor graphs

Document Type : Research Paper

Authors

Department of Mathematics, Malayer University, P.O.Box: 65719-95863, Malayer, Iran.

Abstract

In this paper,
for a distributive lattice $\mathcal L$, we study and compare some lattice theoretic features of $\mathcal L$ and topological properties of the Stone spaces ${\rm Spec}(\mathcal L)$ and ${\rm Max}(\mathcal L)$ with the corresponding graph theoretical aspects of the zero-divisor graph $\Gamma(\mathcal L)$.
Among other things,
we show that the Goldie dimension of $\mathcal L$ is equal to the cellularity of the topological space ${\rm Spec}(\mathcal L)$ which is also equal to the clique number of the zero-divisor graph $\Gamma(\mathcal L)$. Moreover, the domination number of $\Gamma(\mathcal L)$ will be compared with the density and the weight of the topological space ${\rm Spec}(\mathcal L)$.

For a $0$-distributive lattice $\mathcal L$, we investigate the compressed subgraph $\Gamma_E(\mathcal L)$ of the zero-divisor graph $\Gamma(\mathcal L)$ and determine some properties of this subgraph in terms of some lattice theoretic objects such as associated prime ideals of $\mathcal L$.

Keywords

#### References

[1] Anderson, D.F. and Livingston, P.S., The zero-divisor graph of a commutative ring, J. Algebra {217}(2) (1999), 434-447.
[2] Anderson, D.F. and Weber, D., The zero-divisor graph of a commutative ring without identity, Int. Electron. J. Algebra  23 (2018), 176-202.
[3] Azapanah, F. and Motamedi, M., Zero-divisor graph of $C(X)$, Acta Math. Hungarica 108(1-2) (2005), 25-36.
[4] Bagheri, S. and Koohi Kerahroodi, M., The annihilator graph of a $0$-distributive lattice, Trans. Combin.  7(3) (2018), 1-18.
[5] Bagheri, S., Nabaei, F., Rezaei, R., and Samei, K., Reduction graph and its application on algebraic graphs. Rocky Mountain J. Math.  48 (2018), 729-751.
[6] Beck, I., Coloring of commutative rings, J. Algebra 116(1) (1988), 208-226.
[7] Demeyer, F.R., Mckenzie, T., and Schneider, K., The zero divisor graph of a commutative semigroup, Semigroup Forum 65 (2002), 206-214.
[8] Duffus, D. and Rival, I., Path length in the covering graph of a lattice, Discrete Math. 19 (1977), 139-158.
[9] Engelking, R., General Topology'', Heldermann-Verlag, 1989.
[10] Estaji, E. and Khashyarmanesh, K., The zero divisor graph of a lattice, Res. Math. 61 (2012), 1-11.
[11] Facchini, A., Module Theory'': Endomorphism rings and direct sum decompositions in some classes of modules, Springer,  1998.
[12] Gedeonova, E., Lattices whose covering graphs are S-graphs, Colloq. Math. Soc. Janos Bolyai 33 (1980), 407-435.
[13] Gr"{a}tzer, G., Lattice Theory: Foundation'', Birkh"{a}user, 2011.
[14] Hala$check{s}$, R. and Jukl, M., On Beck's coloring of posets, Discrete Math. 309 (2009), 4584-4589.
[15] Joshi, V., Zero divisor graph of a poset with respect to an ideal, Order 29(3) (2012),  499-506.
[16] Joshi, V. and Khiste, A., On the zero divisor graphs of pm-lattices, Discrete Math. 312 (2012), 2076-2082.
[17] Joshi, V. and Sarode, S.,  Diameter and girth of zero divisor graph of multiplicative lattices, Asian-Eur. J. Math. 9(4) (2016), 1650071.
[18] Joshi, V., Waphare, B.N., and Pourali, H.Y., The graph of equivalence classes of zero divisors, Int. Scholarly Res. Not. (2014),  896270, 7 pages.
[19] Mulay, S.B., Cycles and symmetries of zero-divisors, Comm. Algebra (2002), 3533-3558.
[20] Nongsiang, D. and Kumar Saikia, P., Reduced zero-divisor graphs of posets, Trans. Combin, 7(2) (2018), 47-54.
[21] Pawar, Y.S., and Thakare, N.K., pm-lattices, Algebra Universalis, 7 (1977), 259-263.
[22] Samei, K., On the maximal spectrum of commutative semiprimitive rings, Colloq. Math. 83(1) (2000), 5-13.
[23] Samei, K., The zero-divisor graph of a reduced ring, J. Pure Appl. Algebra 209(3) (2007),  813-821.
[24] Spiroff, S. and Wickham, C., A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra 39 (2011), 2338-2348.
[25] Tamizh Chelvam, T. and Nithya, S., A note on the zero divisor graph of a lattice, Trans. Combin. 3(3) (2014),  51-59.
[26] West, D.B., Introduction to Graph Theory'',  Prentice- Hall, 2001.
[27] Wilson, R.J., Introduction to Graph Theory'',  Addison Wesley Longman Limited, 1996.
[28] Xue, Z. and Liu, S., Zero-divisor graphs of partially ordered sets, Appl. Math. Lett. 23(4) (2010), 449-452.