Zero-divisor graphs of lower dismantlable lattices II
-
Avinash Patil
,
B. N. Waphare
Abstract
In this paper, we continue our study of the zero-divisor graphs of lower dismantlable lattices that was started in [PATIL, A.—WAPHARE, B. N.—JOSHI, V.—POURALI, H. Y.: Zero-divisor graphs of lower dismantlable lattices I, Math. Slovaca 67 (2017), 285–296]. The present paper mainly deals with an Isomorphism Problem for the zero-divisor graphs of lattices. In fact, we prove that the zero-divisor graphs of lower dismantlable lattices with the greatest element 1 as join-reducible are isomorphic if and only if the lattices are isomorphic.
Acknowledgement
The authors are grateful to the referee for many fruitful suggestions which improved the presentation of the paper. The first author is financially supported by University Grant Commission, New Delhi, India via minor research project File No. 47-884/14(WRO).
References
[1] Alizadeh, M.—Das, A. K.—Maimani, H. R.—Pournaki, M. R.–Yassemi, S.: On the diameter and girth of zero-divisor graphs of posets, Discrete Appl. Math. 160 (2012), 1319–1324.10.1016/j.dam.2012.01.011Suche in Google Scholar
[2] 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
[3] Beck I.: Coloring of a commutative ring, J. Algebra 116 (1988), 208–226.10.1016/0021-8693(88)90202-5Suche in Google Scholar
[4] 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
[5] Halaš, R.—Länger, H.: The zero divisor graph of a qoset, Order 27 (2010), 343–351.10.1007/s11083-009-9120-1Suche in Google Scholar
[6] Janowitz, M. F.: Section semicomplemented lattices, Math. Z. 108 (1968), 63–76.10.1007/BF01110457Suche in Google Scholar
[7] Joshi, V.: On completion of section semicomplemented posets, Southeast Asian Bull. Math. 31 (2007), 881–892.Suche in Google Scholar
[8] Joshi, V.: Zero divisor graph of a poset with respect to an ideal, Order 29 (2012), 499–506.10.1007/s11083-011-9216-2Suche in Google Scholar
[9] Joshi, V.—Khiste, A. U.: On the zero divisor graph of a Boolean poset, Math. Slovaca 64 (2014), 511–519.10.2478/s12175-014-0221-ySuche in Google Scholar
[10] Joshi, V.—Khiste, A. U.: Complement of the zero-divisor graph of a lattice, Bull. Aust. Math. Soc. 89 (2014), 177–190.10.1017/S0004972713000300Suche in Google Scholar
[11] Joshi, V.—Sarode, S. Beck’s conjecture and multiplicative lattices, Discrete Math. 338 (2015), 93–98.10.1016/j.disc.2014.10.011Suche in Google Scholar
[12] Joshi, V.—Waphare, B. N.—Pourali, H. Y.: On generalized zero divisor graph of a poset, Discrete Appl. Math. 161 (2013), 1490–1495.10.1016/j.dam.2012.12.019Suche in Google Scholar
[13] Joshi, V.—Waphare, B. N.—Pourali, H. Y.: Zero divisor graphs of lattices and primal ideals, Asian-Eur. J. Math 5 (2012), 1250037, 9 pp.10.1142/S1793557112500374Suche in Google Scholar
[14] Joshi, V.—Waphare, B. N.—Pourali, H. Y.: The graph of equivalence classes of zero divisor, ISRN Discrete Math. (2014), Article ID 896270, 7 pages.10.1155/2014/896270Suche in Google Scholar
[15] Kelly D.—Rival, I.: Crowns, fences, and dismantlable lattices, Canad. J. Math. 26 (1974), 1257–1271.10.4153/CJM-1974-120-2Suche in Google Scholar
[16] 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
[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] 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
[19] Nimbhorkar, S. K.—Wasadikar, M. P.—DeMeyer, L.: Coloring of semilattices, Ars Combin. 84 (2007), 97–104.Suche in Google Scholar
[20] Patil, A.—Waphare, B. N.—Joshi, V.—Pourali, H. Y.: Zero-divisor graphs of lower dismantlable lattices I, Math. Slovaca 67 (2017), 285–296.10.1515/ms-2016-0266Suche in Google Scholar
[21] Rival I.: Lattices with doubly irreducible elements, Canad. Math. Bull. 17 (1974), 91–95.10.4153/CMB-1974-016-3Suche in Google Scholar
[22] Thakare, N. K.—Pawar, M. M.—Waphare, B. N.: A structure theorem for dismantlable lattices and enumeration, Period. Math. Hungar. 45 (2002), 147–160.10.1023/A:1022314517291Suche in Google Scholar
[23] West, D. B.: Introduction to Graph Theory, Second Edition, Prentice-Hall of India, New Delhi, 2002.Suche in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- On the number of cycles in a graph
- Characterization of posets for order-convergence being topological
- Classification of posets using zero-divisor graphs
- On a generalized concept of order relations on B(H)
- A study of stabilizers in triangle algebras
- Revealing two cubic non-residues in a quadratic field locally
- Codensity and stone spaces
- Big mapping class groups are not acylindrically hyperbolic
- Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations
- Starlike and convex functions with respect to symmetric conjugate points involving conical domain
- Summations of Schlömilch series containing anger function terms
- Some vector valued sequence spaces of Musielak-Orlicz functions and their operator ideals
- Nuclear operators on Cb(X, E) and the strict topology
- More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism
- Matrix generalized (θ, ϕ)-derivations on matrix Banach algebras
- Nonlinear ∗-Jordan triple derivations on von Neumann algebras
- Addendum to “A sequential implicit function theorem for the chords iteration”, Math. Slovaca 63(5) (2013), 1085–1100
- Comparison of density topologies on the real line
- Rational homotopy of maps between certain complex Grassmann manifolds
- Examples of random fields that can be represented as space-domain scaled stationary Ornstein-Uhlenbeck fields
- Rothe’s method for physiologically structured models with diffusion
- Zero-divisor graphs of lower dismantlable lattices II
- An identity of symmetry for the degenerate Frobenius-Euler Polynomials
Artikel in diesem Heft
- On the number of cycles in a graph
- Characterization of posets for order-convergence being topological
- Classification of posets using zero-divisor graphs
- On a generalized concept of order relations on B(H)
- A study of stabilizers in triangle algebras
- Revealing two cubic non-residues in a quadratic field locally
- Codensity and stone spaces
- Big mapping class groups are not acylindrically hyperbolic
- Applications of Henstock-Kurzweil integrals on an unbounded interval to differential and integral equations
- Starlike and convex functions with respect to symmetric conjugate points involving conical domain
- Summations of Schlömilch series containing anger function terms
- Some vector valued sequence spaces of Musielak-Orlicz functions and their operator ideals
- Nuclear operators on Cb(X, E) and the strict topology
- More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism
- Matrix generalized (θ, ϕ)-derivations on matrix Banach algebras
- Nonlinear ∗-Jordan triple derivations on von Neumann algebras
- Addendum to “A sequential implicit function theorem for the chords iteration”, Math. Slovaca 63(5) (2013), 1085–1100
- Comparison of density topologies on the real line
- Rational homotopy of maps between certain complex Grassmann manifolds
- Examples of random fields that can be represented as space-domain scaled stationary Ornstein-Uhlenbeck fields
- Rothe’s method for physiologically structured models with diffusion
- Zero-divisor graphs of lower dismantlable lattices II
- An identity of symmetry for the degenerate Frobenius-Euler Polynomials
