The Zero-Divisor Graph of a Lattice

Abstract

For a finite bounded lattice £, we associate a zero-divisor graph G(£) which is a natural generalization of the concept of zero-divisor graph for a Boolean algebra. Also, we study the interplay of lattice-theoretic properties of £ with graph-theoretic properties of G(£).

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References

  1. 1

    Anderson D.D., Naseer M.: Beck’s coloring of a commutative ring. J. Algebra 159(2), 500–514 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Anderson, D.F., Frazier, A., Lauve, A., Livingston, P.S.: The zero-divisor graph of a commutative ring, II. In: Ideal Theoretic Methods in Commutative Algebra (Columbia, 1999) Lecture Notes in Pure and Applied Mathematics, vol. 220, pp. 61–72. Dekker, New York (2001)

  3. 3

    Anderson D.F., Levy R., Shapiro J.: Zero-divisor graphs, von Neumann regular rings, and Boolean algebras. J. Pure Appl. Algebra 180(3), 221–241 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Anderson D.F., Livingston P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217(2), 434–447 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5

    Beck I.: Coloring of commutative rings. J. Algebra 116(1), 208–226 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

    DeMeyer, F.R., Schneider, K.: Automorphisms and zero-divisor graphs of commutative rings. Commutative Rings, pp. 25–37. Nova Science Publications, Hauppauge (2002)

  7. 7

    DeMeyer F.R., McKenzie T., Schneider K.: The zero-divisor graph of a commutative semigroup. Semigroup Forum 65(2), 206–214 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8

    Halas R., Jukl M.: On Beck’s coloring of posets. Discrete Math. 309, 4584–4589 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    LaGrange J.D.: Complemented zero-divisor graphs and Boolean rings. J. Algebra 315(2), 600–611 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10

    Levy R., Shapiro J.: The zero-divisor graph of von Neumann regular rings. Commun. Algebra 30(2), 745–750 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Nation J.B.: Notes on Lattice. Theory Cambridge studies in advanced mathematics, vol. 60. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  12. 12

    Mulay S.B.: Cycles and symmetries of zero-divisors. Commun. Algebra 30(7), 3533–3558 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13

    Redmond, S.P.: The zero-divisor graph of a non-commutative ring. Commutative Rings, pp. 203–211. Nova Science Publications, Hauppauge (2002)

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Correspondence to K. Khashyarmanesh.

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Estaji, E., Khashyarmanesh, K. The Zero-Divisor Graph of a Lattice. Results. Math. 61, 1–11 (2012). https://doi.org/10.1007/s00025-010-0067-8

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Mathematics Subject Classification (2010)

  • Primary 05C75
  • 06E99
  • Secondary 13M99

Keywords

  • Zero-divisor graph
  • lattice
  • atom in a lattice