Semi-complement graph of lattice modules

  • Narayan PhadatareEmail author
  • Vilas Kharat
  • Sachin Ballal


Let L be a C-lattice and M be a lattice module over L. In this paper, we introduce the semi-complement graph of M denoted by \(\Gamma (M)\) that is the undirected graph with all semi-complement elements of M as a vertex set, and two vertices X and Y are adjacent if and only if \(X\vee Y\) is a semi-complement element. In this paper, we investigate some properties of \(\Gamma (M)\) under some condition on M. For instance, we characterize non-connected graph \(\Gamma (M)\) of a principally generated comultiplication lattice module M over a C-lattice L.


Minimal element Second element Semi-complement element 



We would like to thank Dr. Rupesh Shewale and Mr N.M. Shaikh for their suggestions for preparation of the present paper. The authors are very much thankful to the referee for valuable guidance.

Compliance with ethical standards:

Conflict of interest

The authors declare that there is no conflict of interests regarding the publishing of this paper.

Ethical approval:

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Ansari-Toroghy H, Farshadifar F (2007) The dual notion of multiplication modules. Taiwanese J Math 11(4):1189–1201MathSciNetCrossRefzbMATHGoogle Scholar
  2. Ansari-Toroghy H, Habibi S (2014) The Zariski topology-graph of modules over commutative rings. Commun Algebra 42:3283–3296MathSciNetCrossRefzbMATHGoogle Scholar
  3. Johnson JA (1970) A-adic completions of Noetherian lattice modules. Fund Math 66:341–371MathSciNetGoogle Scholar
  4. Beck I (1988) Coloring of commutative rings. J Algebra 116:208–226MathSciNetCrossRefzbMATHGoogle Scholar
  5. Ballal S, Gophane M, Kharat V (2016) On weakly primary elements in multiplicative lattices. Southeast Asian Bull Math 40:49–57MathSciNetzbMATHGoogle Scholar
  6. Ballal S, Kharat V (2014) On generalization of prime, weakly prime and almost prime elements in multiplicative lattices. Int J Algebra 8(9):439–449CrossRefGoogle Scholar
  7. Ballal S, Kharat V (2015) Zariski topology on lattice modules. Asian-Eur J Math 8: 1550066 (10 pages). 1142/S1793557115500667Google Scholar
  8. Ballal S, Kharat V (2015) On \(\phi \)-absorbing primary elements in lattice modules. Algebra Volume 2015, Article ID 183930: 6 pages.
  9. Behboodi M, Rakeei Z (2011) The annihilating-ideal graph of commutative rings-\(I\). J Algebra Appl 10(4):727–739MathSciNetCrossRefzbMATHGoogle Scholar
  10. Behboodi M, Rakeei Z (2011) The annihilating-ideal graph of commutative rings-\(II\). J Algebra Appl 10(4):741–753MathSciNetCrossRefzbMATHGoogle Scholar
  11. Callialp F, Tekir U, Ulucak G (2015) Comultiplication lattice modules. Iran J Sci Technol Trans A Sci A2(39):213–220MathSciNetGoogle Scholar
  12. Gratzer G (1998) General lattice theory, 2nd edn. Birkhauser Verlag, BaselzbMATHGoogle Scholar
  13. Harary F (1988) Graph theory. Narosa, New DelhizbMATHGoogle Scholar
  14. Phadatare N, Ballal S, Kharat V (2017) On the second spectrum of lattice modules discuss. Math Gen Algebra Appl 37:59–74. MathSciNetCrossRefGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSavitribai Phule Pune UniversityPuneIndia

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