Document Type : Original Manuscript

Authors

Department of Mathematics, University of Guilan, P.O. Box 1914, Rasht, Iran.

Abstract

Let $L$ be a lattice with the least element $0$ and the greatest element $1$. In this paper, we associate a graph to filters of $L$, in which the vertex set is being the set of all non-trivial filters of $L$, and two distinct vertices $F$ and $E$ are adjacent if and only if $F \cap E \neq \{1\}$. We denote this graph by $\mathcal{G}$ $(L)$. The basic properties and possible structures of $\mathcal{G}$ $(L)$ are studied. Moreover, we characterize the planarity of $\mathcal{G}$ $(L)$.

Keywords

1. D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative rings, J. Algebra, 217 (1999) 434–447.
2. I. Beck, Coloring of commutative rings, J. Algebra, 116 (1988) 208–226.
3. G. Birkhoff, Lattice theory, Amer. Math. Soc., 1973.
4. J. A. Bondy, U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008.
5. J. Bosak, The graphs of semigroups, in Theory of Graphs and its Applications, Academic Press, New York, 1964, pp. 119–125.
6. I. Chakrabarty, S. Ghosh, T. K. Mukherjee, M. K. Sen, Intersection graphs of ideals of rings, Discrete Math., 309 (2009) 5381–5392.
7. B. Csakany, G. Pollak, The graph of subgroups of a finite group, Czechoslovak Math. J., 19 (1969) 241–247.
8. S. Ebrahimi Atani and M. Chenari, Supplemented property in the lattices, Serdica Math. J., 46 (2020), 73–88.
9. S. Ebrahimi Atani, S. Dolati Pish Hesari and M. Khoramdel, The identitysummand graph of commutative semirings, J. Korean Math. Soc., 51 (2014) 189–202.
10. S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel and M. Nikmard Rostamalipour, Semisimple lattice with respect to filter theory, to appear in J. Algebra Relat. Topics.
11. S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel and Z. E. Sarvandi, Intersection graphs of co-ideals of semirings, Commun. Fac. Sci. Univ. Auk. Ser. A1 Math. Stat., 68(1) (2018) 840–851.
12. S. Ebrahimi Atani, S. Dolati Pish Hesari, M. Khoramdel and M. Sedghi Shanbeh Bazari, A semiprime filter-based identity-summand graph of a lattice, Le Matematiche, Vol. LXX III (2018), 297–318.
13. S. Ebrahimi Atani and M. Sedghi Shanbeh Bazari, On 2-absorbing filters of lattices, Discuss. Math. Gen. Algebra Appl., 36 (2016) 157–168.