Twelve Kinds of Graphs of Lattice Implication Algebras Based on Filter and LI- ideal

Yıl 2020, Cilt: 17 Sayı: 1, 11 - 40, 01.05.2020

Atena Tahmasbpour Meikola

Öz

In this paper, at first we introduce the concepts of filter- annihilator, LI- ideal- annihilator, right- filter- annihilator, left- filter- annihilator, right- LI- ideal- annihilator, and left- LI- ideal- annihilator. Then by using of these concepts, are constructed six new types of graphs in a lattice implication algebra(L,˅,˄,´,→,0,I) which are denoted by Ф_F (L),Ф_A (L),∆_F (L),Σ_F (L),∆_A (L), and Σ_A (L), respectively. Then basic properties of graph theory such as connectivity, regularity, and planarity on the structure of these graphs are investigated. Secondly, by utilizing of binary operations ⊕ and ⊗ we construct graphs Ψ_F (L) and Ψ_A (L), respectively. Thirdly, via the binary operations ⊕ and ⊗, concept of annihilator we construct graphs Ω_F (L) and Ω_A (L), respectively. Finally, by utilizing of binary operations ˄ and ˅, we construct graphs Υ_F (L) and Υ_A (L), respectively, some their interesting properties are presented.

Anahtar Kelimeler

Lattice implication algebra, Diameter, Chromatic number, Euler graph

Teşekkür

The author is grateful to the reviewers for many suggestions which improved the presentation of the paper.

Kaynakça

• [1] I. Beck, coloring of commutative rings, Journal of Algebra, 116(1), (1998), 208-226.
• [2] R. Halas, M. Jukl, On Beck’s coloring of posets, Discrete Mathematics, 309, (2009), 4584-4589.
• [3] Z. Xue, S. Liu, Zero- divisor graphs of partially ordered sets, Applied Mathematics Letters 23(4), (2010), 449- 452.
• [4] H. R. Maimani, M. R. Pournaki, S. Yassemi, Zero- divisor graphs with respect to an ideal, Communication in Algebra, 34(3), (2006), 923-929.
• [5] D. Lu, T. Wu, The zero-divisor graphs of posets and application to semigroup, Graphs and Combinatorics, 26(6), (2010), 793-804.
• [6] Y. Xu, Lattice implication algebras, Journal of Southwest Jiaotong University, 1, (1993), 20-27.
• [7] Y. Xu, K. Y. Qin, On filters of lattice implication algebras, Journal of Fuzzy Mathematics, 1(2), 1993, 251-260.
• [8] Y. B. Jun, E. H. Roh, Y. Xu, LI- ideals in lattice implication algebras, Bulletin of the Korean Mathematical Society, 35, (1998), 13-24.
• [9] Y. B. Jun, K. J. Lee, Graphs based on BCI/BCK- algebras, International Journal of Mathematics and Mathematical Sciences, 2011, (2011), 1-8.
• [10] O. Zahiri, R. A. Borzooei, Graph of BCI- algebra, International Journal of Mathematics and Mathematical Sciences, 2012, (2012), 1-16.
• [11] A. Tahmasbpour Meikola, Graphs of BCI/BCK- algebras, Turkish Journal of Mathematics, 42(3), (2018), 1272-1293.
• [12] A. Tahmasbpour Meikola, Graphs of BCK- algebras based on dual ideal, 9th seminar on algebraic hyperstructures and fuzzy mathematics, University of Mazandaran, Babolsar, Iran, (2019).
• [13] A. Tahmasbpour Meikola, Graphs of lattice implication algebras based on LI- ideal, 3th International Conference on Soft Computing, University of guilan, Iran, (2019), http://www.civilica.com/Paper-CSCG03-CSCG03_254.html.
• [14] A. Tahmasbpour Meikola, Graphs of lattice implication algebras based on LI- ideal, Journal of Science and Engineering Elites, 4(6), (2020), 169-179.
• [15] A. Tahmasbpour Meikola, Graphs of BCK- algebras based on fuzzy ideal and fuzzy dual ideal, 1th International Conference on Physics, Mathematics and Development of Basic Sciences, Tehran, Iran, Center for the development of Interdisciplinary studies, (2020), http://www.civilica.com/Paper-FMCBC01-FMCBC01_013.html
• [16] A. Tahmasbpour Meikola, Graphs of lattice implication algebras based on fuzzy filter and fuzzy LI- ideal, 1th International Conference on Physics, Mathematics and Development of Basic Sciences, Tehran, Iran, Center for the development of Interdisciplinary studies, (2020), http://www.civilica.com/Paper-FMCBC01-FMCBC01_014.html.
• [17] M. Behzadi, Z. Torkzadeh, A. Ahadpanah, A graph on residuated lattices, 1th Algebraic Structure Conference, Hakim Sabzevar University, Iran, (2012).
• [18] M. Alizadeh, H. R. Maimani, M. R. Pournaki, S. Yassemi, An ideal theoretic approach to complete partite zero- divisor graphs of posets, Journal of Algebra and Its Applications, 12(2), (2013), 161-180.
• [19] R. Diestel, Graph theory, Springer: New York, NY, USA, (1997).
• [20] M. Afkhami, Z. Barati, K. Khashyarmanesh, A graph associated to lattice, Springer, 63(1), (2014), 67-78.
• [21] F. Shahsavar, M. Afkhami, K. Khashyarmanesh, On End-Regular, Planar and outerplanar of zero-divisor graphs of posets, 6th Algebraic Conference, University of Mazandaran, Iran, (2013).
• [22] M. Afkhami, K. H. Ahadjavaheri, K. Khashyarmanesh, On the comaximal graphs associated to a lattice of genus one, 7th Algebraic Combinatorics Conference, Ferdowsi University of Mashhad, Iran, (2014), 16-17.
• [23] A. Mohammadian, A. Erfanian, M. Farrokhi, Planar, toroidal and projective generalized Peterson graphs, 7th Algebraic Combinatorics Conference, Ferdowsi University of Mashhad, Iran, (2014).
• [24] Y. Xu, D. Ruan, K. Y. Qin, J. Liu, Lattice Valued- Logic- An Alternative Approach to Treat Fuzziness and Incomparability, Springer: New York, NY, USA, (2003).