In this paper, we introduce the concept of (prime) ideals on neutrosophic extended triplet groups (NETGs) and investigate some related properties of them. Firstly, we give characterizations of ideals generated by some subsets, which lead to a construction of a NETG by endowing the set consisting of all ideals with a special multiplication. In addition, we show that the set consisting of all ideals is a distributive lattice. Finally, by introducing the topological structure on the set of all prime ideals on NETGs, we obtain the necessary and sufficient conditions for the prime ideal space to become a $ T_{1} $-space and a Hausdorff space.
Citation: Xin Zhou, Xiao Long Xin. Ideals on neutrosophic extended triplet groups[J]. AIMS Mathematics, 2022, 7(3): 4767-4777. doi: 10.3934/math.2022264
In this paper, we introduce the concept of (prime) ideals on neutrosophic extended triplet groups (NETGs) and investigate some related properties of them. Firstly, we give characterizations of ideals generated by some subsets, which lead to a construction of a NETG by endowing the set consisting of all ideals with a special multiplication. In addition, we show that the set consisting of all ideals is a distributive lattice. Finally, by introducing the topological structure on the set of all prime ideals on NETGs, we obtain the necessary and sufficient conditions for the prime ideal space to become a $ T_{1} $-space and a Hausdorff space.
| [1] |
K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Set. Syst., 20 (1986), 87–96. http://dx.doi.org/10.1016/S0165-0114(86)80034-3 doi: 10.1016/S0165-0114(86)80034-3
|
| [2] | D. S. Dummit, R. M. Foote, Abstract algebra, 3Eds, New Jersey: John Viley & Sons Inc., 2004. ISBN: 0-471-43334-9 |
| [3] | I. N. Herstein, Topics in algebra, Lexington: Xerox college publishing, 1975. |
| [4] |
Y. C. Ma, X. H. Zhang, X. F. Yang, X. Zhou, Generalized neutrosophic extended triplet group, Symmetry, 11 (2019), 327. https://doi.org/10.3390/sym11030327 doi: 10.3390/sym11030327
|
| [5] |
X. D. Peng, J. G. Dai, A bibliometric analysis of neutrosophic set: Two decades review from 1998–2017, Artif. Intell. Rev., 53 (2020), 199–255. https://doi.org/10.1007/s10462-018-9652-0 doi: 10.1007/s10462-018-9652-0
|
| [6] |
X. D. Peng, C. Liu, Algorithms for neutrosophic soft decision making based on EDAS, new similarity measure and level soft set, J. Intell. Fuzzy Syst., 32 (2017), 955–968. https://doi.org/10.3233/jifs-161548 doi: 10.3233/jifs-161548
|
| [7] |
X. D. Peng, J. G. Dai, Approaches to single-valued neutrosophic MADM based on MABAC, TOPSIS and new similarity measure with score function, Neural Comput. Appl., 29 (2018), 939–954. https://doi.org/10.1007/s00521-016-2607-y doi: 10.1007/s00521-016-2607-y
|
| [8] | F. Smarandache, Neutrosophy: Neutrosophic probability, set and logic, Rehoboth: American Research Press, 1998. |
| [9] |
F. Smarandache, Neutrosophic set-a generalization of the intuitionistic fuzzy sets, Int. J. Pure Appl. Math., 24 (2005), 287–297. https://doi.org/10.1109/grc.2006.1635754 doi: 10.1109/grc.2006.1635754
|
| [10] |
D. B. Surowski, The uniqueness aspect of the fundamental theorem of finite Abelian groups, Am. Math. Mon., 102 (1995), 162–163. https://doi.org/10.2307/2975352 doi: 10.2307/2975352
|
| [11] | F. Smarandache, Neutrosophic perspectives: Triplets, duplets, multisets, hybrid operators, modal logic, hedge algebras and applications, Brussels: Pons Publishing House, 2017. |
| [12] |
F. Smarandache, M. Ali, Neutrosophic triplet group, Neural Comput. Appl., 29 (2018), 595–601. https://doi.org/10.1007/s00521-016-2535-x doi: 10.1007/s00521-016-2535-x
|
| [13] |
L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. http://dx.doi.org/10.1016/S0019-9958(65)90241-X doi: 10.1016/S0019-9958(65)90241-X
|
| [14] |
X. H. Zhang, C. X. Bo, F. Smarandache, J. H. Dai, New inclusion relation of neutrosophic sets with applications and related lattice structure, Int. J. Mach. Learn. Cyb., 9 (2018), 1753–1763. http://dx.doi.org/10.1007/s13042-018-0817-6 doi: 10.1007/s13042-018-0817-6
|
| [15] |
X. H. Zhang, C. X. Bo, F. Smarandache, C. Park, New operations of totally dependent-neutrosophic sets and totally dependent-neutrosophic soft sets, Symmetry, 10 (2018), 187. https://doi.org/10.3390/sym10060187 doi: 10.3390/sym10060187
|
| [16] |
X. H. Zhang, Q. Hu, F. Smarandache, X. An, On neutrosophic triplet groups: Basic properties, NT-subgroups and some notes, Symmetry, 10 (2018), 289. https://doi.org/10.3390/sym10070289 doi: 10.3390/sym10070289
|
| [17] |
X. H. Zhang, X. J. Wang, F. Smarandache, T. G. Jaíyéolá, T. Y. Lian, Singular neutrosophic extended triplet groups and generalized groups, Cogn. Syst. Res., 57 (2019), 32–40. https://doi.org/10.1016/j.cogsys.2018.10.009 doi: 10.1016/j.cogsys.2018.10.009
|
| [18] |
X. H. Zhang, X. Y. Wu, F. Smarandache, M. H. Hu, Left (right)-quasi neutrosophic triplet loops (groups) and generalized BE-algebras, Symmetry, 10 (2018), 241. https://doi.org/10.3390/sym10070241 doi: 10.3390/sym10070241
|
| [19] |
X. H. Zhang, F. Smarandache, X. L. Liang, Neutrosophic duplet semi-group and cancellable neutrosophic triplet groups, Symmetry, 9 (2017), 275. https://doi.org/10.3390/sym9110275 doi: 10.3390/sym9110275
|
| [20] |
X. H. Zhang, X. Y. Wu, X. Y. Mao, F. Smarandache, C. Park, On neutrosophic extended triplet groups (loops) and abel-grassmann's groupoids (AG-groupoids), J. Intell. Fuzzy Syst., 37 (2019), 1–11. https://doi.org/10.3233/JIFS-181742 doi: 10.3233/JIFS-181742
|
| [21] |
X. Zhou, P. Li, F. Smarandache, A. M. Khalil, New results on neutrosophic extended triplet groups equipped with a partial order, Symmetry, 11 (2019), 1514. https://doi.org/10.3390/sym11121514 doi: 10.3390/sym11121514
|
Xin Zhou, Xiao Long Xin. Ideals on neutrosophic extended triplet groups[J]. AIMS Mathematics, 2022, 7(3): 4767-4777. doi: 10.3934/math.2022264
DownLoad: