Quotient on some Generalizations of topological group
Main Article Content
Abstract
In this paper, we define some generalizations of topological group namely -topological group, -topological group and -topological group with illustrative examples. Also, we define grill topological group with respect to a grill. Later, we deliberate the quotient on generalizations of topological group in particular -topological group. Moreover, we model a robotic system which relays on the quotient of -topological group.
Received 20/1/2023
Revised 19/2/2023
Accepted 20/2/2023
Published 6/3/2023
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
How to Cite
References
Bosan MS, Khan MD, Kocinac LDR. On S-topological Group. Math Morav. 2014; 18(2): 35-44. https://doi.org/10.5937/MatMor1402035B.
Bohn E, Lee J. Semi-Topological Groups. Bull Amer Math Soc. 1965; 72: 996-998. https://doi.org/10.2307/2313342.
Ram M. On Almost Topological Groups. Math. Morav. 2019; 23(1) : 97-106. https:// doi.org/10.5937/MatMor1901097R.
Jafari S, Gnanachandra P, Muneesh Kumar A. On p-Topological Groups. Math Morav. 2021; 25(2): 13-27. https:// doi.org/10.5937/MatMor2102013J.
Gnanachandra P, Jafari S, Rajesh N. β-Ideal Topological Groups. Casp J Math Sci. 2022; 11(2): 518-525. https:// doi.org/10.22080/CJMS.2022.21139.1577.
Jafari S, Gnanachandra P, Muneesh Kumar A. Indagation on p-Ideal Topological Group. Questions Answers Gen. Topology. 2022; 40: 33-41.
Kuratowski K. Topology I. USA: Academic Press; 1966. https://doi.org/10.1016/C2013-0-11022-7.
Njastad O. On Some Classes of Nearly Open Sets. Pacific J Math. 1965; 15(3): 961-970.
Levine N. Semi-Open Sets and Semi Continuity in Topological Spaces. Am Math Mon. 1963; 70(1): 36-41. https://doi.org/10.2307/2312781.
Mashhour A S, Abd El-Monsef M E, El-Deeb S N. On Precontinuous and Weak Precontinuous Mappings. Proc Math phys Soc. Egypt. 1982; 53: 47-53.
Andrijivic D. On b-Open Sets. Mat Vesnik. 1996; 48: 59-64.
Abd El Monsef ME, El-Deeb SN, Mahmoud RA. β-Open Sets and β-Continuous Mappings. Bull Fac Sci Univ Assuit. 1983; 12: 77-80.
Jankovic D, Hamlett TR. New Topologies From Old via Ideals. Amer Math Monthly. 1990; 97: 295-310. https://doi.org/10.1080/00029890.1990.11995593.
Vaidyanadhaswamy R. The Localisation Theory in Set-Topology. Proc Indian Acad Sci Math Sci. 1945; 20: 51-61. https://doi.org/10.1007/BF03048958.
Choquet G. Sur Les Notions de Filter et Grill. C R Math Acad Sci. Paris. 1947; 224: 171-173.
Duszyński Z. On p-Open Mappings. Bull Math Soc Sci Math Roumanie. 2006; 49(97)(3): 223–238.
Williard S. General Topology, Dover Publications Inc. New York, 2012. 384 p
Davamanoharan C, Pious Missier S, Jafari S. On p-Homeomorphisms in Topological Spaces. Ital J Pure Appl Math. 2013; 30: 195-214.
Kar A, Bhattacharyya P. Some Weak Seperation Axioms. Bull Calcutta Math Soc. 1990; 82: 415-422.
Reilly I, Vamanamurthy MK. On α-continuity in Topological Spaces. Acta Math.Hungar. 1985; 45(1-2): 27-32. https://doi.org/10.1007/BF01955019.
Kandil A, El-Sheikh SA, Abdelhakem M, Hazza SA. On Ideals and Grills in Topological Spaces. South Asian J Math. 2015; 5(6): 233-238.
Ram M. Some Analogues of Topological Groups. Discuss Math Gen Algebra Appl. 2021; 41:171-181. https://doi.org/10.7151/dmgaa.1357.
Reznichenko E, Sipacheva O. Discrete Subsets in Topological Groups and Countable Extremally Disconnected Groups. Proc Amer Math Soc. 2021; 149: 2655-2668. https://doi.org/10.1090/proc/13992.
Scheepers M. A Selection Principle and Products in Topological Groups. Axioms. 2022; 11(286): 1-13. https://doi.org/10.3390/axioms11060286.
Megrelishvili. A Note on the Topological Group c0. Axioms. 2018; 7(77): 1-6. https://doi.org/10.3390/axioms704007.
Peng L, Liu Y. On Lindelof Feathered Topological Groups. Topology Appl. 2020; 285: 1-9. https://doi.org/10.1016/j.topol.2020.107405.
Rodriguez EM, Tkachenko M. D-Independent Topological Groups. Topology Appl. 2021; 300, 107761. https://doi.org/10.1016/j.topol.2021.107761.
Petrakis I. Closed Subsets in Bishop Topological Groups. Theor Comput Sci. 2022; 935: 128-143. https://doi.org/10.1016/j.tcs.2022.09.004.
Muneesh Kumar A, Gnanachandra P. Indagation on p-Grill Topological Groups and Modelling Industry Transportation via Neutrosophic Sets Neuroquantology. 2022; 20(8): 5946-5954. https://doi.org/10.14704/nq.2022.20.8.NQ44621.
Vilalobos AC, Sanchez I. Hattori Topologies on Almost Topological Groups. Topology Appl. 2023; 326: 108411. https://doi.org/10.1016/j.topol.2023.108411.
Ludkowski S.V. Structures and Functions of Topological Metagroups. Axioms. 2020; 9(66): 1-13. https://doi.org/10.3390/axioms902006.
Kazachenko K, Osipov AV. Various Types of Completeness in Topologized Semilattices. Semigr Forum. 2022; 104: 358-375. https://doi.org/10.1007/s00233-022-10259-5.
Chalebgwa TP, Morris SA. Topological Transcendental Fields. Axioms. 2022; 11(118): 1-5. https://doi.org/10.3390/axioms11030118.
Hernandez Arzusa JC. Commutative Topological Semigroups Embedded into Topological Abelian Groups. Axioms. 2020; 9(87): 1-9. https://doi.org/10.3390/axioms903008.
Sanchez I, Sanchis M. Fuzzy Quasi-Pseudometrics on Algebraic Structures. Fuzzy sets Syst. 2018; 330: 79-89. https://doi.org/10.1016/j.fss.2017.05.022.
Sanchez I, Sanchis M. Complete Invariant Fuzzy Metrics on Groups. Fuzzy sets Syst. 2018; 330: 41-51. https://doi.org/10.1016/j.fss.2016.12.019.
Tu JJ, Xie LH. Complete Invariant Fuzzy Metrics on Semigroups and Groups. J Appl. Anal Comput. 2021; 11: 766-771. https://doi.org/10.3390/axioms11100546.
Al. Khafaje MA, Ajeel YJ. On Generalized Continuous Fuzzy Proper Function from a Fuzzy Topological Space to Another Fuzzy Topological Space. Baghdad Sci. J. 2019; 16(1(Suppl.)): 237-241. https://doi.org/10.21123/bsj.2019.16.1(Suppl.).0237.
He SY, Wei JC, Xie LH. Complete Invariant – Metrics on Semigroups and Groups. Axioms. 2022; 11(546): 1-8. https://doi.org/10.3390/axioms11100546.
He W, Peng D, Tkachenko M, Zhang H. M- Factorizable Feathered Topological Groups. Topology Appl. 2021; 289: 1-17. https://doi.org/10.1016/j.topol.2020.107481.
Zhang H, Peng D, He W. On M Factorizable Topological Groups, Topology Appl. 2020; 274: 1-13. https://doi.org/10.1016/j.topol.2020.107126.
Morris SA. The Tubby Torus as a Quotient Groups. Axioms. 2020; 9(1): 1-4. https://doi.org/10.3390/axioms901001.
Leiderman AG, Tkachenko M. Metrizable Quotient of Free Topological Groups. Rev Real Acad. Cienc Exactas Fis Nat. - A: Mat. 2020; 124: 1-16. https://doi.org/10.1007/s13398-020-00855-x.
Banakh T, Kakol J, Sliwa W. Metrizable Quotients of Cp-spaces. Topology Appl. 2018; 249: 95-102. https://doi.org/10.1016/j.topol.2018.09.012.
Leiderman AG, Morris SA. Separability of Topological Groups: a Survey with Open Problems. Axioms. 2019; 8(3): 1-18. https://doi.org/10.3390/axioms801000.
Leiderman AG, Morris SA, Tkachenko MG. The Separable Quotient Problem for Topological Groups. Isr J Math. 2019; 234: 331-369. https://doi.org/10.1007/s11856-019-1931-1.
Morris SA. A Remark on the Separable Quotient Problem for Topological Groups. Bull Aust Math Soc. 2019; 100(3): 453-457. https://doi.org/10.1017/S0004972719000571.
Ferrando JC, Kalkol J, Lopez Pelicer M, Sliwa W. On the Separable Quotient Problem for Banach Spaces. Funct Approx.Comment Math. 2018; 59(2): 153-173. https://doi.org/10.7169/facm/1704.
Kakol J, Sliwa W. Efimov Spaces and the Separable Quotient Problem for Spaces CP (X). J Math Anal Appl. 2018; 457: 104-113. https://doi.org/10.1016/j.jmaa.2017.08.010.
Morris SA, Yost DT. An Observation on the Separable Quotient Problem for Banach Spaces. Axioms. 2020; 9(1): 7. https://doi.org/10.3390/axioms901000.
Gabriyelyan SS, Morris SA. A Topological Group Observation on the Banach-Mazur Separable Quotient Problem. Topology Appl. 2019; 259: 283-286. https://doi.org/10.1016/j.topol.2019.02.036.
He J, Tsaban B, Zhang S. Menger Bounded Groups and Axioms about Filters. Topology Appl. 2022; 309: 107914. https://doi.org/10.1016/j.topol.2021.107914.
Scheepers M. On a Hypothesis for N0 Bounded Groups. Topology Appl. 2019; 258: 229-238. https://doi.org/10.1016/j.topol.2019.02.058.
Megrelishvili M, Shlossberg M. Minimality of Topological Matrix Groups and Fermat Primes. Topology Appl. 2022; 322: 108272. https://doi.org/10.1016/j.topol.2022.108272.
Bagchi S. Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces. Symmetry. 2020; 12(1039): 1-13. https://doi.org/10.3390/sym12061039.
Conner G, Kent C. Fundamental Groups of Locally Connected Subsets of the Plane. Adv Math. 2019; 347: 384-407. https://doi.org/10.1016/j.aim.2019.01.043.
Arhangel’skii A, Choban MM. Rajkov Remainder and Other Group Remainders of a Topological Groups. Topology Appl. 2018; 241: 22-88. https://doi.org/10.1016/j.topol.2018.03.024.
Hernandez J, Hernandez S. Reflections in Topological Algebraic Structures. Topology Appl. 2020; 281: 107204. https://doi.org/10.1016/j.topol.2020.107204.
Dheyab AH, Hussain MQ, Yousif RA. Semihollow-Lifting Modules and Projectivity. Baghdad Sci J. 2022; 19(4): 811-815. https://doi.org/10.21123/bsj.2022.19.4.0811.