Semi-Quotients of Paratopologized Groups

Rafaqat Noreen

doi:https://doi.org/10.14445/22315373/IJMTT-V58P516

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 58 | Number 2 | Year 2018 | Article Id. IJMTT-V58P516 | DOI : https://doi.org/10.14445/22315373/IJMTT-V58P516

Semi-Quotients of Paratopologized Groups

Rafaqat Noreen

Abstract

Continuing the study of paratopologized groups, our focus in this paper is to investigate the semi-quotient mappings for the paratopologized groups. Semi-quotient mappings are stronger than semi-continuous mappings. Various results on semi-quotients of paratopologized groups are proved. It is proved that (G/R,∗,sτQ) is an irresolute paratopological group and (G/R,sτQ) is regular semi-quotient space. Semi-isomorphisms of irresolute paratopological groups are discussed.Semi connectedness of irresolute paratopological groups(G/R,∗,sτQ) is also investigated.It is shown that If R and (G/R,∗,sτQ) are semi connected, then G is semi connected.

Keywords

Semi open set, semi closed set, semi homeomorphism, s-paratopological group, irresolute paratopological group, semi connected space, semi component, semi quotient mapping.

References

[1] D.R. Anderson, J.A. Jensen, Semi-continuity on topological spaces, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat., 42(1967), 782-783.
[2] A.V. Arhangel'skii, E. A. Reznichenko, Paratopological and semitopological groups versus topological groups, Topology Appl. 151 (2005), 107-119.
[3] A.V. Arhangel'skii, M. Tkachenko, Topological Groups and Related Structures, Atlantis Press and World Sci., 2008.
[4] N. Biswas, On some mapping on topological spaces, Bull. cal. Math. Soc., 61(1969), 127-135
[5] E. Bohn, J. Lee, Semi-topological groups, Amer. Math. Monthly, 72(1965), 996-998.
[6] M. S. Bosan, M. Khan, and Ljubisa D.R. Kočinac, On s-topological groups, mathematica moravica,18(2)(2014),35-44.
[7] D.A. Carnahan, Some properties related to compactness in topological spaces, Ph. D. Thesis univ. of Arkansas, 1973.
[8] S.G. Crossley, S.K. Hildebrand, Semi-closure, Texas J. Sci. 22(1971), 99-112.
[9] S.G. Crossley, S.K. Hildebrand, Semi-closed sets and semi-continuity in topological spaces, Texas J. Sci., 22(1971), 123-126.
[10] S.G. Crossley, S.K. Hildebrand, Semi-topological properties, Fund. Math., 74(1972), 233-254.
[11] P. Das, Note on semi connectedness, I.J.M.M., 12(1974), 31-34.
[12] R. Engelking, General Topology (revised and completed edition), HeldermannVerlag, Berlin, 1989.
[13] M.K. Gupta and T. Noiri, C-compactness modulo an ideal, Internat. J. Math. sci. (2006), 1-12.
[14] S. Kempisty, Sur les fonctionsquasicontinues, Fund. Math., 19(1932), 184-197.
[15] M. Khan and M. S. Bosan, A note on s-topological groups, Life Sci J; 11(7s), (2014), 370-374.
[16] M. Khan and R. Noreen, On Paratopologized Groups, Analele Uni. Oradea Fasc. Math. Tom XXIII 2(2016), 147-156.
[17] M. Khan, R. Noreen and M. S. Bosan, Semi-quotient mappings and spaces, Open math. 14(2016), 1014-1022.
[18] J.P. Lee, Onsemihomeomorphisms, Internat. J. Math. Math. Sci. 13(1990), 129-134.
[19] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly., 70(1963), 36-41.
[20] F. Lin and S. Lin, pseudobounded orpseudobounded paratopological groups,Filomat, 25(3)(2011), 93-103.
[21] S. Marcus, Sur les fonctionsquasicontinuesausensde S. Kempisty, Coll. Math., 8(1961), 47-53.
[22] A.Neubrunnov´a, On certain generalizations of the notion of continuity, Mat. Časopis, 23(1973), 374-380.
[23] T. Noiri, On semi continuous mappings, Atti. Accad. Naz. Lin. El. Sci. Fis. mat. Natur. 8(54)(1973), 210-214.
[24] R. Noreen and M. Khan , Quasi-boundedness of Irresolute paratopological groups,(http./doi.org/10.1080/25742558.2018.1458553), cogent Mathematics and Statistics, (2018) 5: 1458553.
[25] T.M. Nour, A note on some applications of semi-open sets, Internat. J. Math. Sci. 21(1998), 205-207.
[26] O. Njasted, On some classes of nearly open sets, Pacific Jr. Math. 15(1965), 961-970.
[27] Z. Piotrowski, On semi-homeomorphisms, Boll. U.M.I. (5)16-A(1979), 501--509.
[28] V. Pipitone , G. Russo, Spazisemiconnesi e spazisemiaperti, Rend. Circ. Mat. Palermo (2)24(1975), 273-285.
[29] O.V. Ravsky, Paratopological groups I, Matem. Studii , 16 (2001), 37-48.
[30] M.S. Sarak, Semi compact sets and associated properties, International journal of mathematics and mathematical sciences, (2009), 475-495.
[31] J.P. Sarker, H. Dasgupta, Locally semi-connectedness in topological spaces, Indian j. pure appl. math., 16(12)(1985), 1488-1494.
[32] R. Shen, Remaks on products of generalized topologies, Acta Math. Hungar, 124(2009), 363-369.

Citation :

Rafaqat Noreen, "Semi-Quotients of Paratopologized Groups," International Journal of Mathematics Trends and Technology (IJMTT), vol. 58, no. 2, pp. 112-119, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V58P516