×

Stratified \(L\)-prefilter convergence structures in stratified \(L\)-topological spaces. (English) Zbl 1402.54015

Summary: In this paper, a new approach to fuzzy convergence theory in the framework of stratified \(L\)-topological spaces is provided. Firstly, the concept of stratified \(L\)-prefilter convergence structures is introduced, and it is shown that the resulting category is a Cartesian closed topological category. Secondly, the relations between the category of stratified \(L\)-prefilter convergence spaces and the category of stratified \(L\)-topological spaces are studied, and it is proved that the latter can be embedded in the former as a reflective subcategory. Finally, the relations between the category of stratified \(L\)-prefilter convergence spaces and the category of stratified \(L\)-Min convergence spaces (fuzzy convergence spaces in the sense of Min) are investigated, and it is shown that the former can be embedded in the latter as a reflective subcategory.

MSC:

54A40 Fuzzy topology
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54E20 Stratifiable spaces, cosmic spaces, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adámek J, Herrlich H, Strecker GE (1990) Abstract and concrete categories. Wiley, New York · Zbl 0695.18001
[2] Chang, CL, Fuzzy topological spaces, J Math Anal Appl, 24, 182-190, (1968) · Zbl 0167.51001 · doi:10.1016/0022-247X(68)90057-7
[3] Fang, JM, Stratified \(L\)-ordered convergence structures, Fuzzy Sets Syst, 161, 2130-2149, (2010) · Zbl 1197.54015 · doi:10.1016/j.fss.2010.04.001
[4] Fang, JM, Relationships between \(L\)-ordered convergence structures and strong \(L\)-topologies, Fuzzy Sets Syst, 161, 2923-2944, (2010) · Zbl 1271.54030 · doi:10.1016/j.fss.2010.07.010
[5] Güloǧlu, M.; Coker, D., Convergence in \(I\)-fuzzy topological spaces, Fuzzy Sets Syst, 151, 615-623, (2005) · Zbl 1069.54006 · doi:10.1016/j.fss.2004.06.016
[6] Höhle, U.; Šostak, AP; Höhle, U. (ed.); Rodabaugh, SE (ed.), Axiomatic foundations of fixed-basis fuzzy topology, No. 3, 123-173, (1999), Boston · Zbl 0977.54006 · doi:10.1007/978-1-4615-5079-2_5
[7] Jäger, G., A category of \(L\)-fuzzy convergence spaces, Quaest Math, 24, 501-517, (2001) · Zbl 0991.54004 · doi:10.1080/16073606.2001.9639237
[8] Jäger, G., Subcategories of lattice-valued convergence spaces, Fuzzy Sets Syst, 156, 1-24, (2005) · Zbl 1086.54006 · doi:10.1016/j.fss.2005.04.013
[9] Jäger, G., Lattice-valued convergence spaces and regularity, Fuzzy Sets Syst, 159, 2488-2502, (2008) · Zbl 1177.54003 · doi:10.1016/j.fss.2008.05.014
[10] Jäger, G., Compactification of lattice-valued convergence spaces, Fuzzy Sets Syst, 161, 1002-1010, (2010) · Zbl 1196.54011 · doi:10.1016/j.fss.2009.10.010
[11] Jäger, G., Compactness in lattice-valued function spaces, Fuzzy Sets Syst, 161, 2962-2974, (2010) · Zbl 1210.54014 · doi:10.1016/j.fss.2010.07.002
[12] Jäger, G., A one-point compactification for lattice-valued convergence spaces, Fuzzy Sets Syst, 190, 21-31, (2012) · Zbl 1244.54016 · doi:10.1016/j.fss.2011.06.007
[13] Jäger, G., Largest and smallest T\(_2\)-compactifications of lattice-valued convergence spaces, Fuzzy Sets Syst, 190, 32-46, (2012) · Zbl 1251.54011 · doi:10.1016/j.fss.2011.09.011
[14] Jäger, G., Connectedness and local connectedness for lattice-valued convergence spaces, Fuzzy Sets Syst, 300, 134-146, (2016) · Zbl 1378.54008 · doi:10.1016/j.fss.2015.11.013
[15] Li, LQ; Jin, Q., On adjunctions between Lim, S\(L\)-Top, and S\(L\)-Lim, Fuzzy Sets Syst, 182, 66-78, (2011) · Zbl 1244.54018 · doi:10.1016/j.fss.2010.10.002
[16] Li, LQ; Jin, Q., On stratified \(L\)-convergence spaces: pretopological axioms and diagonal axioms, Fuzzy Sets Syst, 204, 40-52, (2012) · Zbl 1254.54010 · doi:10.1016/j.fss.2012.02.012
[17] Li, LQ; Jin, Q., \(p\)-Topologicalness and \(p\)-regularity for lattice-valued convergence spaces, Fuzzy Sets Syst, 238, 26-45, (2014) · Zbl 1315.54007 · doi:10.1016/j.fss.2013.08.012
[18] Li, LQ; Jin, Q.; Hu, K., On stratified \(L\)-convergence spaces: Fischer’s diagonal axiom, Fuzzy Sets Syst, 267, 31-40, (2015) · Zbl 1392.54010 · doi:10.1016/j.fss.2014.09.001
[19] Li, LQ; Jin, Q.; Meng, GW; Hu, K., The lower and upper \(p\)-topological (\(p\)-regular) modifications for lattice-valued convergence spaces, Fuzzy Sets Syst, 282, 47-51, (2016) · Zbl 1392.54011 · doi:10.1016/j.fss.2015.03.002
[20] Lowen, R., Convergence in fuzzy topological spaces, Gen Topl Appl, 10, 147-160, (1979) · Zbl 0409.54008 · doi:10.1016/0016-660X(79)90004-7
[21] Lowen, E.; Lowen, R.; Wuyts, P., The categorical topological approach to fuzzy topology and fuzzy convergence, Fuzzy Sets Syst, 40, 347-373, (1991) · Zbl 0728.54001 · doi:10.1016/0165-0114(91)90165-M
[22] Lowen, E.; Lowen, R.; Rodabaugh, SE (ed.); Klement, EP (ed.); Höhle, U. (ed.), A topological universe extension of FTS, (1992), Dordrecht · Zbl 0774.54002
[23] Min, KC, Fuzzy limit spaces, Fuzzy Sets Syst, 32, 343-357, (1989) · Zbl 0691.54003 · doi:10.1016/0165-0114(89)90267-4
[24] Pang, B.; Fang, JM, \(L\)-fuzzy Q-convergence structures, Fuzzy Sets Syst, 182, 53-65, (2011) · Zbl 1244.54019 · doi:10.1016/j.fss.2011.04.003
[25] Pang, B., Further study on \(L\)-fuzzy Q-convergence structures, Iran J Fuzzy Syst, 10, 147-164, (2013) · Zbl 1335.54012
[26] Pang, B., On \((L, M)\)-fuzzy convergence spaces, Fuzzy Sets Syst, 238, 46-70, (2014) · Zbl 1315.54008 · doi:10.1016/j.fss.2013.07.007
[27] Pang, B.; Shi, F-G, Degrees of compactness of \((L, M)\)-fuzzy convergence spaces and its applications, Fuzzy Sets Syst, 251, 1-22, (2014) · Zbl 1334.54029 · doi:10.1016/j.fss.2014.05.002
[28] Pang, B., Enriched \((L, M)\)-fuzzy convergence spaces, J Intell Fuzzy Syst, 27, 93-103, (2014) · Zbl 1307.54010
[29] Pang, B.; Zhao, Y., Stratified \((L, M)\)-fuzzy Q-convergence spaces, Iran J Fuzzy Syst, 14, 95-111, (2016) · Zbl 1359.54007
[30] Pang, B.; Zhao, Y., Several types of enriched \((L, M)\)-fuzzy convergence spaces, Fuzzy Sets Syst, 321, 55-72, (2017) · Zbl 1375.54004 · doi:10.1016/j.fss.2016.09.001
[31] Pu, BM; Liu, YM, Fuzzy topology (I), neighborhood structure of a fuzzy point and Moore-Smith convergence, J Math Anal Appl, 76, 571-599, (1980) · Zbl 0447.54006 · doi:10.1016/0022-247X(80)90048-7
[32] Preuss G (2002) Foundations of topology: an approach to convenient topology. Kluwer, Dordrecht · Zbl 1058.54001 · doi:10.1007/978-94-010-0489-3
[33] Wu, WC; Fang, JM, \(L\)-ordered fuzzifying convergence spaces, Iran J Fuzzy Syst, 9, 147-161, (2012) · Zbl 1260.54026
[34] Xu, LS, Characterizations of fuzzifying topologies by some limit structures, Fuzzy Sets Syst, 123, 169-176, (2001) · Zbl 1005.54005 · doi:10.1016/S0165-0114(00)00103-2
[35] Yao, W., On many-valued stratified \(L\)-fuzzy convergence spaces, Fuzzy Sets Syst, 159, 2503-2519, (2008) · Zbl 1206.54012 · doi:10.1016/j.fss.2008.03.003
[36] Yao, W., On \(L\)-fuzzifying convergence spaces, Iran J Fuzzy Syst, 6, 63-80, (2009) · Zbl 1176.54007
[37] Yao, W., Moore-Smith convergence in \((L, M)\)-fuzzy topology, Fuzzy Sets Syst, 190, 47-62, (2012) · Zbl 1250.54009 · doi:10.1016/j.fss.2011.08.009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.