Completeness for Saturated L-Quasi-Uniform Limit Spaces
الموضوعات : Transactions on Fuzzy Sets and Systems
1 - School of Mechanical Engineering, University of Applied Sciences Stralsund, Stralsund, Germany.
الکلمات المفتاحية: Saturated prefilter, Saturated L-quasi-uniform limit space, Completeness.,
ملخص المقالة :
We define and study two completeness notions for saturated L-quasi-uniform limit spaces. The one, that we term Lawvere completeness, is defined using the concept of promodule and lends a lax algebraic interpretation of completeness also for saturated L-quasi-uniform limit spaces. The other, termed Cauchy completeness, is defined using saturated Cauchy pair prefilters. We show that both concepts coincide with related notions in the case of saturated L-quasi-uniform spaces and that also for saturated L-quasi-uniform limit spaces, both completeness notions are equivalent.
[1] R. Belohlávek, Fuzzy Relation Systems, Foundation and Principles, Kluwer Academic/Plenum Publishers, New York, Boston, Dordrecht, London, Moscow, (2002). DOI: http://doi.org/10.1007/978-1-4615-0633-1
[2] M. M. Clementino and D. Hofmann, Lawvere completeness in topology, Appl. Categor. Struct., 17(2) (2009), 175-210. DOI: http://doi.org/10.1007/s10485-008-9152-5
[3] R. C. Flagg, Quantales and continuity spaces, Algebra Univers, 37(3) (1997), 257-276. DOI: http://doi.org/10.1007/s000120050018
[4] D. Hofmann, G. J. Seal and W. Tholen, Monoidal Topology, Cambridge University Press, (2014). DOI: http://doi.org/10.1017/CBO9781107517288
[5] U. Höhle, Probabilistic topologies induced by L-fuzzy uniformities, Manuscripta Math, 38(3) (1982), 289-323. DOI: http://doi.org/10.1007/BF01170928
[6] G. Jäger, Sequential completeness for T-quasi-uniform spaces and a fixed point theorem, Mathematics, 10(13) (2022), 2285. DOI: http://doi.org/10.3390/math10132285
[7] G. Jäger, Diagonal conditions and uniformly continuous extension in ⊤-uniform limit spaces, Iranian J.Fuzzy Systems, 19(5) (2022), 131-145. DOI: http://doi.org/10.22111/IJFS.2022.7161
[8] G. Jäger, ⊤-quasi-Cauchy spaces-a non-symmetric theory of completeness and completion, Appl. Gen.Topol, 24(1) (2023), 205-227. DOI: http://doi.org/10.4995/agt.2023.18783
[9] G. Jäger and Y. Yue, T-uniform convergence spaces, Iranian J. Fuzzy Systems, 19(2) (2022), 133-149. DOI: http://doi.org/10.22111/IJFS.2022.6795
[10] W. F. Lindgren and P. Fletcher, A construction of the pair completion of a quasi-uniform space, Can. Math. Bull, 21(1) (1978) 53-59. DOI: http://doi.org/10.4153/CMB-1978-009-2
[11] B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, New York, (1983).
[12] L. Sun and Y. Yue, Completeness of L-quasi-uniform convergence spaces, Iranian J. Fuzzy Systems, 20(2) (2023), 57-67. DOI: http://doi.org/10.22111/IJFS.2023.7556
[13] Y. Wang and Y. Yue, Cauchy completion of Fuzzy quasi-uniform spaces, Filomat, 35(12) (2021), 3983-4004. DOI: http://doi.org/10.2298/FIL2112983W
[14] Y. Yue and J. Fang, Completeness in probabilistic quasi-uniform spaces, Fuzzy Sets and Systems, 370(2019), 34-62. DOI: http://doi.org/10.1016/j.fss.2018.08.005
