A new approach for KM-fuzzy partial metric spaces

summary:The main purpose of this paper is to give a new approach for partial metric spaces. We first provide the new concept of KM-fuzzy partial metric, as an extension of both the partial metric and KM-fuzzy metric. Then its relationship with the KM-fuzzy quasi-metric is established. In particularly, we construct a KM-fuzzy quasi-metric from a KM-fuzzy partial metric. Finally, after defining the notion of partial pseudo-metric systems, a one-to-one correspondence between partial pseudo-metric systems and KM-fuzzy partial pseudo-metrics is constructed. Furthermore, a fuzzifying topology $\tau_{P}$ on X deduced from KM-fuzzy partial metric is established and some properties of this fuzzifying topology are discussed

On some results of analysis in metric spaces and fuzzy metric spaces

The notion of a fuzzy metric space due to George and Veeramani has many advantages in analysis since many notions and results from classical metric space theory can be extended and generalized to the setting of fuzzy metric spaces, for instance: the notion of completeness, completion of spaces as well as extension of maps. The layout of the dissertation is as follows: Chapter 1 provide the necessary background in the context of metric spaces, while chapter 2 presents some concepts and results from classical metric spaces in the setting of fuzzy metric spaces. In chapter 3 we continue with the study of fuzzy metric spaces, among others we show that: the product of two complete fuzzy metric spaces is also a complete fuzzy metric space. Our main contribution is in chapter 4. We introduce the concept of a standard fuzzy pseudo metric space and present some results on fuzzy metric identification. Furthermore, we discuss some properties of t-nonexpansive maps.Mathematical SciencesM. Sc. (Mathematics

Fuzzy b-Metric Spaces

Metric spaces and their various generalizations occur frequently in computer science applications. This is the reason why, in this paper, we introduced and studied the concept of fuzzy b-metric space, generalizing, in this way, both the notion of fuzzy metric space introduced by I. Kramosil and J. Michálek and the concept of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-bmetric space, extending the notion of fuzzy quasi metric space recently introduced by V. Gregori and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasipseudo- b-metric into an ascending family of quasi-pseudo-b-metrics is established. The use of fuzzy b-metric spaces and fuzzy quasi-b-metric spaces in the study of denotational semantics and their applications in control theory will be an important next step

A construction of a fuzzy topology from a strong fuzzy metric

[EN] After the inception of the concept of a fuzzy metric by I. Kramosil and J. Michalek, and especially after its revision by A. George and G. Veeramani, the attention of many researches was attracted to the topology induced by a fuzzy metric. In most of the works devoted to this subject the resulting topology is an ordinary, that is a crisp one. Recently some researchers showed interest in the fuzzy-type topologies induced by fuzzy metrics. In particular, in the paper  (J.J. Mi\~{n}ana, A. \v{S}ostak, {\it Fuzzifying topology induced by a strong fuzzy metric}, Fuzzy Sets and Systems,  6938 DOI information: 10.1016/j.fss.2015.11.005.) a fuzzifying topology ${\mathcal T}:2^X \to [0,1]$ induced by a fuzzy metric  $m: X\times X \times [0,\infty)$ was constructed. In this paper we extend  this construction to get the fuzzy topology ${\mathcal T}: [0,1]^X \to [0,1]$ and study some properties of this fuzzy topology.54AGrecova, S.; Sostak, A.; Uljane, I. (2016). A construction of a fuzzy topology from a strong fuzzy metric. Applied General Topology. 17(2):105-116. doi:10.4995/agt.2016.4495.SWORD105116172Chang, C. . (1968). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications, 24(1), 182-190. doi:10.1016/0022-247x(68)90057-7Goguen, J. . (1967). L-fuzzy sets. Journal of Mathematical Analysis and Applications, 18(1), 145-174. doi:10.1016/0022-247x(67)90189-8Goguen, J. . (1973). The fuzzy tychonoff theorem. Journal of Mathematical Analysis and Applications, 43(3), 734-742. doi:10.1016/0022-247x(73)90288-6George, A., & Veeramani, P. (1994). On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64(3), 395-399. doi:10.1016/0165-0114(94)90162-7George, A., & Veeramani, P. (1997). On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems, 90(3), 365-368. doi:10.1016/s0165-0114(96)00207-2V. Gregori, A. López-Crevillén and S. Morillas, On continuity and uniform continuity in fuzzy metric spaces, Proc. Workshop Appl. Topology WiAT'09 (2009), 85-91.Gregori, V., López-Crevillén, A., Morillas, S., & Sapena, A. (2009). On convergence in fuzzy metric spaces. Topology and its Applications, 156(18), 3002-3006. doi:10.1016/j.topol.2008.12.043V. Gregori and J. Mi-ana, Some concepts related to continuity in fuzzy metric spaces, Proc. Workshop Appl. Topology WiAT'13 (2013), 85-91.Gregori, V., Morillas, S., & Sapena, A. (2010). On a class of completable fuzzy metric spaces. Fuzzy Sets and Systems, 161(16), 2193-2205. doi:10.1016/j.fss.2010.03.013Gregori, V., & Romaguera, S. (2004). Characterizing completable fuzzy metric spaces. Fuzzy Sets and Systems, 144(3), 411-420. doi:10.1016/s0165-0114(03)00161-1Höhle, U. (1980). Upper semicontinuous fuzzy sets and applications. Journal of Mathematical Analysis and Applications, 78(2), 659-673. doi:10.1016/0022-247x(80)90173-0I. Kramosil and J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975), 336-344.Kubiak, T., & Sostak, A. P. (2004). A fuzzification of the category of M-valued L-topological spaces. Applied General Topology, 5(2), 137. doi:10.4995/agt.2004.1965Lowen, R. (1976). Fuzzy topological spaces and fuzzy compactness. Journal of Mathematical Analysis and Applications, 56(3), 621-633. doi:10.1016/0022-247x(76)90029-9Lowen, R. (1977). Initial and final fuzzy topologies and the fuzzy Tychonoff theorem. Journal of Mathematical Analysis and Applications, 58(1), 11-21. doi:10.1016/0022-247x(77)90223-2Mardones-Pérez, I., & de Prada Vicente, M. A. (2015). Fuzzy pseudometric spaces vs fuzzifying structures. Fuzzy Sets and Systems, 267, 117-132. doi:10.1016/j.fss.2014.06.003Mardones-Pérez, I., & de Prada Vicente, M. A. (2012). A representation theorem for fuzzy pseudometrics. Fuzzy Sets and Systems, 195, 90-99. doi:10.1016/j.fss.2011.11.008Menger, K. (1951). Probabilistic Geometry. Proceedings of the National Academy of Sciences, 37(4), 226-229. doi:10.1073/pnas.37.4.226Miheţ, D. (2007). On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems, 158(8), 915-921. doi:10.1016/j.fss.2006.11.012Miñana, J.-J., & Šostak, A. (2016). Fuzzifying topology induced by a strong fuzzy metric. Fuzzy Sets and Systems, 300, 24-39. doi:10.1016/j.fss.2015.11.005Sapena Piera, A. (2001). A contribution to the study of fuzzy metric spaces. Applied General Topology, 2(1), 63. doi:10.4995/agt.2001.3016A. Sapena and S. Morillas, On strong fuzzy metrics, Proc. Workshop Appl. Topology WiAT'09 (2009), 135-141.Schweizer, B., & Sklar, A. (1960). Statistical metric spaces. Pacific Journal of Mathematics, 10(1), 313-334. doi:10.2140/pjm.1960.10.313A. Sostak, On a fuzzy topological structure, Suppl. Rend. Circ. Matem. Palermo, Ser II 11 (1985), 125-186.Shostak, A. P. (1989). Two decades of fuzzy topology: basic ideas, notions, and results. Russian Mathematical Surveys, 44(6), 125-186. doi:10.1070/rm1989v044n06abeh002295Šostak, A. P. (1996). Basic structures of fuzzy topology. Journal of Mathematical Sciences, 78(6), 662-701. doi:10.1007/bf02363065Ying, M. (1991). A new approach for fuzzy topology (I). Fuzzy Sets and Systems, 39(3), 303-321. doi:10.1016/0165-0114(91)90100-5Ying, M. (1992). A new approach for fuzzy topology (II). Fuzzy Sets and Systems, 47(2), 221-232. doi:10.1016/0165-0114(92)90181-3Ying, M. (1993). A new approach for fuzzy topology (III). Fuzzy Sets and Systems, 55(2), 193-207. doi:10.1016/0165-0114(93)90132-2Ying, M. (1993). Compactness in fuzzifying topology. Fuzzy Sets and Systems, 55(1), 79-92. doi:10.1016/0165-0114(93)90303-yYue, Y., & Shi, F.-G. (2010). On fuzzy pseudo-metric spaces. Fuzzy Sets and Systems, 161(8), 1105-1116. doi:10.1016/j.fss.2009.10.00

Metric spaces and textures

[EN] Textures are point-set setting for fuzzy sets, and they provide a framework for the complement-free mathematical concepts. Further dimetric on textures is a gener- alization of classical metric spaces. The aim of this paper is to give some properties of dimetric texture space by using categorical approach. We prove that the category of clas- sical metric spaces is isomorphic to a full subcategory of dimetric texture spaces, and give a natural transformation from metric topologies to dimetric ditopologies. Further, it is pre- sented a relation between dimetric texture spaces and quasi-pseudo metric spaces in the sense of J. F. Kelly. Guardar / Salir Siguiente >Dost, S. (2017). Metric spaces and textures. Applied General Topology. 18(1):203-217. doi:10.4995/agt.2017.6889.SWORD203217181Brown, L. M., & Diker, M. (1998). Ditopological texture spaces and intuitionistic sets. Fuzzy Sets and Systems, 98(2), 217-224. doi:10.1016/s0165-0114(97)00358-8Brown, L. M., & Ertürk, R. (2000). Fuzzy sets as texture spaces, I. Representation theorems. Fuzzy Sets and Systems, 110(2), 227-235. doi:10.1016/s0165-0114(98)00157-2Brown, L. M., & Ertürk, R. (2000). Fuzzy sets as texture spaces, II. Subtextures and quotient textures. Fuzzy Sets and Systems, 110(2), 237-245. doi:10.1016/s0165-0114(98)00158-4Brown, L. M., Ertürk, R., & Dost, Ş. (2004). Ditopological texture spaces and fuzzy topology, I. Basic concepts. Fuzzy Sets and Systems, 147(2), 171-199. doi:10.1016/j.fss.2004.02.009Brown, L. M., Ertürk, R., & Dost, Ş. (2004). Ditopological texture spaces and fuzzy topology, II. Topological considerations. Fuzzy Sets and Systems, 147(2), 201-231. doi:10.1016/j.fss.2004.02.010Brown, L. M., Ertürk, R., & Dost, Ş. (2006). Ditopological texture spaces and fuzzy topology—III: Separation axioms. Fuzzy Sets and Systems, 157(14), 1886-1912. doi:10.1016/j.fss.2006.02.001Diker, M., & Altay Uğur, A. (2012). Textures and covering based rough sets. Information Sciences, 184(1), 44-63. doi:10.1016/j.ins.2011.08.012Dost, Ş. (2017). Semi-compactness in ditopological texture spaces and soft fuzzy topological spaces. Journal of Intelligent & Fuzzy Systems, 32(1), 925-936. doi:10.3233/jifs-1614

Tameness in generalized metric structures

We broaden the framework of metric abstract elementary classes (mAECs) in several essential ways, chiefly by allowing the metric to take values in a well-behaved quantale. As a proof of concept we show that the result of Boney and Zambrano on (metric) tameness under a large cardinal assumption holds in this more general context. We briefly consider a further generalization to partial metric spaces, and hint at connections to classes of fuzzy structures, and structures on sheaves