Fuzzy properties in fuzzy convergence spaces

Based on the concept of limit of prefilters and residual implication, several notions in fuzzy topology are fuzzyfied in the sense that, for each notion, the degree to which it is fulfilled is considered. We establish therefore theories of degrees of compactness and relative compactness, of closedness, and of continuity. The resulting theory generalizes the corresponding crisp theory in the realm of fuzzy convergence spaces and fuzzy topology

Quantale-valued Cauchy tower spaces and completeness

[EN] Generalizing the concept of a probabilistic Cauchy space, we introduce quantale-valued Cauchy tower spaces. These spaces encompass quantale-valued metric spaces, quantale-valued uniform (convergence) tower spaces and quantale-valued convergence tower groups. For special choices of the quantale, classical and probabilistic metric spaces are covered and probabilistic and approach Cauchy spaces arise. We also study completeness and completion in this setting and establish a connection to the Cauchy completeness of a quantale-valued metric space.Jäger, G.; Ahsanullah, TMG. (2021). Quantale-valued Cauchy tower spaces and completeness. Applied General Topology. 22(2):461-481. https://doi.org/10.4995/agt.2021.15610OJS461481222J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories, Wiley, New York, 1989.T. M. G. Ahsanullah and G. Jäger, Probabilistic uniform convergence spaces redefined, Acta Math. Hungarica 146 (2015), 376-390. https://doi.org/10.1007/s10474-015-0525-6T. M. G. Ahsanullah and G. Jäger, Probabilistic uniformization and probabilistic metrization of probabilistic convergence groups, Math Slovaca 67 (2017), 985-1000. https://doi.org/10.1515/ms-2017-0027P. Brock and D. C. Kent, Approach spaces, limit tower spaces, and probabilistic convergence spaces, Appl. Cat. Structures 5 (1997), 99-110. https://doi.org/10.1023/A:1008633124960H. R. Fischer, Limesräume, Math. Ann. 137 (1959), 269-303. https://doi.org/10.1007/BF01360965R. C. Flagg, Completeness in continuity spaces, in: Category Theory 1991, CMS Conf. Proc. 13 (1992), 183-199.R. C. Flagg, Quantales and continuity spaces, Algebra Univers. 37 (1997), 257-276. https://doi.org/10.1007/s000120050018L. C. Florescu, Probabilistic convergence structures, Aequationes Math. 38 (1989), 123-145. https://doi.org/10.1007/BF01839999G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove and D. S. Scott, Continuous lattices and domains, Cambridge University Press, 2003. https://doi.org/10.1017/CBO9780511542725D. Hofmann and C. D. Reis, Probabilistic metric spaces as enriched categories, Fuzzy Sets and Systems 210 (2013), 1-21. https://doi.org/10.1016/j.fss.2012.05.005U. Höhle, Commutative, residuated l-monoids, in: Non-classical logics and their applications to fuzzy subsets (U. Höhle, E. P. Klement, eds.), Kluwer, Dordrecht 1995, pp. 53-106. https://doi.org/10.1007/978-94-011-0215-5_5G. Jäger, A convergence theory for probabilistic metric spaces, Quaest. Math. 38 (2015), 587-599. https://doi.org/10.2989/16073606.2014.981734G. Jäger and T. M. G. Ahsanullah, Probabilistic limit groups under a $t$-norm, Topology Proceedings 44 (2014), 59-74.G. Jäger and T. M. G. Ahsanullah, Characterization of quantale-valued metric spaces and quantale-valued partial metric spaces by convergence, Applied Gen. Topology 19, no. 1 (2018), 129-144. https://doi.org/10.4995/agt.2018.7849G. Jäger, Quantale-valued uniform convergence towers for quantale-valued metric spaces, Hacettepe J. Math. Stat. 48, no. 5 (2019), 1443-1453. https://doi.org/10.15672/HJMS.2018.585G. Jäger, The Wijsman structure of a quantale-valued metric space, Iranian J. Fuzzy Systems 17, no. 1 (2020), 171-184.H. H. Keller, Die Limes-Uniformisierbarkeit der Limesräume, Math. Ann. 176 (1968), 334-341. https://doi.org/10.1007/BF02052894D. C. Kent and G. D. Richardson, Completions of probabilistic Cauchy spaces, Math. Japonica 48, no. 3 (1998), 399-407.F. W. Lawvere, Metric spaces, generalized logic, and closed categories, Rendiconti del Seminario Matematico e Fisico di Milano 43 (1973), 135-166. Reprinted in: Reprints in Theory and Applications of Categories} 1 (2002), 1-37. https://doi.org/10.1007/BF02924844R. Lowen, Index Analysis, Springer, London, Heidelberg, New York, Dordrecht 2015. https://doi.org/10.1007/978-1-4471-6485-2R. Lowen and Y. J. Lee, Approach theory in merotopic, Cauchy and convergence spaces. I, Acta Math. Hungarica 83, no. 3 (1999), 189-207. https://doi.org/10.1023/A:1006717022079R. Lowen and Y. J. Lee, Approach theory in merotopic, Cauchy and convergence spaces. II, Acta Math. Hungarica 83, no. 3 (1999), 209-229. https://doi.org/10.1023/A:1006769006149R. Lowen and B. Windels, On the quantification of uniform properties, Comment. Math. Univ. Carolin. 38, no. 4 (1997), 749-759.R. Lowen and B. Windels, Approach groups, Rocky Mountain J. Math. 30, no. 3 (2000), 1057-1073. https://doi.org/10.1216/rmjm/1021477259J. Minkler, G. Minkler and G. Richardson, Subcategories of filter tower spaces, Appl. Categ. Structures 9 (2001), 369-379. https://doi.org/10.1023/A:1011226611840H. Nusser, A generalization of probabilistic uniform spaces, Appl. Categ. Structures 10 (2002), 81-98. https://doi.org/10.1023/A:1013375301613H. Nusser, Completion of probabilistic uniform limit spaces, Quaest. Math. 26 (2003), 125-140. https://doi.org/10.2989/16073600309486049G. Preuß, Seminuniform convergence spaces, Math. Japonica 41, no. 3 (1995), 465-491.G. Preuß, Foundations of topology. An approach to convenient topology, Kluwer Academic Publishers, Dordrecht, 2002. https://doi.org/10.1007/978-94-010-0489-3Q. Pu and D. Zhang, Preordered sets valued in a GL-monoid, Fuzzy Sets and Systems 187 (2012), 1-32. https://doi.org/10.1016/j.fss.2011.06.012G. N. Raney, A subdirect-union representation for completely distributive complete lattices, Proc. Amer. Math. Soc. 4 (1953), 518-512. https://doi.org/10.1090/S0002-9939-1953-0058568-4E. E. Reed, Completion of uniform convergence spaces, Math. Ann. 194 (1971), 83-108. https://doi.org/10.1007/BF01362537G. D. Richardson and D. C. Kent, Probabilistic convergence spaces, J. Austral. Math. Soc. (Series A) 61 (1996), 400-420. https://doi.org/10.1017/S1446788700000483B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North Holland, New York, 1983

The Servitization of European Manufacturing Industries

This paper provides new evidence for the servitization of European manufacturing – the trend that manufacturing firms increasingly offer services along with their physical products. We employ input-output data as well as data from a company survey to give a comprehensive picture of servitization across countries and industries. The share of services in the output of manufacturing industries increased in the large majority of European countries between 1995 and 2005 and between 2000 and 2005. Service output of manu-facturing, however, is still small compared to the output of physical products. The highest service shares are found in small countries with a high degree of openness and R&D intensity. EU-12 Member States have lower shares of service output compared to the EU-15. There is a strong link between servitization and technological innovation at different levels. Countries with the highest shares of services on manufacturing output have also the highest R&D intensities at the aggregate level. The service output of these countries consists predominantly of knowledge-intensive services. Highly innovative sectors reveal also the highest share of firms that offer services and the highest turnover generated with services. Examples are electrical and optical equipment, machinery, or the chemical and pharmaceutical industry. At the firm level, we find a U-shaped relationship between firm size and service output, which indicates that small, but also large manufacturing firms have advantages in servitization. Producers of complex, customized products tend to have a higher share of services in output than producers of simple, mass-produced goods. Moreover, firms which have launched products new to the market during the last two years are more likely to realize higher shares of turnover from services compared to companies which launched no products new to the market

The Servitization of European Manufacturing Industries

This paper provides new evidence for the servitization of European manufacturing – the trend that manufacturing firms increasingly offer services along with their physical products. We employ input-output data as well as data from a company survey to give a comprehensive picture of servitization across countries and industries. The share of services in the output of manufacturing industries increased in the large majority of European countries between 1995 and 2005 and between 2000 and 2005. Service output of manu-facturing, however, is still small compared to the output of physical products. The highest service shares are found in small countries with a high degree of openness and R&D intensity. EU-12 Member States have lower shares of service output compared to the EU-15. There is a strong link between servitization and technological innovation at different levels. Countries with the highest shares of services on manufacturing output have also the highest R&D intensities at the aggregate level. The service output of these countries consists predominantly of knowledge-intensive services. Highly innovative sectors reveal also the highest share of firms that offer services and the highest turnover generated with services. Examples are electrical and optical equipment, machinery, or the chemical and pharmaceutical industry. At the firm level, we find a U-shaped relationship between firm size and service output, which indicates that small, but also large manufacturing firms have advantages in servitization. Producers of complex, customized products tend to have a higher share of services in output than producers of simple, mass-produced goods. Moreover, firms which have launched products new to the market during the last two years are more likely to realize higher shares of turnover from services compared to companies which launched no products new to the market

A Category of L-Fuzzy Convergence Spaces

In this paper we take convergence of stratified L-filters as a primitive notion and construct in this way a Cartesian closed category, which contains the category of stratified L-topological spaces as reflexive subcategory. The class of spaces with non-idempotent stratified fuzzy interior operator is characterized as subclass of the class of our stratified L-fuzzy convergence spaces and a first characterization, which fuzzy convergences stem from stratified L-topologies is established. Mathematics Subject Classification (2000): 54A40 Keywords: category, L-filter, fuzzy convergence, L-Topological space, function space, fuzzy topology, L-topology, convergence, closed categories, categories, interior operator, convergence spaces Quaestiones Mathematicaes 24 (4) 2001, 500–51

‎Completeness for Saturated L\mathsf{L}-Quasi-Uniform Limit Spaces

‎We define and study two completeness notions for saturated $\mathsf{L}$-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 $\mathsf{L}$-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 $\mathsf{L}$-quasi-uniform spaces and that also for saturated $\mathsf{L}$-quasi-uniform limit spaces‎, ‎both completeness notions are equivalent‎

Sequential Completeness for ⊤-Quasi-Uniform Spaces and a Fixed Point Theorem

We define sequential completeness for ⊤-quasi-uniform spaces using Cauchy pair ⊤-sequences. We show that completeness implies sequential completeness and that for ⊤-uniform spaces with countable ⊤-uniform bases, completeness and sequential completeness are equivalent. As an illustration of the applicability of the concept, we give a fixed point theorem for certain contractive self-mappings in a ⊤-uniform space. This result yields, as a special case, a fixed point theorem for probabilistic metric spaces

Probabilistic approach spaces

summary:We study a probabilistic generalization of Lowen's approach spaces. Such a probabilistic approach space is defined in terms of a probabilistic distance which assigns to a point and a subset a distance distribution function. We give a suitable axiom scheme and show that the resulting category is isomorphic to the category of left-continuous probabilistic topological convergence spaces and hence is a topological category. We further show that the category of Lowen's approach spaces is isomorphic to a simultaneously bireflective and bicoreflective subcategory and that the category of probabilistic quasi-metric spaces is isomorphic to a bicoreflective subcategory of the category of probabilistic approach spaces