The bicompletion of fuzzy quasi-metric spaces

Extending the well-known result that every fuzzy metric space, in the sense of Kramosil and Michalek, has a completion which is unique up to isometry, we show that every KM-fuzzy quasi-metric space has a bicompletion which is unique up to isometry, and deduce that for each KM-fuzzy quasi-metric space, the completion of its induced fuzzy metric space coincides with the fuzzy metric space induced by its bicompletion. © 2010 Elsevier B.V.Supported by the Spanish Ministry of Science and Innovation, under Grant MTM2009-12872-C02-01.Castro Company, F.; Romaguera Bonilla, S.; Tirado Peláez, P. (2011). The bicompletion of fuzzy quasi-metric spaces. Fuzzy Sets and Systems. 166(1):56-64. https://doi.org/10.1016/j.fss.2010.12.004S5664166

Fuzzy Quasi-Metric Spaces: Bicompletion, Contractions on Product Spaces, and Applications to Access Predictions

Desde que L.A. Zadeh presentó la teoría de conjuntos difusos en 1965, esta se ha usado en una amplia serie de áreas de las matemáticas y se ha aplicado en una gran variedad de escenarios de la vida real. Estos escenarios cubren procesos complejos sin modelo matemático sencillo tales como dispositivos de control industrial, reconocimiento de patrones o sistemas que gestionen información imprecisa o altamente impredecible. La topología difusa es un importante ejemplo de uso de la teoría de L.A. Zadeh. Durante años, los autores de este campo han buscado obtener la definición de un espacio métrico difuso para medir la distancia entre elementos según grados de proximidad. El presente trabajo trata acerca de la bicompletación de espacios casi-métricos difusos en el sentido de Kramosil y Michalek. Sherwood probó que todo espacio métrico difuso admite completación que es única excepto por isometría basándose en propiedades de la métrica de Lévy. Probamos aquí que todo espacio casi-métrico difuso tiene bicompletación usando directamente el supremo de conjuntos en [0,1] y límites inferiores de secuencias en [0,1] en lugar de usar la métrica de Lévy. Aprovechamos tanto la bicompletitud y bicompletación de espacios casi-métricos difusos como las propiedades de los espacios métricos difusos y difusos intuicionistas para presentar varias aplicaciones a problemas del campo de la informática. Así estudiamos la existencia y unicidad de solución para las ecuaciones de recurrencia asociadas a ciertos algoritmos formados por dos procedimientos recursivos. Para analizar su complejidad aplicamos el principio de contracción de Banach tanto en un producto de casi-métricas no-Arquimedianas en el dominio de las palabras como en la casi-métrica producto de dos espacios de complejidad casi-métricos de Schellekens. Estudiamos también una aplicación de espacios métricos difusos a sistemas de información basados en localidad de accesos.Castro Company, F. (2010). Fuzzy Quasi-Metric Spaces: Bicompletion, Contractions on Product Spaces, and Applications to Access Predictions [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8420Palanci

On completable fuzzy metric spaces

In this paper we construct a non-completable fuzzy metric space in the sense of George and Veeramani which allows to answer an open question related to continuity on the real parameter t. In addition, the constructed space is not strong (non-Archimedean).Juan Jose Minana acknowledges the support of Conselleria de Educacion, Formacion y Empleo (Programa Vali+d para investigadores en formacion) of Generalitat Valenciana, Spain and the support of Universitat Politecnica de Valencia under Grant PAID-06-12 SP20120471.Gregori Gregori, V.; Miñana, J.; Morillas, S. (2015). On completable fuzzy metric spaces. Fuzzy Sets and Systems. 267:133-139. https://doi.org/10.1016/j.fss.2014.07.009S13313926

PROPERTIES OF FUZZY DITANCE ON FUZZY SET

In this paper we introduce the definition of fuzzy distance space on fuzzy set then we study and discuss several properties of  this space after some illustrative examples are given . Furthermore we introduce the definition of fuzzy convergence, fuzzy Cauchy sequence of fuzzy point and fuzzy bounded fuzzy distance space .&nbsp

FUZZY COMPACT FUZZY DISTANCE SPACE

In this paper we recall the definition of fuzzy distance space on a fuzzy set then we define a compact fuzzy distance space and fuzzy totally bounded after that we prove that fuzzy totally bounded fuzzy complete fuzzy distance space is fuzzy compact. Moreover we recall the definition of fuzzy continuous and uniform fuzzy continuous function to prove that fuzzy continuous function and uniform fuzzy continuous functions are equivalent on a fuzzy compact fuzzy distance spaces

The Wijsman topology of a fuzzy metric space

[EN] We introduce and study the notions of lower Wijsman topology, upper Wijsman topology and Wijsman topology of a fuzzy metric space in the sense of Kramosil and Michalek. In particular, quasi-uniformizability, uniformizability, quasi-metrizability and metrizability of these topologies are discussed. Their relations with other hypertopologies are also analyzed. Corresponding results to the Wijsman topology of a metric space are deduced from our approach with the help of the standard fuzzy metric.J. Gutierrez Garcia acknowledges the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-02. J. Rodriguez-Lopez, S. Romaguera and M. Sanchis also acknowledge the support of the Ministry of Economy and Competitiveness of Spain, Grant MTM2012-37894-C02-01.Gutierrez Garcia, J.; Rodríguez López, J.; Romaguera Bonilla, S.; Sanchis, M. (2016). The Wijsman topology of a fuzzy metric space. Fuzzy Sets and Systems. 300:57-71. https://doi.org/10.1016/j.fss.2015.08.005S577130

Topological and Computational Models for Fuzzy Metric Spaces via Domain Theory

This doctoral thesis is devoted to investigate the problem of establishing connections between Domain Theory and the theory of fuzzy metric spaces, in the sense of Kramosil and Michalek, by means of the notion of a formal ball, and then constructing topological and computational models for (complete) fuzzy metric spaces. The antecedents of this research are mainly the well-known articles of A. Edalat and R. Heckmann [A computational model for metric spaces, Theoret- ical Computer Science 193 (1998), 53-73], and R. Heckmann [Approximation of metric spaces by partial metric spaces, Applied Categorical Structures 7 (1999), 71-83], where the authors obtained nice and direct links between Do- main Theory and the theory of metric spaces - two crucial tools in the study of denotational semantics - by using formal balls. Since every metric induces a fuzzy metric (the so-called standard fuzzy metric), the problem of extending Edalat and Heckmann's works to the fuzzy framework arises in a natural way. In our study we essentially propose two di erent approaches. For the rst one, valid for those fuzzy metric spaces whose continuous t-norm is the minimum, we introduce a new notion of fuzzy metric completeness (the so-called standard completeness) that allows us to construct a (topological) model that includes the classical theory as a special case. The second one, valid for those fuzzy metric spaces whose continuous t-norm is greater or equal than the Lukasiewicz t-norm, allows us to construct, among other satisfactory results, a fuzzy quasi-metric on the continuous domain of formal balls whose restriction to the set of maximal elements is isometric to the given fuzzy metric. Thus we obtain a computational model for complete fuzzy metric spaces. We also prove some new xed point theorems in complete fuzzy metric spaces with versions to the intuitionistic case and the ordered case, respec- tively. Finally, we discuss the problem of extending the obtained results to the asymmetric framework.Ricarte Moreno, L. (2013). Topological and Computational Models for Fuzzy Metric Spaces via Domain Theory [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/34670TESI