Aggregation of Weak Fuzzy Norms

[EN] Aggregation is a mathematical process consisting in the fusion of a set of values into a unique one and representing them in some sense. Aggregation functions have demonstrated to be very important in many problems related to the fusion of information. This has resulted in the extended use of these functions not only to combine a family of numbers but also a family of certain mathematical structures such as metrics or norms, in the classical context, or indistinguishability operators or fuzzy metrics in the fuzzy context. In this paper, we study and characterize the functions through which we can obtain a single weak fuzzy (quasi-)norm from an arbitrary family of weak fuzzy (quasi-)norms in two different senses: when each weak fuzzy (quasi-)norm is defined on a possibly different vector space or when all of them are defined on the same vector space. We will show that, contrary to the crisp case, weak fuzzy (quasi-)norm aggregation functions are equivalent to fuzzy (quasi-)metric aggregation functions.J.R.-L. acknowledges financial support from the research project PGC2018-095709-B-C21 funded by MCIN/AEI/10.13039/501100011033 and FEDER Una manera de hacer Europa.Pedraza Aguilera, T.; Ramos-Canós, J.; Rodríguez López, J. (2021). Aggregation of Weak Fuzzy Norms. Symmetry (Basel). 13(10):1-16. https://doi.org/10.3390/sym13101908116131

The State-of-the-Art of Set Visualization

Sets comprise a generic data model that has been used in a variety of data analysis problems. Such problems involve analysing and visualizing set relations between multiple sets defined over the same collection of elements. However, visualizing sets is a non-trivial problem due to the large number of possible relations between them. We provide a systematic overview of state-of-the-art techniques for visualizing different kinds of set relations. We classify these techniques into six main categories according to the visual representations they use and the tasks they support. We compare the categories to provide guidance for choosing an appropriate technique for a given problem. Finally, we identify challenges in this area that need further research and propose possible directions to address these challenges. Further resources on set visualization are available at http://www.setviz.net

Aggregation operators on partially ordered sets and their categorical foundations

summary:In spite of increasing studies and investigations in the field of aggregation operators, there are two fundamental problems remaining unsolved: aggregation of $L$-fuzzy set-theoretic notions and their justification. In order to solve these problems, we will formulate aggregation operators and their special types on partially ordered sets with universal bounds, and introduce their categories. Furthermore, we will show that there exists a strong connection between the category of aggregation operators on partially ordered sets with universal bounds (Agop) and the category of partially ordered groupoids with universal bounds (Pogpu). Moreover, the subcategories of Agop consisting of associative aggregation operators, symmetric and associative aggregation operators and associative aggregation operators with neutral elements are, respectively, isomorphic to the subcategories of Pogpu formed by partially ordered semigroups, commutative partially ordered semigroups and partially ordered monoids in the sense of Birkhoff. As a justification of the present notions and results, some relevant examples for aggregations operators on partially ordered sets are given. Particularly, aggregation process in probabilistic metric spaces is also considered

Aggregation functions of topological structures

[ES] Las funciones de agregación son un tipo especial de funciones que permiten combinar un número finito de entradas en una única salida que debe resumir de algún modo la información de todas las entradas. Como ejemplo paradigmático de estas funciones podemos destacar la media aritmética aunque el abanico disponible es muy amplio. El estudio de este tipo de funciones se ha convertido en un área muy activa de las matemáticas debido a su utilidad tanto en matemática pura (ecuaciones funcionales, teoría de integración, ...) como en matemática aplicada (toma de decisiones, inteligencia artificial). Además de aglutinar valores, las funciones de agregación permiten también fusionar estructuras topológicas. Por ejemplo, Dobos y sus colaboradores han estudiado el problema de determinar aquellas funciones que permiten obtener, a partir de una familia arbitraria de espacios métricos, una métrica en el producto cartesiano de dichos espacios. Este problema también se ha analizado en el caso de casi-métricas, es decir, métricas que no satisfacen necesariamente el axioma de simetría. Un problema muy interesante relacionado con este es el de caracterizar las funciones que no solo permiten hacer esta agregación de métricas sino que también conservan la topología producto, es decir, caracterizar las funciones que al agregar métricas devuelven una métrica compatible con la topología producto. Aparte del estudio de agregación de estructuras topológicas clásicas, también existen resultados sobre cómo agregar estructuras difusas. Por otra parte, también se ha investigado la agregación de otras estructuras topológicas como las normas y las normas asimétricas pero no existen resultados sobre la agregación de su correspondiente estructura difusa. Así, en este trabajo se propone al estudiante que haga un análisis de los diferentes resultados que existen sobre funciones que agregan estructuras topológicas tanto en el contexto clásico como en el difuso. Además, también se abordará el problema de obtener la caracterización de aquellas funciones que agregan normas difusas débiles. De este modo, los objetivos principales de este trabajo son: revisar el concepto de función de agregación y sus propiedades fundamentales; revisar los resultados existentes en la literatura sobre la agregación de métricas y casi-métricas; estudiar las caracterizaciones que existen sobre funciones que agregan normas y normas asimétricas; revisar algunos conceptos de la topología difusa; caracterizar las funciones que permitan agregar normas difusas y establecer una relación adecuada con las funciones que agregan métricas difusas.[EN] Aggregation functions are a type of functions that allow us to combine a finite number of inputs into one output that in some way summarize de information of all the inputs. An example of this functions is the arithmetic mean, but there are lots of different possibilities. The study of these functions has become in an important area of mathematics because of their utility in pure mathematics (funcional equations, integration theory) and in applies mathematics (making decisions, artificial intelligence). Moreover, we can use aggregation functions to combine topological structures. For example, Dobos and his collaborators have solved the problem of when a function allow us to obtain from a family of metric spaces, a new metric in the Cartesian product of that spaces. This problem has been analyzed in the case of quasi-metrics, i. e. metrics that don't hold the axiom of symmetry. An interesting problem related to this one is characterizing functions that aggregate metrics and preserve the product topology. That is to say, characterize functions that when they aggregate metrics, the output is a metric compatible with the product topology. In addition to the results of functions that aggregate topological structures in the classic sense, there exist results of fuzzy structures. Moreover, there are papers about norms and asymmetric norms, but there are not results about aggregation of norms and quasi-norms in the fuzzy context. So, in this paper there is an analysis of the existing literature about functions that aggregate topological structures in the classic sense and in the fuzzy context. Moreover, we solve the problem of characterize weak fuzzy norm agrgegation functions. So the principal objectives of this paper are: Make a review of the concept of an aggregation function and its principal properties. Make a review of the existing results in the literature about aggregation functions of metrics and quasi-metrics. Study the existing characterizations about norms and asymmetric norms. Make a review about some concepts of fuzzy topology. Characterize fuzzy norm aggregation function and establish a relation between these functions and fuzzy metric aggregation functions.Ramos Canos, J. (2021). Aggregation functions of topological structures. Universitat Politècnica de València. http://hdl.handle.net/10251/175057TFG