Probabilistic uniformities of uniform spaces

[EN] Usually, fuzzy metric spaces are endowed with crisp topologies or crisp uniformities. Nevertheless, some authors have shown how to construct in this context different kinds of fuzzy uniformities like a Hutton [0, 1]- quasi-uniformity or a probabilistic uniformity. In 2010, J. Guti´errez Garc´ıa, S. Romaguera and M. Sanchis [7] proved that the category of uniform spaces is isomorphic to a category whose objects are sets endowed with a fuzzy uniform structure, i. e. a family of fuzzy pseudometrics satisfying certain conditions. We will show here that, by means of this isomorphism, we can obtain several methods to endow a uniform space with a probabilistic uniformity. Furthermore, we obtain a factorization of some functors introduced in [6].The first and third authors are supported by the grant MTM2015-64373-P (MINECO/FEDER, UE).Rodríguez López, J.; Romaguera Bonilla, S.; Sanchis, M. (2017). Probabilistic uniformities of uniform spaces. En Proceedings of the Workshop on Applied Topological Structures. Editorial Universitat Politècnica de València. 103-111. http://hdl.handle.net/10251/128037OCS10311

Fuzzy uniform structures

[EN] The concept of fuzzy uniform structure was introduced in [7] as a fuzzy counterpart of the concept of gauge associated with a uniformity. In fact, the category of fuzzy uniform structures is isomorphic to that of uniform spaces. Here, we introduce two other concepts of fuzzy uniform structures which allow to establish two categories isomorphic to the categories of probabilistic uniform spaces and Lowen uniform spaces, respectively. This sheds light on the relationship between these fuzzy uniformities and classical uniformities. Furthermore, we obtain a factorization of Lowen's adjoint functors omega(*) and iota which establish a relationship between the categories of uniform spaces and Lowen uniform spaces.Research supported under grant MTM2015-64373-P (MINECO/FEDER, UE).Rodríguez López, J. (2017). Fuzzy uniform structures. Filomat. 31(15):4763-4779. https://doi.org/10.2298/FIL1715763R47634779311

Hyperspace of a fuzzy quasi-uniform space

[EN] The aim of this paper is to present a fuzzy counterpart method of constructing the Hausdorff quasi-uniformity of a crisp quasi-uniformity. This process, based on previous works due to Morsi [25] and Georgescu [9], allows to extend probabilistic and Hutton [0, 1]-quasi-uniformities on a set X to its power set. In this way, we obtain an endofunctor for each one of the categories of those objects. We will demonstrate the commutativity of these endofunctors with Lowen and Katsaras' functors. Furthermore, we will prove the compatibility of our construction with the Hausdorff fuzzy quasi-pseudometric introduced in [33].The second author is supported by the grant MTM2015-64373-P (MINECO/FEDER, UE). The authors are grateful to the reviewers for useful comments which have improved the first version of the paperPedraza Aguilera, T.; Rodríguez López, J. (2020). Hyperspace of a fuzzy quasi-uniform space. Iranian Journal of Fuzzy Systems. 17(2):97-114. https://doi.org/10.22111/IJFS.2020.5222S9711417

Aggregation of L-probabilistic quasi-uniformities

[EN] The problem of aggregating fuzzy structures, mainly fuzzy binary relations, has deserved a lot of attention in the last years due to its application in several fields. Here, we face the problem of studying which properties must satisfy a function in order to merge an arbitrary family of (bases of) L-probabilistic quasi-uniformities into a single one. These fuzzy structures are special filters of fuzzy binary relations. Hence we first make a complete study of functions between partially-ordered sets that preserve some special sets, such as filters. Afterwards, a complete characterization of those functions aggregating bases of L-probabilistic quasi-uniformities is obtained. In particular, attention is paid to the case L={0,1}, which allows one to obtain results for functions which aggregate crisp quasi-uniformities. Moreover, we provide some examples of our results including one showing that Lowen's functor iota which transforms a probabilistic quasi-uniformity into a crisp quasi-uniformity can be constructed using this aggregation procedure.J. Rodriguez-Lopez acknowledges financial support from FEDER/Ministerio de Ciencia, Innovacion y Universidades-Agencia Estatal de Investigacion Proyecto PGC2018-095709-B-C21.Pedraza Aguilera, T.; Rodríguez López, J. (2020). Aggregation of L-probabilistic quasi-uniformities. Mathematics. 8(11):1-21. https://doi.org/10.3390/math8111980S12181

On fuzzy uniformities induced by a fuzzy metric space

[EN] Different types of fuzzy uniformities have been introduced in the literature standing out the notions due to Hutton, Hohle and Lowen. The main purpose of this paper is to study several methods to endow a fuzzy metric space (X, M, *), in the sense of George and Veeramani, with a probabilistic uniformity and with a Hutton [0, 1](-quasi)-uniformity. We will show the functorial behavior of these constructions as well as its relation with respect to Lowen's functors and Katsaras's functors, which establish a relationship between the categories of probabilistic uniformities and Hutton [0, 1](-quasi)-uniformities with the category of classical uniformities respectively. Furthermore, we also study the fuzzy topologies associated with these fuzzy uniformities. (C) 2017 Elsevier B.V. All rights reserved.The first named author acknowledges the support of the grants MTM2015-63608-P (MINECO/FEDER, UE) and IT974-16 (Basque Government).The second named author is supported by the grant MTM2015-64373-P (MINECO/FEDER, UE).Gutierrez Garcia, J.; Rodríguez López, J.; Romaguera Bonilla, S. (2018). On fuzzy uniformities induced by a fuzzy metric space. Fuzzy Sets and Systems. 330:52-78. https://doi.org/10.1016/j.fss.2017.05.001S527833