Approximation of proximities by aggregating T-indistinguishability operators

For a continuous Archimedean t-norm T a method to approximate a proximity relation R (i.e. a reflexive and symmetric fuzzy relation) by a T-transitive one is provided. It consists of aggregating the transitive closure R of R with a (maximal) T-transitive relation B contained in R using a suitable weighted quasi-arithmetic mean to maximize the similarity or minimize the distance to R.Peer Reviewe

ET-Lipschitzian aggregation operators

Lipschitzian and kernel aggregation operators with respect to the natural Tindistinguishability operator ET and their powers are studied. A t-norm T is proved to be ET -lipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an archimedean t-norm T with additive generator t, the quasi-arithmetic mean generated by t is proved to be the more stable aggregation operator with respect to T.Peer ReviewedPostprint (published version

Finding close T-indistinguishability operators to a given proximity

Two ways to approximate a proximity relation R (i.e. a reflexive and symmetric fuzzy relation) by a T-transitive one where T is a continuous archimedean t-norm are given. The first one aggregates the transitive closure R of R with a (maximal) T-transitive relation B contained in R. The second one modifies the values of R or B to better fit them with the ones of R.Peer ReviewedPostprint (published version

Aggregating fuzzy subgroups and T-vague groups

Fuzzy subgroups and T-vague groups are interesting fuzzy algebraic structures that have been widely studied. While fuzzy subgroups fuzzify the concept of crisp subgroup, T-vague groups can be identified with quotient groups of a group by a normal fuzzy subgroup and there is a close relation between both structures and T-indistinguishability operators (fuzzy equivalence relations). In this paper the functions that aggregate fuzzy subgroups and T-vague groups will be studied. The functions aggregating T-indistinguishability operators have been characterized [9] and the main result of this paper is that the functions aggregating T-indistinguishability operators coincide with the ones that aggregate fuzzy subgroups and T-vague groups. In particular, quasi-arithmetic means and some OWA operators aggregate them if the t-norm is continuous Archimedean.Peer ReviewedPostprint (author's final draft

Aggregation operators and lipschitzian conditions

Lipschitzian aggregation operators with respect to the natural T - indistin- guishability operator Et and their powers, and with respect to the residuation ! T with respect to a t-norm T and its powers are studied. A t-norm T is proved to be E T -Lipschitzian and -Lipschitzian, and is interpreted as a fuzzy point and a fuzzy map as well. Given an Archimedean t-norm T with additive generator t , the quasi- arithmetic mean generated by t is proved to be the most stable aggregation operator with respect to TPeer Reviewe

A map characterizing the fuzzy points and columns of a T-indistinguishability operator

A new map (ΛE) between fuzzy subsets of a universe X endowed with a T-indistinguishability operator E is introduced. The main feature of ΛE is that it has the columns of E as fixed points, and thus it provides us with a new criterion to decide whether a generator is a column. Two well known maps (φE and ψE) are also reviewed, in order to compare them with ΛE. Interesting properties of the fixed points of ΛE and Λ2 E are studied. Among others, the fixed points of ΛE (Fix(ΛE)) are proved to be the maximal fuzzy points of (X,E) and the fixed points of Λ2 E coincide with the Image of ΛE. An isometric embedding of X into Fix(ΛE) is established and studied

La Hoja de cálculo : un entorno para la enseñanza y estudio de relaciones borrosas

En este trabajo se estudia la posibilidad de introduir conceptos de teoría de conjuntos borrosos en los currículos correspondientes a distintos niveles de enseñanza. Se hace especial hincapié en la enseñanza de las relaciones borrosas presentando un entorno Excel© como soporte docente y de experimentación.Postprint (published version

On the problem of relaxed indistinguishability operators aggregation

[EN] In this paper we focus our attention on exploring the aggregation of relaxed indistinguishability operators. Concretely we characterize, in terms of triangular triplets with respect to a t-norm, those functions that allow to merge a collection of relaxed indistinguishability operators into a single one.This research was funded by the Spanish Ministry of Economy and Competitiveness under Grants TIN2014-56381-REDT (LODISCO), TIN2016-81731-REDT (LODISCO II) and AEI/FEDER, UE funds.Calvo Sánchez, T.; Fuster Parra, P.; Valero, O. (2017). On the problem of relaxed indistinguishability operators aggregation. En Proceedings of the Workshop on Applied Topological Structures. Editorial Universitat Politècnica de València. 19-26. http://hdl.handle.net/10251/128050OCS192

T-generable indistinguishability operators and their use for feature selection and classification.

Peer ReviewedPostprint (author's final draft

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