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Often, in situations of uncertainty in portfolio management, it is difficult to apply the numerical methods based on the linearity principle. When this happens it is possible to use nonnumeric techniques to assess the situations with a non linear attitude. One of the concepts that can be used in these situations is the concept of grouping. In the last thirty years, several studies have tried to give good solutions to the problems of homogeneous groupings. For example, we could mention the Pichat algorithm, the affinities algorithms and several studies developed by the authors of this work. In this paper, we use some topological axioms in order to develop an algorithm that is able to reduce the number of elements of the power sets of the related sets by connecting them to the sets that form the topologies. We will apply this algorithm in the grouping of titles listed in the Stock Exchange or in its dual perspective.

Extensionality is explored form different points of view. Extensional fuzzy subsets from a fuzzy equivalence relation E are considered as observable subsets with respect to the granularity generated by E. Interestingly, they are characterized as the fuzzy subsets that can be obtained as combinations of the fuzzy equivalence classes of E. Extensional mappings are characterized topologically and the set of extensional mappings between two universes are algebraically determined. Specifying the results to fuzzy mappings from a universe X onto [0, 1] an interpretation of type-2 fuzzy subsets of X as fuzzification of its type-1 fuzzy subsets is provided.

A general and a comprehensive theory of fuzzy topological spaces on the basis of a fixed quadruple M = (L, ≤, ⊗, *), where (L, ≤), ⊗ and *, respectively, denote a complete lattice and binary operations on L satisfying some further axioms, was introduced by Höhle and Šostak. L-topological spaces, convergence structure of L-topological spaces and L-continuous functions form an important part of their work. The present paper continues the study in this area, and provides new results on the convergence structure of L-topological spaces and the continuity in L-topological spaces.

The present paper studies several compactness and continuity notions based on the quadruple M = (L,≤,⊗,*), where (L,≤), ⊗ and * respectively denote a complete lattice and binary operations on L, satisfying some further axioms, was introduced by Höhle and Sostak.^1,2

We introduce a category of fuzzy convergence groups, FCONVGRP a subcategory of the category of fuzzy convergence spaces, FCONV. Viewing $I$ as a complete Heyting algebra, we prove that the category of $I$-tolerance groups, $I$-TOLGRP is isomorphic to a subcategory of FCONVGRP. Since FCONV is a topological universe, and thereby possesses function space structure, upon invoking this, we are able, among others, to show that FCONVGRP is topological, and more importantly, it enables us to obtain a compatible fuzzy convergence function space structure on group of homeomorphisms. It is noticeable, however, that the category of fuzzy neighborhood convergence groups, FNCONVGRP — a supercategory of the well-known category FNS, of fuzzy neighborhood spaces, as well as the category of fuzzy neighborhood groups, FNGRP — a subcategory of FNCONVGRP exhibit nice relationships with FCONVGRP. It is important to note that the objects of FCONVGRP are homogeneous, this paves the way to present two pertinent characterization theorems on fuzzy convergence groups. Finally, introducing a category PSTOPGRP, of pseudotopological groups, we reveal the embeddings of FTOPGRP and PSTOPGRP into FCONVGRP.

In order to cope with the incapability of classical yes-or-no mathematics to capture with incomplete information, several mathematical models to represent and to process imprecise and uncertain information were introduced. We mention explicitly : fuzzy set theory, flou set theory, L-fuzzy set theory, rough set theory, intuitionistic fuzzy set theory, twofold fuzzy sets, … As fuzzy set theory being the most developed one, we will focus on the enrichment of the existing mathematical structures and their domains of applications by the introduction of fuzzy set theory. More precisely we will give some examples of the basic and applied research performed at the Fuzziness & Uncertainty Modelling research unit of Ghent University.

An L-valed category is a category-like conglomerate in which “potential objects” and “potential morphisms” could be respectively objects and morphisms only to a certain degree; this degree is represented by an element of a fixed GL-monoid L = (L,≤, ∧, ∨, *). In this paper, after brief discussing basic concepts related to L-valued categories, we cosider a series of examples of L-valued categories mainly related to algebra and topology.