Some characteristics of matroids through rough sets

At present, practical application and theoretical discussion of rough sets are two hot problems in computer science. The core concepts of rough set theory are upper and lower approximation operators based on equivalence relations. Matroid, as a branch of mathematics, is a structure that generalizes linear independence in vector spaces. Further, matroid theory borrows extensively from the terminology of linear algebra and graph theory. We can combine rough set theory with matroid theory through using rough sets to study some characteristics of matroids. In this paper, we apply rough sets to matroids through defining a family of sets which are constructed from the upper approximation operator with respect to an equivalence relation. First, we prove the family of sets satisfies the support set axioms of matroids, and then we obtain a matroid. We say the matroids induced by the equivalence relation and a type of matroid, namely support matroid, is induced. Second, through rough sets, some characteristics of matroids such as independent sets, support sets, bases, hyperplanes and closed sets are investigated.Comment: 13 page

Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

Fuzzy games with a countable space of actions and applications to systems of generalized quasi-variational inequalities

In this paper, we introduce an abstract fuzzy economy (generalized fuzzy game) model with a countable space of actions and we study the existence of the fuzzy equilibrium. As applications, two types of results are obtained. The first ones concern the existence of the solutions for systems of generalized quasi-variational inequalities with random fuzzy mappings which we define. The last ones are new random fixed point theorems for correspondences with values in complete countable metric spaces.Comment: 18 page