Covering matroid

In this paper, we propose a new type of matroids, namely covering matroids, and investigate the connections with the second type of covering-based rough sets and some existing special matroids. Firstly, as an extension of partitions, coverings are more natural combinatorial objects and can sometimes be more efficient to deal with problems in the real world. Through extending partitions to coverings, we propose a new type of matroids called covering matroids and prove them to be an extension of partition matroids. Secondly, since some researchers have successfully applied partition matroids to classical rough sets, we study the relationships between covering matroids and covering-based rough sets which are an extension of classical rough sets. Thirdly, in matroid theory, there are many special matroids, such as transversal matroids, partition matroids, 2-circuit matroid and partition-circuit matroids. The relationships among several special matroids and covering matroids are studied.Comment: 15 page

Parametric matroid of rough set

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a combinatorial generalization of linear independence in vector spaces. In this paper, we define a parametric set family, with any subset of a universe as its parameter, to connect rough sets and matroids. On the one hand, for a universe and an equivalence relation on the universe, a parametric set family is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore it can generate a matroid, called a parametric matroid of the rough set. Three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, since partition-circuit matroids were well studied through the lower approximation number, we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.Comment: 15 page

Contribution of František Matúš to the research on conditional independence

summary:An overview is given of results achieved by F. Matúš on probabilistic conditional independence (CI). First, his axiomatic characterizations of stochastic functional dependence and unconditional independence are recalled. Then his elegant proof of discrete probabilistic representability of a matroid based on its linear representability over a finite field is recalled. It is explained that this result was a basis of his methodology for constructing a probabilistic representation of a given abstract CI structure. His embedding of matroids into (augmented) abstract CI structures is recalled and his contribution to the theory of semigraphoids is mentioned as well. Finally, his results on the characterization of probabilistic CI structures induced by four discrete random variables and by four regular Gaussian random variables are recalled. Partial probabilistic representability by binary random variables is also mentioned

Modeling Complex Systems by Structural Invariants Approach

When modeling complex systems, we usually encounter the following difficulties: partiality, large amount of data, and uncertainty of conclusions. It can be said that none of the known approaches solves these difficulties perfectly, especially in cases where we expect emergences in the complex system. The most common is the physical approach, sometimes reinforced by statistical procedures. The physical approach to modeling leads to a complicated description of phenomena associated with a relatively simple geometry. If we assume emergences in the complex system, the physical approach is not appropriate at all. In this article, we apply the approach of structural invariants, which has the opposite properties: a simple description of phenomena associated with a more complicated geometry (in our case pregeometry). It does not require as much data and the calculations are simple. The price paid for the apparent simplicity is a qualitative interpretation of the results, which carries a special type of uncertainty. Attention is mainly focused (in this article) on the invariant matroid and bases of matroid (M, BM) in combination with the Ramsey graph theory. In addition, this article introduces a calculus that describes the emergent phenomenon using two quantities-the power of the emergent phenomenon and the complexity of the structure that is associated with this phenomenon. The developed method is used in the paper for modeling and detecting emergent situations in cases of water floods, traffic jams, and phase transition in chemistry

Una aproximación a la noción de homotopía entre espacios toplógicos finitos desde las funciones submodulares

En este trabajo se estudian las conexiones entre las FD relaciones con soporte finito y los preórdenes. Se demuestra que existe una correspondencia biunívoca entre las FD relaciones con soporte finito y los espacios pretopológicos finitos, y se aprovecha dicho vínculo para interpretar, en terminos de las funciones submodulares, aquellos conceptos relacionados con la clasificación por tipo de homotopía de los espacios topológicos finitos: función continua, espacio conexo, espacio T0 y beat points. Ademas, se presentan algoritmos que calculan los valores de algunas funciones sub-modulares relacionadas con espacios topológicos finitos y se interpreta el algoritmo de reducción´ de Stong [13] por medio de dichas funciones. Los resultados obtenidos se basan principalmente en [2], [4], [11] y [14].Abstract In this work the connections between the FD relations with finite support and preorders are studied. We show that there is a one to one correspondence between the FD relations with finite support and pretopologicos finite spaces , and that link is used to interpret, in terms of functions submodulares ´ those concepts related to homotopy type classification of finite topological spaces continuous function, connected space, space T0 and beat points. Furthermore , algorithms that compute the values of some functions related submodulares finite topological spaces and Stong reduction algorithm [13] through interprets these functions are presented . The results are mainly based on [2], [4], [11] and [14].Maestrí

Una nueva construcción de los espacios topológicos finitos desde las funciones submodulares

En este trabajo se estudian las conexiones entre las FD-relaciones, las funciones submodulares y los espacios topológicos finitos. Los resultados obtenidos están basados en [7] y [18]. Se interpretan las propiedades de los operadores de clausura de un espacio topológico en términos de las FD-relaciones y las funciones submodulares, como también algunos conceptos y propiedades tales como: puntos de acumulación, punto exterior, conjunto cerrado, conjunto denso y axiomas de separación. Por último, se muestran algunos algoritmos que calculan los valores de las funciones submodulares relacionadas con topologías.Abstract. In this paper we study the connections among FD-relations, submodular functions and finite topological spaces. The results that we show are based in [7] and [18]. We interpret the properties of closure operators of finite topological spaces, in terms of FD-relations and submodular functions, we also characterize concepts and properties such that: accumulation points, exterior point, closed set, dense set and separation axioms. Finally, we show some algorithms that determine the values of the submodular functions related with topologies.Maestrí