On the Use of F-transform on the Reduction of Concept Lattices

In this paper, we show that F-transform can be used to re- duce relational databases. Subsequently, we show that the respective concept lattice is reduced significantly as well. Moreover, we present a clarifying example of the procedure.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. The research was supported by the European Regional Development Fund by projects (CZ.1.05/1.1.00/02.0070) and (TIN12-39353- C04-04)

Rough matroids based on coverings

The introduction of covering-based rough sets has made a substantial contribution to the classical rough sets. However, many vital problems in rough sets, including attribution reduction, are NP-hard and therefore the algorithms for solving them are usually greedy. Matroid, as a generalization of linear independence in vector spaces, it has a variety of applications in many fields such as algorithm design and combinatorial optimization. An excellent introduction to the topic of rough matroids is due to Zhu and Wang. On the basis of their work, we study the rough matroids based on coverings in this paper. First, we investigate some properties of the definable sets with respect to a covering. Specifically, it is interesting that the set of all definable sets with respect to a covering, equipped with the binary relation of inclusion $\subseteq$, constructs a lattice. Second, we propose the rough matroids based on coverings, which are a generalization of the rough matroids based on relations. Finally, some properties of rough matroids based on coverings are explored. Moreover, an equivalent formulation of rough matroids based on coverings is presented. These interesting and important results exhibit many potential connections between rough sets and matroids.Comment: 15page

Crossed simplicial groups and structured surfaces

We propose a generalization of the concept of a Ribbon graph suitable to provide combinatorial models for marked surfaces equipped with a G-structure. Our main insight is that the necessary combinatorics is neatly captured in the concept of a crossed simplicial group as introduced, independently, by Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category leads to Ribbon graphs while other crossed simplicial groups naturally yield different notions of structured graphs which model unoriented, N-spin, framed, etc, surfaces. Our main result is that structured graphs provide orbicell decompositions of the respective G-structured moduli spaces. As an application, we show how, building on our theory of 2-Segal spaces, the resulting theory can be used to construct categorified state sum invariants of G-structured surfaces.Comment: 86 pages, v2: revised versio

Rough sets based on Galois connections

Rough set theory is an important tool to extract knowledge from relational databases. The original definitions of approximation operators are based on an indiscernibility relation, which is an equivalence one. Lately. different papers have motivated the possibility of considering arbitrary relations. Nevertheless, when those are taken into account, the original definitions given by Pawlak may lose fundamental properties. This paper proposes a possible solution to the arising problems by presenting an alternative definition of approximation operators based on the closure and interior operators obtained from an isotone Galois connection. We prove that the proposed definition satisfies interesting properties and that it also improves object classification tasks

Geometry on all prime Three Manifolds

The point of this work is to construct geometric structures on the oriented closed prime three-manifolds that don't at present already have them. One knows these compound prime three-manifolds, have canonically up to deformation from the identity, incompressible torus walls whose complementary components are diffeomorphic to "elemental" prime three-manifolds carrying single Thurston geometries. These geometric elementary parts have finite volume or linear volume growth. This metric geometry is generalized here to Lie geometry meaning an open cover by special coordinate charts in a model space whose transition mappings are related by one of several finite dimensional Lie groups acting on the model space. The Lie group is allowed to vary in a constrained manner from region to region in the manifold. Our geometric version of torus wall crossing after fixing finitely many parameters is rigid using covering spaces, pushouts, and sliding flat toroidal cylinders rigidly together. A new concept, Lie generated geometry, describes abstractly what these constructions produce. These Lie generated geometries are determined by special coordinate chart coverings as suggested above, but structurally they consist of sheaves of germs of charts into the upper half space which are related by four Lie groups acting there. The key point beside the Lie group generating system is that each germ of the structure can be analytically continued along any path, like solving a classical ODE in the complex plane. This defines for each Lie geometrized manifold a developing map of its universal cover into upper half space. The Theorem solves a 44 year old question from a 1976 Princeton math department preprint motivated by the Poincar\'{e} Conjecture and finally documented in the 1983 reference of Thurston and the second author

The structure of oppositions in rough set theory and formal concept analysis - Toward a new bridge between the two settings

Rough set theory (RST) and formal concept analysis (FCA) are two formal settings in information management, which have found applications in learning and in data mining. Both rely on a binary relation. FCA starts with a formal context, which is a relation linking a set of objects with their properties. Besides, a rough set is a pair of lower and upper approximations of a set of objects induced by an indistinguishability relation; in the simplest case, this relation expresses that two objects are indistinguishable because their known properties are exactly the same. It has been recently noticed, with different concerns, that any binary relation on a Cartesian product of two possibly equal sets induces a cube of oppositions, which extends the classical Aristotelian square of oppositions structure, and has remarkable properties. Indeed, a relation applied to a given subset gives birth to four subsets, and to their complements, that can be organized into a cube. These four subsets are nothing but the usual image of the subset by the relation, together with similar expressions where the subset and / or the relation are replaced by their complements. The eight subsets corresponding to the vertices of the cube can receive remarkable interpretations, both in the RST and the FCA settings. One facet of the cube corresponds to the core of RST, while basic FCA operators are found on another facet. The proposed approach both provides an extended view of RST and FCA, and suggests a unified view of both of them. © 2014 Springer International Publishing

Representative Set of Objects in Rough Sets Based on Galois Connections

This paper introduces a novel definition, called representative set of objects of a decision class, in the framework of decision systems based on rough sets. The idea behind such a notion is to consider subsets of objects that characterize the different classes given by a decision system. Besides the formal definition of representative set of objects of a decision class, we present different mathematical properties of such sets and a relationship with classification tasks based on rough sets. © 2020, Springer Nature Switzerland AG

The CUBIST Project: Combining and Uniting Business Intelligence with Semantic Technologies

As a preface to this Special 'CUBIST' Edition of the International Journal of Intelligent Information Technologies IJIIT, this article describes the European Framework Seven Combining and Unifying Business Intelligence with Semantic Technologies CUBIST project, which ran from October 2010 to September 2013. The project aimed to combine the best elements of traditional BI with the newer, semantic, technologies of the Sematic Web, in the form of the Resource Description Framework RDF, and Formal Concept Analysis FCA. CUBIST's purpose was to provide end-users with "conceptually relevant and user friendly visual analytics" to allow them to explore their data in new ways, discovering hidden meaning and solving hitherto difficult problems. To this end, three of the partners in CUBIST were use-cases: recruitment consultancy, computational biology and the space industry. Each use-case provided their own requirements and problems that were finally addressed by the prototype CUBIST visual-analytics developed in the project

The Resemblance Structure of Natural Kinds: A Formal Model for Resemblance Nominalism

278 p.The aim of this thesis is to better understand the ways natural kinds are related to each other by species-genus relations and the ways in which the members of the kind are related to each other by resemblance relations, by making use of formal models of kinds. This is done by first analysing a Minimal Conception of Natural Kinds and then reconstructing it from the ontological assumptions of Resemblance Nominalism. The questions addressed are:(1) What is the external structure of kinds' In what ways are kinds related to each other by species-genus relations'(2) What is the internal structure of kinds' In what sense are the instances of a kind similar enough to each other'According to the Minimal Conception of Kinds, kinds have two components, a set of members of the kind (the extension) and a set of natural attributes common to these objects (the intension). Several interesting features of this conception are discussed by making use of the mathematical theory of concept lattices. First, such structures provide a model for contemporary formulations of syllogistic logic. Second, kinds are ordered forming a complete lattice that follows Kant's law of the duality between extension and intension, according to which the extension of a kind is inversely related to its intension. Finally, kinds are shown to have Aristotelian definitions in terms of genera and specific differences. Overall this results in a description of the specificity relations of kinds as an algebraic calculus.According to Resemblance Nominalism, attributes or properties are classes of similar objects. Such an approach faces Goodman's companionship and imperfect community problems. In order to deal with these, a specific nominalism, namely Aristocratic Resemblance Nominalism, is chosen. According to it, attributes are classes of objects resembling a given paradigm. A model for it is introduced by making use of the mathematical theory of similarity structures and of some results on the topic of quasianalysis. Two other models (the polar model and an order-theoretic model) are considered and shown to be equivalent to the previous one.The main result is that the class of lattices of kinds that a nominalist can recover uniquely by starting from these assumptions is that of complete coatomistic lattices. Several other related results are obtained, including a generalization of the similarity model that allows for paradigms with several properties and properties with several paradigms. The conclusion is that, under nominalist assumptions, the internal structure of kinds is fixed by paradigmatic objects and the external structure of kinds is that of a coatomistic lattice that satisfies the Minimal Conception of Kinds