7 research outputs found

    Fuzzy algebras of concepts

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    Preconcepts are basic units of knowledge that form the basis of formal concepts in formal concept analysis (FCA). This paper investigates the relations among different kinds of preconcepts, such as protoconcepts, meet and join-semiconcepts and formal concepts. The first contribution of this paper, is to present a fuzzy powerset lattice gradation, that coincides with the preconcept lattice at its 1-cut. The second and more significant contribution, is to introduce a preconcept algebra gradation that yields different algebras for protoconcepts, semiconcepts, and concepts at different cuts. This result reveals new insights into the structure and properties of the different categories of preconcepts.Partial funding for open access charge: Universidad de Málag

    Formal Contexts, Formal Concept Analysis, and Galois Connections

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    Formal concept analysis (FCA) is built on a special type of Galois connections called polarities. We present new results in formal concept analysis and in Galois connections by presenting new Galois connection results and then applying these to formal concept analysis. We also approach FCA from the perspective of collections of formal contexts. Usually, when doing FCA, a formal context is fixed. We are interested in comparing formal contexts and asking what criteria should be used when determining when one formal context is better than another formal context. Interestingly, we address this issue by studying sets of polarities.Comment: In Proceedings Festschrift for Dave Schmidt, arXiv:1309.455

    On the isomorphism problem of concept algebras

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    Weakly dicomplemented lattices are bounded lattices equipped with two unary operations to encode a negation on {\it concepts}. They have been introduced to capture the equational theory of concept algebras \cite{Wi00}. They generalize Boolean algebras. Concept algebras are concept lattices, thus complete lattices, with a weak negation and a weak opposition. A special case of the representation problem for weakly dicomplemented lattices, posed in \cite{Kw04}, is whether complete {\wdl}s are isomorphic to concept algebras. In this contribution we give a negative answer to this question (Theorem \ref{T:main}). We also provide a new proof of a well known result due to M.H. Stone \cite{St36}, saying that {\em each Boolean algebra is a field of sets} (Corollary \ref{C:Stone}). Before these, we prove that the boundedness condition on the initial definition of {\wdl}s (Definition \ref{D:wdl}) is superfluous (Theorem \ref{T:wcl}, see also \cite{Kw09}).Comment: 15 page

    On the isomorphism problem of concept algebras

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    Weakly dicomplemented lattices are bounded lattices equipped with two unary operations to encode a negation on concepts. They have been introduced to capture the equational theory of concept algebras (Wille 2000; Kwuida 2004). They generalize Boolean algebras. Concept algebras are concept lattices, thus complete lattices, with a weak negation and a weak opposition. A special case of the representation problem for weakly dicomplemented lattices, posed in Kwuida (2004), is whether complete weakly dicomplemented lattices are isomorphic to concept algebras. In this contribution we give a negative answer to this question (Theorem4). We also provide a new proof of a well known result due to M.H. Stone(Trans Am Math Soc 40:37-111, 1936), saying that each Boolean algebra is a field of sets (Corollary4). Before these, we prove that the boundedness condition on the initial definition of weakly dicomplemented lattices (Definition1) is superfluous (Theorem1, see also Kwuida (2009)

    Generalized pattern extraction from concept lattices

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    Conceptual Factors and Fuzzy Data

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    With the growing number of large data sets, the necessity of complexity reduction applies today more than ever before. Moreover, some data may also be vague or uncertain. Thus, whenever we have an instrument for data analysis, the questions of how to apply complexity reduction methods and how to treat fuzzy data arise rather naturally. In this thesis, we discuss these issues for the very successful data analysis tool Formal Concept Analysis. In fact, we propose different methods for complexity reduction based on qualitative analyses, and we elaborate on various methods for handling fuzzy data. These two topics split the thesis into two parts. Data reduction is mainly dealt with in the first part of the thesis, whereas we focus on fuzzy data in the second part. Although each chapter may be read almost on its own, each one builds on and uses results from its predecessors. The main crosslink between the chapters is given by the reduction methods and fuzzy data. In particular, we will also discuss complexity reduction methods for fuzzy data, combining the two issues that motivate this thesis.Komplexitätsreduktion ist eines der wichtigsten Verfahren in der Datenanalyse. Mit ständig wachsenden Datensätzen gilt dies heute mehr denn je. In vielen Gebieten stößt man zudem auf vage und ungewisse Daten. Wann immer man ein Instrument zur Datenanalyse hat, stellen sich daher die folgenden zwei Fragen auf eine natürliche Weise: Wie kann man im Rahmen der Analyse die Variablenanzahl verkleinern, und wie kann man Fuzzy-Daten bearbeiten? In dieser Arbeit versuchen wir die eben genannten Fragen für die Formale Begriffsanalyse zu beantworten. Genauer gesagt, erarbeiten wir verschiedene Methoden zur Komplexitätsreduktion qualitativer Daten und entwickeln diverse Verfahren für die Bearbeitung von Fuzzy-Datensätzen. Basierend auf diesen beiden Themen gliedert sich die Arbeit in zwei Teile. Im ersten Teil liegt der Schwerpunkt auf der Komplexitätsreduktion, während sich der zweite Teil der Verarbeitung von Fuzzy-Daten widmet. Die verschiedenen Kapitel sind dabei durch die beiden Themen verbunden. So werden insbesondere auch Methoden für die Komplexitätsreduktion von Fuzzy-Datensätzen entwickelt

    Aspectos topológicos en el análisis de conceptos formales

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    En este trabajo se presenta un desarrollo teórico desde un enfoque topológico al análisis de conceptos formales (FCA en inglés). Con esto se busca combinar el FCA y un estudio topológico, el cuál permita encontrar información subyacente en tablas de datos binarios y difusos; información oculta sin el uso de herramientas topológicas. De esta manera, se obtiene un método para realizar análisis de datos de forma más completa que con el uso exclusivo del FCA. Se estudian los principales resultados en el FCA como área de la matemática aplicada sobre bases de datos, entre ellos el Teorema Básico sobre el retículo concepto, que garantiza que los conceptos formales tienen estructura de retículo completo. La estructura topológica para los contextos formales es propuesta a partir de bases topológicas para el conjunto de objetos y atributos. Para determinar relaciones entre objetos y atributos, se caracterizan algunos operadores topológicos tales como el interior, la clausura y la frontera para los datos. Con la estructura topológica se explora además, la continuidad entre contextos formales. Por otra parte, se describe la representación de un contexto formal como grafo bipartito y se exponen topologías para su retículo asociado. Se presenta una generalización para el Análisis de Conceptos Formales Difusos (FFCA en inglés) mostrando los resultados que se conservan del FCA clásico y se extiende la estructura topológica del caso binario al caso difuso. Finalmente, se muestran algunos ejemplos ilustrativos hallados en el estado del arte como aplicaciones de los resultados, se presentan las conclusiones, entre ellas, el hecho de que conociendo los conceptos formales de un contexto, se puede extraer rápidamente las bases topológicas propuestas para dotar de estructura topológica la tabla, así mismo, se concluye que la generalización para los datos difusos es posible, pero tiene grandes restricciones por falta de software especializado para realizar los cálculos necesarios. Por otra parte, como posibles trabajos futuros se plantea el desarrollo de algoritmos para cálculos en grandes volúmenes de datos difusos, el uso de otras topologías y la exploración de más relaciones entre el FCA, la teoría de retículos, de grafos y la topologíaAbstract : This work presents a theoretical development from a topological approach to formal concept analysis (FCA). This seeks to combine the FCA and a topological study, which enables find information in tables underlying binary and fuzzy data, and hidden information without the use of topological tools. We propose, a method for the analysis of data more accurate in contrast of using only FCA. The basic theorem on concept lattices ensures that formal concepts have complete lattice structure, is discussed alongside the main results of FCA as an area of applied mathematics on databases. The topological structure for formal contexts is proposed from topological basis for the set of objects and attributes. With this in mind to determine relationships between objects and attributes, some topological operators such as interior, closure and boundary for the data are characterized. Also, it is studied the continuity between formal contexts with its topological structure and it is described the representation of formal context as a bipartite graph and the topologies of its associated lattice. We present a generalization for fuzzy formal concept analysis (FFCA) showing that the results of the classical FCA are preserved and it is extended the topological structure from binary case to fuzzy case. Finally, we apply our methodology in examples of the state of the art. The conclusions are presented, including the fact that knowing the formal concepts of a context, you can quickly extract the topological bases proposed to provide topological structure for the table, also concluded that the generalization for fuzzy data is possible, but has great limited by the lack of specialized software to perform the necessary computations. As possible future work we propose to develop algorithms for computations in fuzzy large volumes of data, using other topologies and exploring more relationships between the FCA, Lattice, Graph and Topology theoryMaestrí
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