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1141:. The lines allow to read off the subconcept-superconcept hierarchy. Each object and attribute name is used as a label exactly once in the diagram, with objects below and attributes above concept circles. This is done in a way that an attribute can be reached from an object via an ascending path if and only if the object has the attribute.
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A bicluster of similar values in a numerical object-attribute data-table is usually defined as a pair consisting of an inclusion-maximal set of objects and an inclusion-maximal set of attributes having similar values for the objects. Such a pair can be represented as an inclusion-maximal rectangle in
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TCA generalizes the above mentioned case by considering temporal data bases with an arbitrary key. That leads to the notion of distributed objects which are at any given time at possibly many places, as for example, a high pressure zone on a weather map. The notions of 'temporal objects', 'time' and
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In the simplest case TCA considers objects that change in time like a particle in physics, which, at each time, is at exactly one place. That happens in those temporal data where the attributes 'temporal object' and 'time' together form a key of the data base. Then the state (of a temporal object at
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Denis
Ponomaryov; Nadezhda Omelianchuk; Victoria Mironova; Eugene Zalevsky; Nikolay Podkolodny; Eric Mjolsness; Nikolay Kolchanov (2011), Karl Erich Wolff; Dmitry E. Palchunov; Nikolay G. Zagoruiko; Urs Andelfinger (eds.), "From Published Expression and Phenotype Data to Structured Knowledge: The
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triclustering include OA-biclustering and OAC-triclustering (here O stands for object, A for attribute, C for condition); to generate patterns these methods use prime operators only once being applied to a single entity (e.g. object) or a pair of entities (e.g. attribute-condition), respectively.
402:
This aim traces back to the educationalist
Hartmut von Hentig, who in 1972 pleaded for restructuring sciences in view of better teaching and in order to make sciences mutually available and more generally (i.e. also without specialized knowledge) critiqueable. Hence, by its origins formal concept
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Temporal concept analysis (TCA) is an extension of Formal
Concept Analysis (FCA) aiming at a conceptual description of temporal phenomena. It provides animations in concept lattices obtained from data about changing objects. It offers a general way of understanding change of concrete or abstract
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biological processes, the latter should possibly overlap, since a gene may be involved in several processes. The same remark applies for recommender systems where one is interested in local patterns characterizing groups of users that strongly share almost the same tastes for a subset of items.
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There are a number of simple and fast algorithms for generating formal concepts and for constructing and navigating concept lattices. For a survey, see
Kuznetsov and Obiedkov or the book by Ganter and Obiedkov, where also some pseudo-code can be found. Since the number of formal concepts may be
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Since only non-incident object-attribute pairs can be related, these relations can conveniently be recorded in the table representing a formal context. Many lattice properties can be read off from the arrow relations, including distributivity and several of its generalizations. They also reveal
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in a hierarchy called more formally the context's "concept lattice". The concept lattice can be graphically visualized as a "line diagram", which then may be helpful for understanding the data. Often however these lattices get too large for visualization. Then the mathematical theory of formal
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is to group together some objects having similar values of some attributes. For example, in gene expression data, it is known that genes (objects) may share a common behavior for a subset of biological situations (attributes) only: one should accordingly produce local patterns to characterize
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The original formal context can be reconstructed from the labelled diagram, as well as the formal concepts. The extent of a concept consists of those objects from which an ascending path leads to the circle representing the concept. The intent consists of those attributes to which there is an
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as part of general lattice theory. Other previous approaches to the same idea arose from various French research groups, but the
Darmstadt group normalised the field and systematically worked out both its mathematical theory and its philosophical foundations. The latter refer in particular to
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Of course, formal concepts can be considered as "rigid" biclusters where all objects have all attributes and vice versa. Hence, it is not surprising that some bicluster definitions coming from practice are just definitions of a formal concept. Relaxed FCA-based versions of biclustering and
386:
In his article "Restructuring
Lattice Theory" (1982), initiating formal concept analysis as a mathematical discipline, Wille starts from a discontent with the current lattice theory and pure mathematics in general: The production of theoretical results—often achieved by "elaborate mental
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a time in a view) is formalized as a certain object concept of the formal context describing the chosen view. In this simple case, a typical visualization of a temporal system is a line diagram of the concept lattice of the view into which trajectories of temporal objects are embedded.
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The formal concept analysis can be used as a qualitative method for data analysis. Since the early beginnings of FCA in the early 1980s, the FCA research group at TU Darmstadt has gained experience from more than 200 projects using the FCA (as of 2005). Including the fields of:
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the numerical table, modulo rows and columns permutations. In it was shown that biclusters of similar values correspond to triconcepts of a triadic context where the third dimension is given by a scale that represents numerical attribute values by binary attributes.
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Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice
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Real-world data is often given in the form of an object-attribute table, where the attributes have "values". Formal concept analysis handles such data by transforming them into the basic type of a ("one-valued") formal context. The method is called
465:
The aim and meaning of Formal
Concept Analysis as mathematical theory of concepts and concept hierarchies is to support the rational communication of humans by mathematically developing appropriate conceptual structures which can be logically
4451:
Beate Kohler-Koch; Frank Vogt; Gerhard Stumme; Rudolf Wille (2000), "Normen- und
Regelgeleitete internationale Kooperationen: Quoted in: Peter Becker et al. The ToscanaJ Suite for Implementing Conceptual Information Systems",
3932:
Dominik Endres; Ruth Adam; Martin A. Giese; Uta
Noppeney (2012), Florent Domenach; Dmitry I. Ignatov; Jonas Poelmans (eds.), "Understanding the Semantic Structure of Human fMRI Brain Recordings with Formal Concept Analysis",
80:, is a useful starting point for translations, but translators must revise errors as necessary and confirm that the translation is accurate, rather than simply copy-pasting machine-translated text into the English Knowledge.
4249:
Jens Illig; Andreas Hotho; Robert Jäschke; Gerd Stumme (2011), Karl Erich Wolff; Dmitry E. Palchunov; Nikolay G. Zagoruiko; Urs
Andelfinger (eds.), "A Comparison of Content-Based Tag Recommendations in Folksonomy Systems",
1631:") is not concept forming in the same way as defined above. For this reason, the values 1 and 0 or TRUE and FALSE are usually avoided when representing formal contexts, and a symbol like × is used to express incidence.
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Boumedjout Lahcen and Leonard Kwuida. "Lattice Miner: A Tool for Concept Lattice Construction and Exploration". In: Supplementary Proceeding of International Conference on Formal concept analysis (ICFCA'10),
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exponential in the size of the formal context, the complexity of the algorithms usually is given with respect to the output size. Concept lattices with a few million elements can be handled without problems.
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assumed that negated attributes are available for concept formation. But pairs of attributes which are negations of each other often naturally occur, for example in contexts derived from conceptual scaling.
3315:
4394:
Nada Mimouni; Adeline Nazarenko; Sylvie Salotti (2015), Jaume Baixeries; Christian Sacarea; Manuel Ojeda-Aciego (eds.), "A Conceptual Approach for Relational IR: Application to Legal Collections",
4362:
Dieter Eschenfelder; Wolfgang Kollewe; Martin Skorsky; Rudolf Wille (2000), Gerd Stumme; Rudolf Wille (eds.), "Ein Erkundungssystem zum Baurecht: Methoden der Entwicklung Eines TOSCANA-Systems",
2452:'place' are represented as formal concepts in scales. A state is formalized as a set of object concepts. That leads to a conceptual interpretation of the ideas of particles and waves in physics.
90:
418:
had been reduced to its extent. Now again, the philosophy of concepts should become less abstract by considering the intent. Hence, formal concept analysis is oriented towards the categories
2422:. Weak opposition is a dual weak complementation. A (bounded) lattice such as a concept algebra, which is equipped with a weak complementation and a dual weak complementation, is called a
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concept analysis may be helpful, e.g., for decomposing the lattice into smaller pieces without information loss, or for embedding it into another structure which is easier to interpret.
66:
4075:
Aurélie Bertaux; Florence Le Ber; Agnès Braud; Michèle Trémolières (2009), Sébastien Ferré; Sebastian Rudolph (eds.), "Identifying Ecological Traits: A Concrete FCA-Based Approach",
2590:-dimensional concepts. This reduction allows one to use standard definitions and algorithms from multidimensional concept analysis for computing multidimensional clusters.
3280:
Wolff, Karl Erich (2004), "'Particles' and 'Waves' as Understood by Temporal Concept Analysis.", in Wolff, Karl Erich; Pfeiffer, Heather D.; Delugach, Harry S. (eds.),
4158:
2168:
3259:
Wolff, Karl Erich (2019), "Temporal Concept Analysis with SIENA", in Cristea, Diana; Le Ber, Florence; Missaoui, Rokia; Kwuida, Léonard; Sertkaya, Bariş (eds.),
4818:
1665:
by the inclusion of extents, or, equivalently, by the dual inclusion of intents. An order ≤ on the concepts is defined as follows: for any two concepts (
3282:
Conceptual Structures at Work. 12th International Conference on Conceptual Structures, ICCS 2004. Huntsville, AL, USA, July 2004, LNAI 3127. Proceedings
2515:
in that graph. The mathematical and algorithmic results of formal concept analysis may thus be used for the theory of maximal bicliques. The notion of
4480:
2130:
replaces the binary incidence relation between objects and attributes by a ternary relation between objects, attributes, and conditions. An incidence
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Do not translate text that appears unreliable or low-quality. If possible, verify the text with references provided in the foreign-language article.
4823:
3657:
Kaytoue, M.; Kuznetsov, S.; Macko, J.; Wagner Meira Jr., Napoli A. (2011). "Mining Biclusters of Similar Values with Triadic Concept Analysis".
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183:. Each concept in the hierarchy represents the objects sharing some set of properties; and each sub-concept in the hierarchy represents a
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Wolff, Karl Erich (2010), "Temporal Relational Semantic Systems", in Croitoru, Madalina; Ferré, Sébastien; Lukose, Dickson (eds.),
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Many FCA software applications are available today. The main purpose of these tools varies from formal context creation to formal
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submatrix (not necessarily contiguous) all of whose elements equal 1. It is however misleading to consider a formal context as
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1980:
Formal concept analysis has elaborate mathematical foundations, making the field versatile. As a basic example we mention the
2640:
1807:. Conversely, it can be shown that every complete lattice is the concept lattice of some formal context (up to isomorphism).
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formed by them is much less developed than that of concept lattices, and seems to be difficult. Voutsadakis has studied the
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ascending path from that concept circle (in the diagram). In this diagram the concept immediately to the left of the label
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gymnastics"—were impressive, but the connections between neighboring domains, even parts of a theory were getting weaker.
4235:
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Prelic, S.; Bleuler, P.; Zimmermann, A.; Wille, P.; Buhlmann, W.; Gruissem, L.; Hennig, L.; Thiele, E.; Zitzler (2006).
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250:. One such possibility of very general nature is that data tables can be transformed into algebraic structures called
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Ordered Sets. Proceedings of the NATO Advanced Study Institute held at Banff, Canada, August 28 to September 12, 1981
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Content in this edit is translated from the existing German Knowledge article at ]; see its history for attribution.
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Wort und Feld: wortsemantische Fragestellungen mit besonderer Berücksichtigung des Wortfeldbegriffes: Dissertation
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Formal Concept Analysis: 7th International Conference, ICFCA 2009 Darmstadt, Germany, May 21–24, 2009 Proceedings
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objects in continuous, discrete or hybrid space and time. TCA applies conceptual scaling to temporal data bases.
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1964:, an irredundant set of implications from which all valid implications can be derived by the natural inference (
116:
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of the objects (as well as a superset of the properties) in the concepts above it. The term was introduced by
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Richard Cole; Gerd Stumme (2000), Bernhard Ganter; Guy W. Mineau (eds.), "CEM – A Conceptual Email Manager",
3565:"Toward the next generation of recommender systems: a survey of the state-of-the-art and possible extensions"
4808:
2751:
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Kuznetsov, S.; Obiedkov, S. (2002). "Comparing Performance of Algorithms for Generating Concept Lattices".
2908:
254:, and that these can be utilized for data visualization and interpretation. A data table that represents a
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2241:) is in general not a concept. However, since the concept lattice is complete one can consider the join (
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between sets of objects and of attributes. This is why in French a concept lattice is sometimes called a
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The original motivation of formal concept analysis was the search for real-world meaning of mathematical
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structural information and can be used for determining, e.g., the congruence relations of the lattice.
48:
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in which the rows correspond to the objects, the columns correspond to the attributes, and each entry
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Wille, Rudolf. "Formal Concept Analysis as Mathematical Theory of Concepts and Concept Hierarchies".
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and generating the concepts lattice of a given formal context and the corresponding implications and
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The above line diagram consists of circles, connecting line segments, and labels. Circles represent
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Susanne Motameny; Beatrix Versmold; Rita Schmutzler (2008), Raoul Medina; Sergei Obiedkov (eds.),
3770:"Assessment of discretization techniques for relevant pattern discovery from gene expression data"
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were systematically categorized by their attributes. For the purpose here it has been simplified.
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With these derivation operators, Wille gave an elegant definition of a formal concept: a pair (
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3900:"Formal Concept Analysis for the Identification of Combinatorial Biomarkers in Breast Cancer"
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3003:"Familles minimales d'implications informatives résultant d'un tableau de données binaires"
2361:. The concept lattice equipped with the two additional operations Δ and 𝛁 is known as the
1788:, or meet. Its extent consists of those objects that are common to all extents of the set.
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1984:, which are simple and easy to compute, but very useful. They are defined as follows: For
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Proceedings of the 4th ACM SIGKDD Workshop on Data Mining in Bioinformatics (BIOKDD 2004)
3607:"A Systematic Comparison and Evaluation of Biclustering Methods for Gene Expression Data"
441:
objects. In his late philosophy, Peirce assumed that logical thinking aims at perceiving
433:
Formal concept analysis aims at the clarity of concepts according to Charles S. Peirce's
3519:"Discovery of optimal factors in binary data via a novel method of matrix decomposition"
3284:, Lecture Notes in Artificial Intelligence, vol. 3127, Springer, pp. 126–141,
3230:, Lecture Notes in Artificial Intelligence, vol. 6208, Springer, pp. 165–180,
351:
The theory in its present form goes back to the early 1980s and a research group led by
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2465:
2213:: Modelling negation of formal concepts is somewhat problematic because the complement
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The data in the example is taken from a semantic field study, where different kinds of
363:. Its basic mathematical definitions, however, were already introduced in the 1930s by
356:
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3754:
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3201:
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2527:(of the concept lattice) and has applications e.g. for Boolean matrix factorization.
2482:
1785:
1662:
494:
458:
438:
4203:, vol. 40, no. 1, Medford, NJ 09855: Information Today, pp. 521–543,
4116:
4102:
3870:
3713:
3622:
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3336:
3166:
Wille, Rudolf (2000), "Boolean Concept Logic", in Ganter, B.; Mineau, G. W. (eds.),
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4349:
3968:
3729:"Triadic Formal Concept Analysis and triclustering: searching for optimal patterns"
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223:
196:
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4522:"International Conferences On Conceptual Structures – Conferences and Workshops"
4024:"Mining gene expression data with pattern structures in formal concept analysis"
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17:
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Magier oder Magister? Über die Einheit der Wissenschaft im Verständigungsprozeß
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4326:, LNAI, vol. 1867, Berlin Heidelberg: Springer-Verlag, pp. 438–452,
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4084:
4079:, LNAI, vol. 5548, Berlin Heidelberg: Springer-Verlag, pp. 224–236,
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3493:"GALACTIC GAlois LAttices, Concept Theory, Implicational system and Closures"
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One can find a non exhaustive list of FCA tools in the FCA software website:
3227:
Conceptual Structures: From Information to Intelligence. ICCS 2010. LNAI 6208
3168:
ICCS 2000 Conceptual Structures: Logical, Linguistic and Computational Issues
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2205:: Extensive work has been done on a fuzzy version of formal concept analysis.
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of attributes and expresses that every object possessing each attribute from
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Davey, B.A.; Priestley, H. A. (2002), "Chapter 3. Formal Concept Analysis",
4521:
4208:
4193:
4022:
Mehdi Kaytoue; Sergei Kuznetsov; Amedeo Napoli; Sébastien Duplessis (2011),
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2807:"Restructuring lattice theory: An approach based on hierarchies of concepts"
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For computing purposes, a formal context may be naturally represented as a
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It corrects the starting point of lattice theory during the development of
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can be defined in analogy to the formal concepts above, the theory of the
1960:. For each finite formal context, the set of all valid implications has a
4398:, LNAI, vol. 9113, Heidelberg New York: Springer, pp. 303–318,
4254:, LNCS, vol. 6581, Heidelberg New York: Springer, pp. 136–149,
3986:, LNCS, vol. 6581, Heidelberg New York: Springer, pp. 101–120,
3583:
2628:
2620:
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analysis aims at interdisciplinarity and democratic control of research.
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3194:
Dicomplemented Lattices. A contextual generalization of Boolean algebras
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Formal concept analysis finds practical application in fields including
3906:, LNAI, vol. 4933, Berlin Heidelberg: Springer, pp. 229–240,
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3068:
2632:
1611:." In this matrix representation, each formal concept corresponds to a
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Applying either derivation operator and then the other constitutes two
894:
442:
235:
119:
to the source of your translation. A model attribution edit summary is
4553:, Lecture Notes in Artificial Intelligence, vol. 3626, Springer,
3937:, LNCS, vol. 7278, Berlin Heidelberg: Springer, pp. 96–111,
2606:
is union-closed. The complements of knowledge states therefore form a
2472:. Most of these tools are academic open-source applications, such as:
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4331:
1001:
928:
731:
624:
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between objects and attributes, tabulating pairs of the form "object
184:
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Formal Concept Analysis. ICFCA International Conference Proceedings
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Conceptual Structures: Logical, Linguistic, and Computational Issues
3808:
International Workshop on Knowledge Discovery in Inductive Databases
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Wille, R. (1995). "The basic theorem of triadic concept analysis"".
343:
The formal concepts of any formal context can—as explained below—be
3982:
Arabidopsis Gene Net Supplementary Database and Its Applications",
3401:
2813:. Nato Science Series C. Vol. 83. Springer. pp. 445–470.
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Priss, Uta, "Linguistic Applications of Formal Concept Analysis",
4157:
Gerd Stumme; Alexander Maedche (2001), Universität Leipzig (ed.),
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2903:
860:
566:
480:
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3262:
Supplementary Proceedings of ICFCA 2019, Conference and Workshops
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Besson, J.; Robardet, C.; Raedt, L.D.; Boulicaut, J.-F. (2007).
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2012 IEEE 12th International Conference on Data Mining Workshops
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697:
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77:
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Journal of Experimental and Theoretical Artificial Intelligence
453:. Mathematics is an abstraction of logic, develops patterns of
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Pensa, R.G.; Leschi, C.; Besson, J.; Boulicaut, J.-F. (2004).
3727:
Ignatov, D.; Gnatyshak, D.; Kuznetsov, S.; Mirkin, B. (2015).
2975:, Linguistische Arbeiten 103 (in German), Tübingen: Niemeyer,
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and may be represented as the extents of some formal context.
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A bicluster in a binary object-attribute data-table is a pair
1831:, the extent of which is just the complement of the extent of
962:
266:", is considered as a basic data type. It is referred to as a
29:
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Cerf, L.; Besson, J.; Robardet, C.; Boulicaut, J.-F. (2009).
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These meet and join operations satisfy the axioms defining a
4547:
Ganter, Bernhard; Stumme, Gerd; Wille, Rudolf, eds. (2005),
4366:(in German), Berlin Heidelberg: Springer, pp. 254–272,
4123:, vol. 23, no. 6, New York: ACM, pp. 99–110,
4033:, vol. 181, no. 10, Elsevier, pp. 1989–2001,
2535:
Given an object-attribute numerical data-table, the goal of
2519:(of the complemented bipartite graph) translates to that of
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Begriffliche Wissensverarbeitung – Methoden und Anwendungen
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Begriffliche Wissenverarbeitung – Methoden und Anwendungen
2835:
Ferré, Sébastien; Rudolph, Sebastian, eds. (12 May 2009).
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For possible negations of formal concepts see the section
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to this template: there are already 1,886 articles in the
2426:. Weakly dicomplemented lattices generalize distributive
4117:"Reengineering class hierarchies using concept analysis"
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it is assumed that in any knowledge space the family of
1972:, a knowledge acquisition method based on implications.
3684:"Concept-Based Biclustering for Internet Advertisement"
3775:. In Zaki, M.J.; Morishita, S.; Rigoutsos, I. (eds.).
2301:, respectively. This can be expressed in terms of the
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by unfolding observable, elementary properties of the
4579:, translated by C. Franzke, Springer-Verlag, Berlin,
4550:
Formal Concept Analysis: Foundations and Applications
4477:"International Conference on Formal Concept Analysis"
2672:
Formal Concept Analysis. Foundations and Applications
2511:. The formal concepts then correspond to the maximal
2138:
306:
consists of all objects that share the attributes in
2674:, conference papers at regular conferences such as:
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328:
In this way, formal concept analysis formalizes the
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consists of all attributes shared by the objects in
73:
4201:
Annual Review of Information Science and Technology
3572:
IEEE Transactions on Knowledge and Data Engineering
3352:"Formal Concept Analysis Software and Applications"
2883:
2881:
2676:
International Conference on Formal Concept Analysis
2507:A formal context can naturally be interpreted as a
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attributes shared by all objects from A, and dually
69:
a machine-translated version of the German article.
3810:. LNCS. Vol. 4747. Springer. pp. 11–23.
2891:Attribute Exploration of Gene Regulatory Processes
2547:consisting of an inclusion-maximal set of objects
2162:
1784:In this order, every set of formal concepts has a
191:in 1981, and builds on the mathematical theory of
4577:Formal Concept Analysis: Mathematical Foundations
4436:
4290:Claudio Carpineto; Giovanni Romano, eds. (2004),
3886:
3851:ACM Transactions on Knowledge Discovery from Data
3682:Ignatov, D.; Poelmans, J.; Kuznetsov, S. (2012).
2956:
2931:Formal Concept Analysis: Mathematical Foundations
2684:International Conference on Conceptual Structures
485:Line diagram corresponding to the formal context
4620:SoftStat'93: Advances in Statistical Software 4.
4194:"Formal Concept Analysis in Information Science"
3025:
3023:
2614:Hands-on experience with formal concept analysis
516:Example for a formal context: "bodies of water"
2800:
2798:
463:
389:
4595:Concept Data Analysis: Theory and Applications
4501:"CLA: Concept Lattices and Their Applications"
4293:Concept Data Analysis: Theory and Applications
3088:
3086:
115:accompanying your translation by providing an
60:Click for important translation instructions.
47:expand this article with text translated from
4593:Carpineto, Claudio; Romano, Giovanni (2004),
3837:
3835:
3652:
3650:
1190:puddle, maar, lake, pond, tarn, pool, lagoon,
457:realities and therefore may support rational
8:
4159:"FCA-Merge: Bottom-up merging of ontologies"
2531:Biclustering and multidimensional clustering
4613:"A first course in Formal Concept Analysis"
4168:, Leipzig, pp. 225–230, archived from
3517:Belohlavek, Radim; Vychodil, Vilem (2010).
3030:Ganter, Bernhard; Obiedkov, Sergei (2016).
2582:-dimensional clusters of similar values in
2551:and an inclusion-maximal set of attributes
2670:Many more examples are e.g. described in:
2365:of a context. Concept algebras generalize
4622:, Gustav Fischer Verlag, pp. 429–438
4456:(in German), Springer, pp. 325–340,
4038:
3882:
3880:
3744:
3666:
3630:
3170:, LNAI 1867, Springer, pp. 317–331,
3136:"Formal Concept Analysis and Fuzzy Logic"
2902:
2137:
1619:, because the negated incidence ("object
127:{{Translated|de|Formale Begriffsanalyse}}
4575:Ganter, Bernhard; Wille, Rudolf (1998),
2929:Ganter, Bernhard; Wille, Rudolf (1999).
2333:, and weak opposition can be written as
514:
3526:Journal of Computer and System Sciences
2794:
2680:Concept Lattices and their Applications
410:in the 19th century. Then—and later in
382:Motivation and philosophical background
4227:
3844:"Closed patterns meet n-ary relations"
3806:. In Dzeroski, S.; Struyf, J. (eds.).
2897:(PhD). University of Jena. p. 9.
1537:) is a formal concept precisely when:
1368:objects sharing all attributes from B.
1144:In the diagram shown, e.g. the object
94:
2586:-dimensional data are represented by
2369:. Weak negation on a concept lattice
1792:, every set of formal concepts has a
1158:temporary, running, natural, maritime
520:
461:. On this background, Wille defines:
7:
3001:Guigues, J.L.; Duquenne, V. (1986).
2924:
2922:
2920:
2918:
2293:. These two operations are known as
1859:
1529:Equivalently and more intuitively, (
1100:
1068:
1034:
1000:
961:
927:
893:
859:
830:
798:
764:
730:
696:
662:
623:
594:
565:
4819:Formal semantics (natural language)
4115:Gregor Snelting; Frank Tip (1998),
3467:"FcaBedrock Formal Context Creator"
2909:urn:nbn:de:gbv:27-20120103-132627-0
2747:Formal semantics (natural language)
1635:Concept lattice of a formal context
512:. Formal definitions follow below.
4659:A Formal Concept Analysis Homepage
4630:Introduction to Lattices and Order
3801:"Mining bi-sets in numerical data"
3010:Mathématiques et Sciences Humaines
2555:such that almost all objects from
2305:. Weak negation can be written as
1579:that does not have that attribute.
1445:The derivation operators define a
25:
2864:. Klett (1972), Suhrkamp (1974).
2574:This fact can be generalized to
2559:have almost all attributes from
1164:has exactly the characteristics
1122:
1113:
1106:
1090:
1081:
1074:
1058:
1051:
1044:
1024:
1015:
1008:
992:
985:
978:
971:
951:
942:
935:
917:
908:
901:
883:
874:
867:
849:
842:
820:
813:
806:
788:
781:
774:
752:
745:
736:
720:
713:
706:
686:
679:
672:
654:
647:
640:
633:
613:
602:
584:
573:
430:and classical conceptual logic.
361:Technische Universität Darmstadt
286:is a set of objects (called the
34:
4437:Ganter, Stumme & Wille 2005
3887:Ganter, Stumme & Wille 2005
2957:Ganter, Stumme & Wille 2005
2777:Statistical relational learning
2523:(of the formal context) and of
1281:of attributes, one defines two
4824:Ontology (information science)
2157:
2139:
1564:that the object does not have,
1261:which attributes. For subsets
125:You may also add the template
27:Method of deriving an ontology
1:
4524:. New Mexico State University
4121:Proceeding. SIGSOFT '98/FSE-6
3623:10.1093/bioinformatics/btl060
2971:Lutzeier, Peter Rolf (1981),
2860:Hentig, von, Hartmut (1972).
2498:Related analytical techniques
2424:weakly dicomplemented lattice
2279:) of all concepts satisfying
1895:also has each attribute from
1823:The negation of an attribute
1811:Attribute values and negation
1560:, there is some attribute in
1257:that expresses which objects
1204:A formal context is a triple
4404:10.1007/978-3-319-19545-2_19
4372:10.1007/978-3-642-57217-3_12
1253:is a binary relation called
1200:Formal contexts and concepts
500:The data table represents a
359:and Peter Burmeister at the
294:is a set of attributes (the
3816:10.1007/978-3-540-75549-4_2
3290:10.1007/978-3-540-27769-9_8
3095:"Polyadic Concept Analysis"
2888:Wollbold, Johannes (2011).
2782:Schema (genetic algorithms)
2767:Inductive logic programming
2392:which satisfies the axioms
1123:
1114:
1107:
1091:
1082:
1075:
1059:
1052:
1045:
1025:
1016:
1009:
993:
986:
979:
972:
952:
943:
936:
918:
909:
902:
884:
875:
868:
850:
843:
821:
814:
807:
789:
782:
775:
753:
746:
737:
721:
714:
707:
687:
680:
673:
655:
648:
641:
634:
614:
603:
585:
574:
97:will aid in categorization.
4845:
4635:Cambridge University Press
4611:Wolff, Karl Erich (1994),
3538:10.1016/j.jcss.2009.05.002
3268:, Springer, pp. 94–99
2428:orthocomplemented lattices
1786:greatest common subconcept
1575:, there is some object in
1408:⊆ G (extent closure), and
489:shown in the example table
72:Machine translation, like
4786:10.1007/978-3-030-77867-5
4776:10.1007/978-3-030-21462-3
4766:10.1007/978-3-319-59271-8
4756:10.1007/978-3-319-19545-2
4746:10.1007/978-3-319-07248-7
4736:10.1007/978-3-642-38317-5
4726:10.1007/978-3-642-29892-9
4716:10.1007/978-3-642-20514-9
4706:10.1007/978-3-642-11928-6
4696:10.1007/978-3-642-01815-2
4686:10.1007/978-3-540-78137-0
4676:10.1007/978-3-540-70901-5
4559:10.1007/978-3-540-31881-1
4296:, John Wiley & Sons,
4260:10.1007/978-3-642-22140-8
4085:10.1007/978-3-642-01815-2
4049:10.1016/j.ins.2010.07.007
3992:10.1007/978-3-642-22140-8
3943:10.1007/978-3-642-29892-9
3746:10.1007/s10994-015-5487-y
3329:10.1080/09528130210164170
3236:10.1007/978-3-642-14197-3
2839:. Springer. p. 314.
2819:10.1007/978-94-009-7798-3
2697:Association rule learning
2578:-dimensional case, where
2439:Temporal concept analysis
1794:least common superconcept
1156:, but not the attributes
523:
445:, by the triade concept,
274:is defined to be a pair (
203:and others in the 1930s.
49:the corresponding article
4618:, in F. Faulbaum (ed.),
4234:: CS1 maint: location (
3406:Toscanaj.sourceforge.net
3191:Kwuida, Léonard (2004),
3093:Voutsadakis, G. (2002).
2981:10.1515/9783111678726.fm
2809:. In Rival, Ivan (ed.).
2128:Triadic concept analysis
2122:Extensions of the theory
1935:), then the implication
1915:is a formal context and
4209:10.1002/aris.1440400120
3863:10.1145/1497577.1497580
3114:10.1023/A:1021252203599
2752:General Concept Lattice
2732:Correspondence analysis
2163:{\displaystyle (g,m,c)}
1923:are subsets of the set
1603:equals to 1 if "object
1567:for every attribute in
1545:has every attribute in
1271:of objects and subsets
157:formal concept analysis
136:For more guidance, see
3696:10.1109/ICDMW.2012.100
3381:Conexp.sourceforge.net
3377:"The Concept Explorer"
3032:Conceptual Exploration
2805:Wille, Rudolf (1982).
2271:; or dually the meet (
2203:Fuzzy concept analysis
2164:
1441:⊆ M (intent closure).
490:
474:
400:
256:heterogeneous relation
199:that was developed by
4129:10.1145/291252.288273
2717:Conceptual clustering
2707:Commonsense reasoning
2653:office administration
2233:of a formal concept (
2165:
1970:attribute exploration
1927:of attributes (i.e.,
508:next to it shows its
484:
175:from a collection of
138:Knowledge:Translation
109:copyright attribution
4031:Information Sciences
3690:. pp. 123–130.
3584:10.1109/TKDE.2005.99
2637:software engineering
2456:Algorithms and tools
2375:weak complementation
2303:derivation operators
2182:under the condition
2172:then expresses that
2136:
1552:for every object in
1283:derivation operators
414:—a concept as unary
270:. In this theory, a
242:Overview and history
228:software development
220:knowledge management
4252:Kont 2007, KPP 2007
4192:Priss, Uta (2006),
3984:Kont 2007, KPP 2007
3497:galactic.univ-lr.fr
3402:"ToscanaJ: Welcome"
2712:Conceptual analysis
2517:bipartite dimension
2249:) of all concepts (
1968:). This is used in
1850:. It is in general
1663:(partially) ordered
1166:temporary, stagnant
1148:has the attributes
517:
153:information science
4829:Semantic relations
4440:, pp. 149–160
3632:20.500.11850/23740
3437:"The Coron System"
3069:10.1007/BF01108624
2178:has the attribute
2160:
1818:conceptual scaling
1453:(Galois lattice).
1451:treillis de Galois
515:
491:
372:, but also to the
117:interlanguage link
4644:978-0-521-78451-1
4604:978-0-470-85055-8
4463:978-3-540-66391-1
4413:978-3-319-19544-5
4269:978-3-642-22139-2
4094:978-3-642-01814-5
4001:978-3-642-22139-2
3952:978-3-642-29891-2
3913:978-3-540-78136-3
3825:978-3-540-75549-4
3705:978-1-4673-5164-5
3559:Adomavicius, C.;
3447:on 16 August 2022
3299:978-3-540-22392-4
3245:978-3-642-14196-6
3211:978-3-8322-3350-1
3177:978-3-540-67859-5
3041:978-3-662-49290-1
2828:978-94-009-7800-3
2737:Description logic
2665:political science
2598:In the theory of
2521:Ferrers dimension
2470:association rules
1883:relates two sets
1827:is an attribute ¬
1739:. Equivalently, (
1725:) precisely when
1447:Galois connection
1374:closure operators
1364:, i.e., a set of
1322:, i.e., a set of
1132:
1131:
370:Charles S. Peirce
252:complete lattices
169:concept hierarchy
149:
148:
61:
57:
16:(Redirected from
4836:
4804:Machine learning
4647:
4623:
4617:
4607:
4589:
4571:
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4530:
4529:
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4391:
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4359:
4353:
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4332:10.1007/10722280
4319:
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3829:
3805:
3796:
3790:
3789:
3787:
3786:
3781:. pp. 24–30
3774:
3765:
3759:
3758:
3748:
3739:(1–3): 271–302.
3724:
3718:
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3679:
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3452:
3443:. Archived from
3433:
3427:
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3417:
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3414:
3412:
3398:
3392:
3391:
3389:
3387:
3373:
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3366:
3364:
3363:
3354:. Archived from
3347:
3341:
3340:
3323:(2–3): 189–216.
3310:
3304:
3302:
3277:
3271:
3269:
3267:
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3147:
3141:. Archived from
3140:
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3052:
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2885:
2876:
2875:
2857:
2851:
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2832:
2802:
2727:Concept learning
2722:Conceptual space
2702:Cluster analysis
2649:library sciences
2604:knowledge states
2600:knowledge spaces
2594:Knowledge spaces
2563:and vice versa.
2432:Boolean algebras
2421:
2391:
2360:
2332:
2292:
2270:
2232:
2212:
2210:Concept algebras
2189:triadic concepts
2185:
2181:
2177:
2171:
2169:
2167:
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2161:
2113:
2102:
2098:
2091:
2084:
2055:
2044:
2040:
2033:
2003:
1993:
1958:
1951:
1914:
1860:concept algebras
1848:
1843:= G \
1842:
1805:complete lattice
1541:every object in
1519:
1508:
1483:
1436:
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1425:
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588:
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518:
472:
398:
375:Port-Royal Logic
365:Garrett Birkhoff
216:machine learning
201:Garrett Birkhoff
128:
122:
96:
95:|topic=
93:, and specifying
78:Google Translate
59:
55:
38:
37:
30:
21:
18:Concept analysis
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4040:10.1.1.457.8879
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2847:
2846:978-364201814-5
2834:
2833:, reprinted in
2829:
2804:
2803:
2796:
2791:
2786:
2762:Grounded theory
2757:Graphical model
2742:Factor analysis
2692:
2616:
2596:
2533:
2525:order dimension
2509:bipartite graph
2505:
2500:
2458:
2441:
2393:
2382:
2379:order-reversing
2363:concept algebra
2334:
2306:
2299:weak opposition
2280:
2258:
2257:) that satisfy
2214:
2208:
2183:
2179:
2175:
2134:
2133:
2131:
2124:
2100:
2096:
2089:
2082:
2063:
2042:
2038:
2031:
2008:
1995:
1985:
1982:arrow relations
1978:
1976:Arrow relations
1966:Armstrong rules
1962:canonical basis
1956:
1949:
1900:
1868:
1846:
1840:
1835:, i.e., with (¬
1813:
1780:
1773:
1766:
1759:
1752:
1745:
1738:
1731:
1724:
1717:
1710:
1703:
1697:, we say that (
1692:
1685:
1678:
1671:
1657:) of a context
1656:
1647:
1637:
1627:have attribute
1602:
1571:that is not in
1556:that is not in
1517:
1506:
1484:provided that:
1469:
1434:
1429:
1423:
1419:
1401:
1396:
1390:
1386:
1334:
1331:
1292:
1289:
1272:
1262:
1240:
1205:
1202:
1188:and the extent
1180:has the intent
1160:. Accordingly,
1139:formal concepts
1134:
563:
521:bodies of water
510:concept lattice
495:bodies of water
487:bodies of water
479:
473:
470:
435:pragmatic maxim
399:
396:
384:
357:Bernhard Ganter
244:
145:
144:
143:
126:
120:
62:
56:(February 2012)
39:
35:
28:
23:
22:
15:
12:
11:
5:
4842:
4840:
4832:
4831:
4826:
4821:
4816:
4811:
4809:Lattice theory
4806:
4796:
4795:
4792:
4791:
4790:
4789:
4779:
4769:
4759:
4749:
4739:
4729:
4719:
4709:
4699:
4689:
4679:
4666:
4661:
4654:
4653:External links
4651:
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4149:
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4107:
4093:
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4014:
4000:
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3912:
3890:
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3831:
3824:
3791:
3760:
3719:
3704:
3674:
3646:
3611:Bioinformatics
3597:
3578:(6): 734–749.
3551:
3509:
3484:
3473:. 12 June 2014
3458:
3441:Coron.loria.fr
3428:
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3251:
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3158:
3127:
3108:(3): 295–304.
3082:
3063:(2): 149–158.
3047:
3040:
3019:
2993:
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2946:
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2914:
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2871:978-3518067079
2870:
2852:
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2790:
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2779:
2774:
2772:Pattern theory
2769:
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2759:
2754:
2749:
2744:
2739:
2734:
2729:
2724:
2719:
2714:
2709:
2704:
2699:
2693:
2691:
2688:
2615:
2612:
2608:closure system
2595:
2592:
2532:
2529:
2504:
2501:
2499:
2496:
2495:
2494:
2491:
2488:
2485:
2480:
2477:
2466:concept mining
2457:
2454:
2440:
2437:
2436:
2435:
2206:
2200:
2159:
2156:
2153:
2150:
2147:
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2123:
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2057:
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1977:
1974:
1867:
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1778:
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1743:
1736:
1729:
1722:
1715:
1708:
1701:
1690:
1683:
1676:
1669:
1652:
1643:
1639:The concepts (
1636:
1633:
1607:has attribute
1594:
1581:
1580:
1565:
1550:
1527:
1526:
1466:formal concept
1443:
1442:
1410:
1409:
1370:
1369:
1328:
1327:
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1198:
1130:
1129:
1127:
1120:
1118:
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772:
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542:
537:
532:
526:
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502:formal context
478:
475:
471:Rudolf Wille,
468:
397:Rudolf Wille,
394:
383:
380:
326:
325:
314:
272:formal concept
268:formal context
262:has attribute
243:
240:
167:of deriving a
165:principled way
147:
146:
142:
141:
134:
123:
101:
98:
86:adding a topic
81:
70:
63:
44:
43:
42:
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33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4841:
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4609:
4606:
4600:
4596:
4591:
4588:
4586:3-540-62771-5
4582:
4578:
4573:
4570:
4568:3-540-27891-5
4564:
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4523:
4517:
4514:
4502:
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4459:
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4430:
4427:
4423:
4419:
4415:
4409:
4405:
4401:
4397:
4390:
4387:
4383:
4381:3-540-66391-6
4377:
4373:
4369:
4365:
4358:
4355:
4351:
4347:
4343:
4341:3-540-67859-X
4337:
4333:
4329:
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4303:0-470-85055-8
4299:
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4206:
4202:
4195:
4188:
4185:
4175:on 2016-02-13
4171:
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4138:1-58113-108-9
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4025:
4018:
4015:
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4007:
4003:
3997:
3993:
3989:
3985:
3977:
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3970:
3966:
3962:
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3944:
3940:
3936:
3928:
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3915:
3909:
3905:
3901:
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3888:
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3856:
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3821:
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3813:
3809:
3802:
3795:
3792:
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3779:
3771:
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3752:
3747:
3742:
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3723:
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3701:
3697:
3693:
3689:
3685:
3678:
3675:
3669:
3664:
3660:
3653:
3651:
3647:
3642:
3638:
3633:
3628:
3624:
3620:
3617:(9): 1122–9.
3616:
3612:
3608:
3601:
3598:
3593:
3589:
3585:
3581:
3577:
3573:
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3403:
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3358:on 2010-04-16
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3213:
3207:
3203:
3202:Shaker Verlag
3196:
3195:
3187:
3184:
3179:
3173:
3169:
3162:
3159:
3148:on 2017-12-09
3144:
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2990:
2986:
2982:
2978:
2974:
2967:
2964:
2959:
2958:
2950:
2947:
2942:
2940:3-540-62771-5
2936:
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2900:
2893:
2892:
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2483:Lattice Miner
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2326:
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2318:
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2310:
2304:
2300:
2296:
2295:weak negation
2291:
2287:
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2278:
2274:
2269:
2265:
2261:
2256:
2252:
2248:
2244:
2240:
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2218:
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2201:
2198:
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1468:of a context
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1234:
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1199:
1197:
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1155:
1151:
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1128:
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1119:
1112:
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1103:
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1089:
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797:
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744:
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91:main category
88:
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58:
52:
50:
45:You can help
41:
32:
31:
19:
4629:
4619:
4594:
4576:
4549:
4526:. Retrieved
4516:
4505:. Retrieved
4495:
4484:. Retrieved
4471:
4453:
4446:
4435:
4429:
4395:
4389:
4363:
4357:
4323:
4317:
4307:, retrieved
4292:
4285:
4251:
4244:
4220:, retrieved
4200:
4187:
4177:, retrieved
4170:the original
4165:
4152:
4142:, retrieved
4120:
4110:
4076:
4070:
4060:, retrieved
4030:
4017:
3983:
3976:
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1866:Implications
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1798:
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1282:
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1235:is a set of
1232:
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1227:is a set of
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506:line diagram
505:
501:
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486:
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432:
412:model theory
408:formal logic
405:
401:
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385:
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353:Rudolf Wille
350:
342:
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317:
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298:) such that
295:
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283:
279:
275:
271:
267:
263:
259:
251:
248:order theory
245:
224:semantic web
205:
197:ordered sets
189:Rudolf Wille
168:
160:
156:
150:
113:edit summary
104:
84:
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46:
4814:Data mining
3857:(1): 1–36.
3733:Mach. Learn
3661:: 175–190.
3532:(1): 3–20.
3477:27 December
3451:27 December
3411:27 December
3386:27 December
2661:linguistics
2645:information
2193:trilattices
2187:. Although
2174:the object
2059:and dually
1873:implication
1767:) whenever
524:attributes
428:linguistics
332:notions of
316:the intent
302:the extent
212:text mining
208:data mining
4798:Categories
4540:References
4528:2016-02-14
4507:2015-11-14
4486:2016-02-14
4396:Icfca 2015
4309:2016-02-04
4222:2016-02-04
4179:2016-02-13
4144:2016-02-04
4077:Icfca 2009
4062:2016-02-13
3935:Icfca 2012
3919:2016-01-29
3904:Icfca 2008
3785:2022-07-20
3502:2 February
3362:2010-06-10
3152:2017-12-08
2682:(CLA), or
2490:FcaBedrock
2377:, i.e. an
2367:power sets
2199:-ary case.
1237:attributes
466:activated.
451:conclusion
181:properties
179:and their
171:or formal
4788:2021 16th
4778:2019 15th
4768:2017 14th
4758:2015 13th
4748:2014 12th
4738:2013 11th
4728:2012 10th
4597:, Wiley,
4422:0302-9743
4278:0302-9743
4217:0066-4200
4057:215797283
4035:CiteSeerX
4010:0302-9743
3961:0302-9743
3755:254738363
3668:1111.3270
3592:206742345
3077:122657534
2904:1204.1995
2678:(ICFCA),
2513:bicliques
2503:Bicliques
1255:incidence
1178:reservoir
1146:reservoir
832:reservoir
530:temporary
447:judgement
424:intension
420:extension
416:predicate
338:intension
334:extension
282:), where
232:chemistry
131:talk page
83:Consider
51:in German
4718:2011 9th
4708:2010 8th
4698:2009 7th
4688:2008 6th
4678:2007 5th
4230:citation
4103:26304023
3871:11148363
3714:32701053
3641:16500941
3563:(2005).
3546:15659185
3337:10784843
3122:17738011
2690:See also
2686:(ICCS).
2641:ontology
2629:genetics
2621:medicine
2493:GALACTIC
2479:ToscanaJ
2104:, then (
2046:, then (
1944:is valid
1354:for all
1312:for all
1223:, where
1182:stagnant
1154:constant
1150:stagnant
1101:trickle
1069:torrent
555:maritime
550:constant
545:stagnant
469:—
455:possible
439:subsumed
395:—
330:semantic
193:lattices
173:ontology
107:provide
4350:5942241
3969:6256292
3016:: 5–18.
2989:8205166
2633:ecology
2430:, i.e.
2170:
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2079:and if
2024:and if
1899:. When
1862:below.
1801:lattice
1679:) and (
1661:can be
1617:boolean
1613:maximal
1514:, and
1464:) is a
1229:objects
1186:natural
1170:natural
895:rivulet
596:channel
562:objects
540:natural
535:running
477:Example
443:reality
345:ordered
236:biology
177:objects
163:) is a
129:to the
111:in the
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1162:puddle
1002:stream
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732:puddle
625:lagoon
504:, the
392:theory
312:dually
310:, and
296:intent
290:) and
288:extent
185:subset
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4503:. CLA
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3965:S2CID
3867:S2CID
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3804:(PDF)
3773:(PDF)
3751:S2CID
3710:S2CID
3663:arXiv
3588:S2CID
3568:(PDF)
3542:S2CID
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3333:S2CID
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3198:(PDF)
3146:(PDF)
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3118:S2CID
3102:Order
3098:(PDF)
3073:S2CID
3057:Order
3006:(PDF)
2899:arXiv
2895:(PDF)
2789:Notes
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