Formal Concept Analysis: Mathematical Foundations

by Bernhard Ganter, C. Franzke (Translator), Rudolf Wille

Overview

Formal Concept AllalY.5is is a field of applied mathematics based on the math­ ematization of concept and conceptual hierarchy. It thereby activates math­ ematical thinking for conceptual data analysis and knowledge processing. The underlying notion of "concept" evolved early in the philosophical theory of concepts and still has effects today. For example, it has left its mark in the German standards DIN 2:)30 and DIN 2;3:)1. In mathematics it played a special role during the emergence of mathematical logic in the 19th century. Subsequently, however, it had virtually no impact on mathematical thinking. It was not until 1979 that the topic was revisited and treated more thoroughly. Since then, through a large number of contributions, Formal Concept Analysis has obtained such breadth that a systematic presentation is urgently needed, but can no longer be realized in one volume. Therefore, the present book foruse:':! on the mathematical foundations of Formal Concept Analysis, which ran be regarded chiefly as a branch of ap­ plied lattice theory. A series of examples serves to demonstrate the utility of the lnathematical definitions and results; in particular, to show how Formal Concept Analysis can be used for the conceptual unfolding of data contexts. These examples do not play the role of case studies in data analysis. A is intended for a comprehensive treatment of methods of separate volume conceptual data and knowledge processing. The general foundations of For­ mal Concept Analysis will also be treated separately.

Product Details

ISBN-13:	9783540627715
Publisher:	Springer Berlin Heidelberg
Publication date:	12/22/1998
Edition description:	1999
Pages:	284
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

About the Author

Bernhard Ganter is Emeritus Professor of Mathematics at Technische Universität Dresden, Germany. His main research field is Formal Concept Analysis. Before being appointed to Dresden in 1993, he was a member of R. Wille’s working group and contributed to the development of Formal Concept Analysis. He is co-founder of the “Mathematics Adventure Land” exhibition in Dresden, which has been a success since 2008.

Rudolf Wille (2 November 1937 – 22 January 2017) was professor of Mathematics (General Algebra) from 1970 to 2003 at Technische Universität Darmstadt, Germany. He is cofounder of the celebrated theory of Formal Concept Analysis, a field of mathematics that applies mathematical lattice theory to organize data based on objects and their shared attributes. An accomplished musician, he also made contributions to Mathematics in Music, Mathematical Pedagogy and the Philosophy of Science, and was a leading scholar in the concept lattice research community.

Table of Contents

0. Order-theoretic Foundations.- 0.1 Ordered Sets.- 0.2 Complete Lattices.- 0.3 Closure Operators.- 0.4 Galois Connections.- 0.5 Hints and References.- 1. Concept Lattices of Contexts.- 1.1 Context and Concept.- 1.2 Context and Concept Lattice.- 1.3 Many-valued Contexts.- 1.4 Context Constructions and Standard Scales.- 1.5 Hints and References.- 2. Determination and Representation.- 2.1 All Concepts of a Context.- 2.2 Diagrams.- 2.3 Implications between Attributes.- 2.4 Dependencies between Attributes.- 2.5 Hints and References.- 3. Parts and Factors.- 3.1 Subcontexts.- 3.2 Complete Congruences.- 3.3 Closed Subrelations.- 3.4 Block Relations and Tolerances.- 3.5 Hints and References.- 4. Decompositions of Concept Lattices.- 4.1 Subdirect Decompositions.- 4.2 Atlas-decompositions.- 4.3 Substitution.- 4.4 Tensorial Decompositions.- 4.5 Hints and References.- 5. Constructions of Concept Lattices.- 5.1 Subdirect Product Constructions.- 5.2 Gluings.- 5.3 Local Doubling.- 5.4 Tensorial Constructions.- 5.5 Hints and References.- 6. Properties of Concept Lattices.- 6.1 Distributivity.- 6.2 Semimodularity and Modularity.- 6.3 Semidistributivity and Local Distributivity.- 6.4 Dimension.- 6.5 Hints and References.- 7. Context Comparison and Conceptual Measurability.- 7.1 Automorphisms of Contexts.- 7.2 Morphisms and Bonds.- 7.3 Scale Measures.- 7.4 Measurability Theorems.- 7.5 Hints and References.- References.