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**Formal concept analysis. Mathematical foundations.
(Formale Begriffsanalyse. Mathematische Grundlagen.)**
*(German)*
Zbl 0861.06001

Berlin: Springer. x, 286 p. (1996).

The authors state in the introduction to this book that Formal Concept Analysis, based on a mathematical interpretation of concepts and hierarchies of concepts, is an area of applied mathematics. While Formal Concept Analysis played a special role in the development of mathematical logic in the nineteenth century, the area of study has not received much attention until its revitalization around 1980. Since then a large number of published articles have given the field considerable breadth, and it has become necessary to publish a systematic treatise on the subject.

This book provides a systematic treatise for the mathematical foundations of concept analysis, and it interprets these foundations in the context of lattice theory. The content of the book is mathematical and does not attempt to explain the mechanisms of human conceptual thought processes.

The material is organized into eight chapters, starting with a brief introduction of ordered sets, lattices, complete lattices, and Galois connections in Chapter 0. The goal of Chapter 1 is to introduce formal contexts, formal concepts of contexts and concept lattices.

A formal context \(K:=(G,M,I)\) consists of two sets \(G\) and \(M\) and a relation \(I\) between \(G\) and \(M\). The elements of \(G\) are interpreted as subjects and the elements of \(M\) as properties, and \(gIm\) means that the subject \(g\) has property \(m\).

For a subset \(A\) of \(G\), \(A'=\{m\in M\mid gIm \text{ for all }g\in A\}\), and for a subset \(B\) of \(M\), \(B'=\{g\in G\mid gIm\) for all \(m\in B\}\). These two derivation operators give rise to a Galois connection between the power sets of \(G\) and \(M\).

A partial ordering for formal concepts is defined as follows: \((A,B)\leq (C,D)\) if \(A\) is a subset of \(C\) (which is equivalent to \(D\) being a subset of \(B\)). With this ordering, the set of all formal concepts is a complete lattice, the concept lattice of the context \((G,M,I)\).

While this may sound like standard lattice theory with a different verbal spin, the examples given in the text are very interesting. They range from a concept lattice describing a Hungarian video about organisms and water to a concept lattice interpreting the sites and temples of Rome/Italy according to the number of stars awarded to them by various travel guides (Baedecker, Les Guides Bleus, Michelin, Polyglott).

In Chapters 2-8 the mathematical theory of concept lattices is rigorously developed. Topics include decomposition (subdirect, atlas, tensor), constructions (gluing, doubling, tensor), distributivity and modularity, automorphisms and measures. Each chapter concludes with a section of references that are relevant to the topics of the chapter, and a very extensive bibliography at the end of the book gives a complete overview of the literature (221 titles).

This book provides a systematic treatise for the mathematical foundations of concept analysis, and it interprets these foundations in the context of lattice theory. The content of the book is mathematical and does not attempt to explain the mechanisms of human conceptual thought processes.

The material is organized into eight chapters, starting with a brief introduction of ordered sets, lattices, complete lattices, and Galois connections in Chapter 0. The goal of Chapter 1 is to introduce formal contexts, formal concepts of contexts and concept lattices.

A formal context \(K:=(G,M,I)\) consists of two sets \(G\) and \(M\) and a relation \(I\) between \(G\) and \(M\). The elements of \(G\) are interpreted as subjects and the elements of \(M\) as properties, and \(gIm\) means that the subject \(g\) has property \(m\).

For a subset \(A\) of \(G\), \(A'=\{m\in M\mid gIm \text{ for all }g\in A\}\), and for a subset \(B\) of \(M\), \(B'=\{g\in G\mid gIm\) for all \(m\in B\}\). These two derivation operators give rise to a Galois connection between the power sets of \(G\) and \(M\).

A partial ordering for formal concepts is defined as follows: \((A,B)\leq (C,D)\) if \(A\) is a subset of \(C\) (which is equivalent to \(D\) being a subset of \(B\)). With this ordering, the set of all formal concepts is a complete lattice, the concept lattice of the context \((G,M,I)\).

While this may sound like standard lattice theory with a different verbal spin, the examples given in the text are very interesting. They range from a concept lattice describing a Hungarian video about organisms and water to a concept lattice interpreting the sites and temples of Rome/Italy according to the number of stars awarded to them by various travel guides (Baedecker, Les Guides Bleus, Michelin, Polyglott).

In Chapters 2-8 the mathematical theory of concept lattices is rigorously developed. Topics include decomposition (subdirect, atlas, tensor), constructions (gluing, doubling, tensor), distributivity and modularity, automorphisms and measures. Each chapter concludes with a section of references that are relevant to the topics of the chapter, and a very extensive bibliography at the end of the book gives a complete overview of the literature (221 titles).

Reviewer: M.Höft (Dearborn)

### MSC:

06-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures |

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

06B23 | Complete lattices, completions |

### Keywords:

formal concept analysis; glueing; hierarchies of concepts; lattice theory; complete lattices; Galois connections; formal contexts; concept lattices; partial ordering; decomposition; atlas; tensor; distributivity; modularity; automorphisms; measures
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\textit{B. Ganter} and \textit{R. Wille}, Formale Begriffsanalyse. Mathematische Grundlagen. Berlin: Springer (1996; Zbl 0861.06001)