Appendix 3: Formal Concept Analysis

Exercise 13 of Chapter 2 is to show that a binary relation R ⊆ A×B induces apair of closure operators, described as follows. For X ⊆ A, let

σ(X) = {b ∈ B : x R b for all x ∈ X}.

Similarly, for Y ⊆ B, let

π(Y ) = {a ∈ A : a R y for all y ∈ Y }.

Then the composition πσ : P(A) → P(A) is a closure operator on A, given by

πσ(X) = {a ∈ A : a R b whenever x R b for all x ∈ X}.

Likewise, σπ is a closure operator on B, and for Y ⊆ B,

σπ(Y ) = {b ∈ B : a R b whenever a R y for all y ∈ Y }.

In this situation, the lattice of closed sets Cπσ ⊆ P(A) is dually isomorphic toCσπ ⊆ P(B), and we say that R establishes a Galois connection between the πσ-closed subsets of A and the σπ-closed subsets of B.

Of course, Cπσ is a complete lattice. Moreover, every complete lattice can berepresented via a Galois connection.

Theorem. Let L be a complete lattice, A a join dense subset of L and B a meet

dense subset of L. Define R ⊆ A × B by a R b if and only if a ≤ b. Then, with σand π defined as above, L ∼= Cπσ (and L is dually isomorphic to Cσπ).

In particular, for an arbitrary complete lattice, we can always take A = B = L. IfL is algebraic, a more natural choice is A = Lc and B = M∗(L) (compact elementsand completely meet irreducibles). If L is finite, the most natural choice is A = J(L)and B = M(L). Again the proof of this theorem is elementary.

Formal Concept Analysis is a method developed by Rudolf Wille and his col-leagues in Darmstadt (Germany), whereby the philosophical Galois connection be-tween objects and their properties is used to provide a systematic analysis of cer-tain very general situations. Abstractly, it goes like this. Let G be a set of “ob-jects” (Gegenstande) and M a set of relevant “attributes” (Merkmale). The relationI ⊆ G × M consists of all those pairs 〈g,m〉 such that g has the property m. Aconcept is a pair 〈X,Y 〉 with X ⊆ G, Y ⊆ M , X = π(Y ) and Y = σ(X). Thus

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〈X,Y 〉 is a concept if X is the set of all elements with the properties of Y , andY is exactly the set of properties shared by the elements of X. It follows (as inexercise 12, Chapter 2) that X ∈ Cπσ and Y ∈ Cσπ. Thus if we order concepts by〈X,Y 〉 ≤ 〈U, V 〉 iff X ⊆ U (which is equivalent to Y ⊇ V ), then we obtain a latticeB(G,M, I) isomorphic to Cπσ.

A small example will illustrate how this works. The rows of Table A1 correspondto seven fine musicians, and the columns to eight possible attributes (chosen by amusically trained sociologist). An × in the table indicates that the musician hasthat attribute.1 The corresponding concept lattice is given in Figure A2, where themusicians are abbreviated by lower case letters and their attributes by capitals.

Instrument Classical Jazz Country Black White Male Female

J. S. Bach × × × ×

Rachmaninoff × × × ×

King Oliver × × × ×

W. Marsalis × × × × ×

B. Holiday × × ×

Emmylou H. × × ×

Chet Atkins × × × × ×

Table A1.

b = r m

o

a h e

Cl Co

JI=M

F

WB

Figure A2

1To avoid confusion, androgynous rock stars were not included.

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Formal concept analysis has been applied to hundreds of real situations outsideof mathematics (e.g., law, medicine, psychology), and has proved to be a useful toolfor understanding the relation between the concepts involved. Typically, these ap-plications involve large numbers of objects and attributes, and computer programshave been developed to navigate through the concept lattice. A good brief intro-duction to concept analysis may be found in Wille [2] or [3], and the whole businessis explained thoroughly in Ganter and Wille [1]. For online introductions, see thewebsite of Uta Priss,

/www.upriss.org/fca/fca.html

Likewise, the representation of a finite lattice as the concept lattice induced bythe order relation between join and meet irreducible elements (i.e., ≤ restricted toJ(L)×M(L)) provides and effective and tractable encoding of its structure. As anexample of the method, let us show how one can extract the ordered set QL suchthat Con L ∼= O(Q(L)) from the table.

Given a finite lattice L, for g ∈ J(L) and m ∈ M(L), define

g ր m if g � m but g ≤ m∗, i.e., g ≤ n for all n > m,

m ց g if m � g but m ≥ g∗, i.e., m ≥ h for all h < g,

g l m if g ր m and m ց g.

Note that these relations can easily be added to the table of J(L)×M(L).

These relations connect with the relation D of Chapter 10 as follows.

Lemma. Let L be a finite lattice and g, h ∈ J(L). Then g D h if and only if there

exists m ∈ M(L) such that g ր m ց h.

Proof. If g D h, then there exists x ∈ L such that g ≤ h ∨ x but g � h∗ ∨ x. Let mbe maximal such that m ≥ h∗ ∨ x but m � g. Then m ∈ M(L), g ≤ m∗, m ≥ h∗

but m � h. Thus g ր m ց h.

Conversely, suppose g ր m ց h. Then g ≤ m∗ ≤ h ∨m while g � m = h∗ ∨m.Therefore g D h. �

As an example, the table for the lattice in Figure A2 is given in Table A3. This isa reduction of the original Table A1: J(L) is a subset of the original set of objects,and likewise M(L) is contained in the original attributes. Arrows indicating therelations ր, ց and l have been added. The Lemma allows us to calculate Dquickly, and we find that |QL| = 1, whence L is simple.

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I=M Cl J Co B W F

b=r × × l l ց × l

o × l × × ր ր

m × × × ց × l l

h l ց × ց × l ×

e l ց l × l × ×

a × l × × l × l

Table A3.

References

1. B. Ganter and R. Wille, Formale Begriffsanalyse: Mathematische Grundlagen, Springer-Verlag, Berlin-Heidelberg, 1996. Translated by C. Franzke as Formal Concept Analysis: Math-

ematical Foundations, Springer, 1998.

2. R. Wille, Restructuring lattice theory: an approach based on hierarchies of concepts, Ordered

Sets, I. Rival, ed., Reidel, Dordrecht-Boston, 1982, pp. 445–470.

3. R. Wille, Concept lattices and conceptual knowledge systems, Computers and Mathematics

with Applications 23 (1992), 493–515.

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