Appendix 3: Formal Concept Analysis
نویسنده
چکیده
Exercise 13 of Chapter 2 is to show that a binary relation R ⊆ A × B induces a pair 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 C πσ ⊆ P(A) is dually isomorphic to C σπ ⊆ 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, C πσ is a complete lattice. Moreover, every complete lattice can be represented 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 σ and π defined as above, L ∼ = C πσ (and L is dually isomorphic to C σπ). In particular, for an arbitrary complete lattice, we can always take A = B = L. If L is algebraic, a more natural choice is A = L c and B = M * (L) (compact elements and 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 colleagues in Darmstadt (Germany), whereby the philosophical Galois connection between objects and their properties is used to provide a systematic analysis of certain very general situations. Abstractly, it goes like this. Let G be a set of " objects " (Gegenstände) and M a set of relevant " attributes " (Merkmale). The relation I ⊆ G × M consists of all those pairs g, m such that g has the …
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تاریخ انتشار 2008