2. Mathematical Foundations

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Among the mathematical theories found recently with important applications in computer science, lattice theory has a specific place for data organization, information engineering, data mining and reasoning. It may be considered as the mathematical tool that unifies data and knowledge, also information retrieval and reasoning [9 13]. In this section, we define a binary context, a formal concept and the lattice of concepts associated with the binary context.

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2.1 Definition of a Binary Context

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A binary context (or binary relation) is a subset of the product of two sets O (set of objects) and P (set of properties).

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Example 1 [10]:

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O = Leech, Bream, Frog, Dog, Spike-weed, Reed, Bean, Maize and P= a b c d e f g h i where O is a set of some living things, and P the set of the following properties: a = needs water; b = lives in water; c = lives on land; d = needs chlorophyll to produce food; e = is two seed leaves; f = one seed leaf; g = Can move around; h = has limbs; i = suckles its offspring. A binary context R may be defined by Table 23.1.

Mathematical Foundations

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Table 23.1 An example of a binary context R. a 1 2 3 4 5 6 7 8 Leech Bream Frog Dog Spike-weed Reed Bean Maize 1 1 1 1 1 1 1 1 b 1 1 1 0 1 1 0 0 c 0 0 1 1 0 1 1 1 d 0 0 0 0 1 1 1 1 e 0 0 0 0 0 0 1 0 f 0 0 0 0 1 1 0 1 g 1 1 1 1 0 0 0 0 h 0 1 1 1 0 0 0 0 i 0 0 0 1 0 0 0 0

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Let f be a function from the powerset of the set of objects O (i.e. 20 ) to the powerset of the set of properties P (i.e. 2P ), such that: f A = m g A g m R (23.1)

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f A is the set of all properties shared by all objects of A (subset of O) with respect to the context R. Let g be a function from 2P to 2O , such that: g B = g m B g m R (23.2)

g B is the set of objects sharing all the properties B (subset of P) with respect to the binary context R. We also define closure A = g f A = A , and closure B = f g B = B . The meaning of A is that a set of objects A shares the same set of properties f A with other objects A A , relative to the context R. A is the maximal set of objects sharing the same properties as objects A. In Example 1, if A = Leech, Bream, Frog, Spike-weed then A = Leech, Bream, Frog, Spike-weed, Reed . This means that the shared properties a and b of living things in A, are also shared by a reed, the only element in A A. The meaning of B is that if an object x of the context R verifies properties B, then x also verifies some number of additional properties B B . B is the maximal set of properties shared by all objects verifying properties B. In Example 1, if B = a h , then B = a h g . This means that any animal that needs water (a) and has limbs (h), can move around (g). For each subset B, we may create an association rule B B B. The number of these rules depends on the binary context R. In [10], we find different algorithms for extracting the minimal set of such association rules.

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