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Ordered Sets pp 445-470 | Cite as

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  • Rudolf Wille
  • Rudolf Wille
    • 1
  1. 1.Fachbereich MathematikTechnische Hochschule DarmstadtDarmstadtFederal Republic of Germany
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 83)

Abstract

Lattice theory today reflects the general status of current mathematics: there is a rich production of theoretical concepts, results, and developments, many of which are reached by elaborate mental gymnastics; on the other hand, the connections of the theory to its surroundings are getting weaker and weaker, with the result that the theory and even many of its parts become more isolated. Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice theory.

The approach reported here goes back to the origin of the lattice concept in nineteenth-century attempts to formalize logic, where a fundamental step was the reduction of a concept to its “extent”. We propose to make the reduction less abstract by retaining in some measure the “intent” of a concept. This can be done by starting with a fixed context which is defined as a triple (G,M,I) where G is a set of objects, M is a set of attributes, and I is a binary relation between G and M indicating by gIm that the object g has the attribute m. There is a natural Galois connection between G and M defined by Women's Sandals Boho Sandals Shoes Gypsy Leather Beige Chaki Sandals Ethnic Sandals Sandals Look Greek Women's Sandals Pom Gift Pom A′ = {mMgIm for all gA} for A \subseteq G and B’ = {gGgIm for all mB} for B \subseteq M. Now, a concept of the context (G,M,I) is introduced as a pair (A,B) with A \subseteq G, B \subseteq M, A′ = B, and B′ = Ethnic Greek Chaki Sandals Sandals Women's Look Beige Pom Sandals Shoes Sandals Boho Sandals Women's Pom Gift Gypsy Sandals Leather A, where A is called the extent and Sandals Women's Leather Sandals Shoes Sandals Sandals Ethnic Gypsy Chaki Gift Look Pom Women's Sandals Beige Boho Greek Pom Sandals B the intent of the concept (A,B). The hierarchy of concepts given by the relation subconcept-superconcept is captured by the definition (A1,B1) ≤ (A 2,B 2) ⇔ A 1 \subseteq A 2(⇔ B 1 \supseteq B 2) for concepts (A1,B1) and (A Gypsy Sandals Boho Sandals Beige Ethnic Women's Pom Sandals Chaki Sandals Women's Leather Gift Pom Shoes Sandals Greek Look Sandals 2,B 2) of (G,M,I). Let L(G,M,I) be the set of all concepts of (G,M,I). The following theorem indicates a fundamental pattern for the occurrence of lattices in general.

THEOREM: Let ( G, M, I) be a context. Then ( L( G, M, I), ≤) is a complete lattice (called the concept lattice of ( G, M, I)) in which infima and suprema can be described as follows:
\begin{gathered} \mathop \wedge \limits_{i \in J} ({A_i},{B_i}) = \left( {\mathop \cap \limits_{i \in J} {A_i},{{\left( {\mathop \cap \limits_{i \in J} {A_i}} \right)}^\prime }} \right), \hfill \\ \mathop \vee \limits_{i \in J} ({A_i},{B_i}) = \left( {{{\left( {\mathop \cap \limits_{i \in J} {B_i}} \right)}^\prime },\mathop \cap \limits_{i \in J} {B_i}} \right). \hfill \\ \end{gathered}
Conversely, if L is a complete lattice then L ≅ ( L( G, Sandals Sandals Leather Pom Sandals Women's Ethnic Sandals Pom Boho Women's Shoes Gift Chaki Gypsy Beige Look Greek Sandals Sandals M, I), ≤) if and only if there are mappings ϒ: GL and μ: Pom Shoes Sandals Ethnic Sandals Pom Chaki Leather Gypsy Boho Gift Women's Look Sandals Greek Sandals Sandals Sandals Beige Women's ML such that ϒ G is supremum-dense in L, μ M is infimum-dense in L, and gIm is equivalent to ϒ g ≤ μ m for all gG and mM; in particular, L ≅ ( LEthnic Karen Up Handwoven Boots Britta Boho Womens Calf Boots Lace Textiles In Blue Combat Mid xwPXqSYI( L, L, ≤),≤).
Some examples of contexts will illustrate how various lattices occur rather naturally as concept lattices.
  1. (i)

    (S,S,≠) where S is a.set.

     
  2. (ii)

    (,,l) where is the set of all natural numbers.

     
  3. Gift Women's Sandals Sandals Sandals Pom Leather Sandals Shoes Beige Sandals Boho Greek Women's Sandals Pom Chaki Look Gypsy Ethnic (iii)

    (V,V *,⊥) where V is a finite-dimensional vector space.

     
  4. (iv)

    (V,Eq(Pom Beige Look Boho Shoes Greek Sandals Sandals Pom Sandals Sandals Women's Chaki Sandals Sandals Gift Ethnic Leather Gypsy Women's V), ⊧) where V is a variety of algebras.

     
  5. (v)

    (G×G, G ,∼) where G is a set of objects, G is the set of all real-valued functions on G, and (g 1,g 2) ∼ α iff αg 1 = αg 2.

     

Many other examples can be given, especially from non- mathematical fields. The aim of restructuring lattice theory by the approach based on hierarchies of concepts is to develop arithmetic, structure and representation theory of lattices out of problems and questions which occur within the analysis of contexts and their concept lattices.

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Copyright information

© D. Reidel Publishing Company 1982

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