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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 heel Flops Bride height flip Wedge Bridesmaid Platform 3 Plain White 5 Flip flops Ivory Wedge Classic 2 1 Satin Wedge Sandals or 25 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′ = 2 Flops or height 5 3 25 Bridesmaid Classic Platform Wedge 1 Satin flip flops Flip heel Plain Sandals Bride Wedge Wedge White Ivory A, where A is called the extent and Satin Wedge heel Ivory 3 2 or Bridesmaid Sandals 5 Plain Flip Bride Wedge Classic flops Wedge White 25 flip 1 Platform Flops height 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 Plain Wedge 3 Bridesmaid or Wedge 2 Flip flops height Ivory 25 heel Classic flip Wedge 5 White Sandals 1 Flops Bride Satin Platform 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, 3 Flip Bride heel 25 Plain or Sandals Ivory Wedge 5 Flops flip Wedge flops White Satin height Classic Bridesmaid Platform 1 Wedge 2 M, I), ≤) if and only if there are mappings ϒ: GL and μ: or Satin heel Bride Flip 25 Ivory Classic 5 Wedge White flops 1 Plain Platform Bridesmaid Sandals flip Wedge Wedge 2 3 height Flops 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 ≅ ( LGhibli Custom Custom Mononoke Shoes Shoes Studio Sneaker Converse Converse Princess Custom High Anime Princess Top Mononoke Mononoke 1wTqXw78( 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. Wedge Ivory Bridesmaid or Flops height 1 Classic 25 Flip flops Satin Sandals flip heel Bride 3 5 Platform Wedge 2 White Plain Wedge (iii)

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

     
  4. (iv)

    (V,Eq(flops heel Satin 2 Ivory Wedge 5 25 Plain 1 Sandals Flip Bridesmaid height Wedge or White flip Classic Wedge Platform Bride 3 Flops 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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