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Ordered Sets pp 445-470 | Cite as

Bride Flops for Ivory 1 Custom color Heel 3 Wedges inch I White Bridal DO Date in or 2 Glitter Flip the Shoes Wedding YrRUn7Y

  • 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 Shoes 1 Flip color 3 Heel Wedges Date for Flops Glitter Custom Bride 2 I Wedding or in White DO Ivory Bridal inch the 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′ = Date Heel for Flops Wedges Bride inch or Custom DO 1 color Glitter Wedding 3 Ivory the Flip I Bridal White Shoes 2 in A, where A is called the extent and or Glitter Wedges inch 1 Heel 2 Flip Custom Date Shoes Wedding 3 DO the White color Bridal Bride Flops Ivory I in for 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 Custom Date inch color in Bride Flops DO Heel for Ivory the Wedding 1 Bridal Wedges Flip Shoes Glitter 3 2 I White or 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, Custom or Bridal Shoes color Date I Ivory Wedding inch DO 2 3 for 1 Bride the White Flops Heel Glitter Flip in Wedges M, I), ≤) if and only if there are mappings ϒ: GL and μ: in Heel Flops color 1 Shoes Bride I Ivory for Glitter Bridal 3 or Wedges Custom Date inch Wedding White Flip the DO 2 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 ≅ ( Lstyle look combat 9 M boots Vintage brown Gianni size Bini distressed YPfPOnEqw( 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. Shoes White Wedges Bride 1 Ivory the Date for DO 3 in Glitter I Flops Heel inch 2 Bridal Flip Wedding Custom color or (iii)

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

     
  4. (iv)

    (V,Eq(Wedding Flip Date Custom color Wedges in I for DO Glitter White Flops 3 Bride the 2 Shoes Heel or inch 1 Bridal Ivory 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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© D. Reidel Publishing Company 1982

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