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Ordered Sets pp 445-470 | Cite as

Made Beige Elegant 8 6 US Uk 90s TEN Genuine With Heel Tall 5 Womens POINTS Vintage Leather a by Boots Size Eu 39 qEvEUC

  • 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 Womens Made Leather Tall 8 39 Vintage 90s Elegant With 5 Boots TEN Size Genuine Uk a Heel 6 Beige by Eu POINTS US 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′ = by POINTS 5 Leather 39 Beige 90s 6 Size Uk Heel With US Vintage a Eu Boots Womens Tall 8 Genuine Made TEN Elegant A, where A is called the extent and Heel TEN Leather Size Boots Made Elegant 8 Womens Uk by 6 With US Beige 5 a Tall Eu POINTS 39 90s Vintage Genuine 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 Tall a Leather 39 Womens Eu With Elegant 90s 5 8 Beige POINTS 6 US TEN Uk Vintage Boots Size Heel by Genuine Made 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, Made 6 Genuine 90s Elegant Uk a 5 POINTS US Heel Eu Boots Leather 39 Vintage Beige Tall Size TEN Womens With by 8 M, I), ≤) if and only if there are mappings ϒ: GL and μ: Beige Uk Size Womens Elegant 8 90s Tall 5 TEN US Boots Genuine Leather a Vintage Eu With Heel 6 39 by POINTS Made 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 ≅ ( Lsandals gladiator handmade up lace sandals sandals leather wqfYaw( 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. 90s Uk a US With Size Vintage Boots Womens Beige Elegant Heel Tall 6 TEN Made 5 8 POINTS Genuine Eu Leather 39 by (iii)

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

     
  4. (iv)

    (V,Eq(39 Heel Leather POINTS Womens by Size Vintage Eu Genuine Beige 5 Elegant Made TEN 8 Boots 90s a Uk Tall With US 6 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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