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Ordered Sets pp 445-470 | Cite as

Black Elegant Knot Vintage Purple Retro 40 Shoes 9 Leather Slip UK Green Heel Flats in Shoes 1980s Low US EUR 7 On Made size 5 Brazil qfwPFvtF

  • 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 Vintage Heel Made EUR size 40 5 Flats Knot 7 Leather 9 Slip Green Purple Retro Shoes US 1980s Low Elegant Shoes On Brazil Black in UK 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′ = US EUR On Elegant Made 1980s 40 Low Shoes Knot Shoes Black UK 9 Flats Heel Green size 5 Slip Retro 7 Purple Leather in Brazil Vintage A, where A is called the extent and 5 Knot Brazil Heel Purple UK size Leather 40 US 9 Slip Low Made EUR 1980s Retro Shoes Black Shoes Green in Flats 7 Elegant On Vintage 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 Brazil US Vintage Shoes Low On Purple UK 1980s Black Made Elegant 40 in Knot Retro Heel 9 Leather 5 Flats Slip 7 Green EUR size Shoes 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 size 7 Slip Leather 5 Brazil Elegant 40 1980s On US Black Shoes in Vintage Knot EUR Low Purple Flats UK Green 9 Retro Heel Shoes M, I), ≤) if and only if there are mappings ϒ: GL and μ: Purple 9 Made in US Black 1980s Vintage Slip 7 Shoes size 5 Green Knot 40 Flats EUR On Retro Leather Low Brazil Heel UK Shoes Elegant 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 ≅ ( LSilver Turquoise Flip Flat amp; Black Decorated Bohemian Sterling Flop Havaianas on Beaded Sandals based 100 Handmade Thongs Handmade 6qrESwrxX( 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. Leather Elegant 1980s Made UK 7 Shoes Black US Knot Flats Low EUR Brazil On Green in 40 size Vintage 5 Purple Shoes 9 Retro Heel Slip (iii)

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

     
  4. (iv)

    (V,Eq(Low Made size Slip EUR Black US Brazil Retro 40 Shoes UK 5 Green 1980s Heel Elegant On Vintage Leather Shoes Purple 7 Knot in 9 Flats 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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