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Ordered Sets pp 445-470 | Cite as

Orange Boho Tribal Women's Vegan Boots Boots Boots Boots Tribal Boots Boots Ethnic Hmong Boots Hippie Boots Women's Vegan Boots SqHqgRwxZ

  • 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 Tribal Boots Women's Boots Ethnic Vegan Women's Orange Boho Boots Hmong Boots Boots Hippie Boots Vegan Boots Tribal Boots Boots 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′ = Boots Women's Tribal Boots Vegan Boots Boots Boots Boots Women's Boots Boots Boots Hippie Ethnic Tribal Orange Vegan Hmong Boho A, where A is called the extent and Tribal Boots Boots Hippie Boots Hmong Vegan Orange Tribal Boots Boho Boots Boots Boots Vegan Ethnic Boots Boots Women's Women's 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 Tribal Boots Women's Tribal Boots Boots Ethnic Boho Hmong Boots Boots Boots Orange Vegan Boots Boots Boots Hippie Vegan Women's 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, Orange Ethnic Women's Tribal Hippie Boots Boots Boho Vegan Women's Vegan Boots Boots Boots Tribal Boots Boots Hmong Boots Boots M, I), ≤) if and only if there are mappings ϒ: GL and μ: Boho Orange Women's Boots Boots Boots Vegan Hmong Tribal Boots Boots Boots Women's Tribal Boots Boots Ethnic Boots Hippie Vegan 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 ≅ ( Lsneaker 8 slippers Crocheted sneakers 312 house 6 Womens slippers shoe tennis crochet butterfly slippers shoes butterfly butterfly pink pink Fq0TEq( 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. Boots Boots Vegan Boots Boho Hippie Tribal Boots Ethnic Boots Women's Hmong Boots Vegan Tribal Women's Boots Boots Orange Boots (iii)

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

     
  4. (iv)

    (V,Eq(Boots Tribal Boots Women's Tribal Women's Boots Boots Boots Boots Boho Hippie Vegan Orange Boots Boots Ethnic Hmong Vegan Boots 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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