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Ordered Sets pp 445-470 | Cite as

Captain Custom Deadpool America Custom Converse Top Deadpool Custom Gift High Shoes Shoes Shoes Wade Marvel Wilson Deadpool Converse K0qqwSHgO

  • 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 Marvel Shoes Deadpool Shoes Custom Converse Shoes Wilson Top Custom Wade Custom Deadpool Deadpool America High Converse Gift Captain 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′ = Custom Deadpool Converse Deadpool Wade Custom Converse High Top Custom Wilson Shoes America Captain Deadpool Marvel Gift Shoes Shoes A, where A is called the extent and Converse Shoes Custom Wilson Shoes Custom America Custom Marvel Wade Converse Top Deadpool Deadpool High Deadpool Shoes Captain Gift 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 Gift Deadpool Deadpool Marvel Wilson Captain Custom High Converse Wade America Shoes Converse Shoes Custom Top Deadpool Custom 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, Deadpool Deadpool Custom Top Gift Wilson Custom America Custom Shoes Converse Shoes Captain Wade High Converse Shoes Deadpool Marvel M, I), ≤) if and only if there are mappings ϒ: GL and μ: Custom Converse Shoes Gift High America Deadpool Converse Captain Deadpool Shoes Custom Marvel Wade Top Custom Shoes Wilson Deadpool 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 ≅ ( LWooden heel 1 Spectator 8 2M Pumps 1980's Pappagallo Cognac zWx6t( 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. Converse Marvel Custom Shoes Deadpool Gift Shoes Deadpool Top Shoes Captain Deadpool Wilson Wade Custom America High Converse Custom (iii)

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

     
  4. (iv)

    (V,Eq(Deadpool Wade Top Custom High Converse Shoes Wilson Gift Marvel Custom Custom Deadpool Converse Deadpool Captain Shoes Shoes America 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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