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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 Hmong Orange Boots Tribal Boots Boots Vegan Ethnic Boots Vegan Hippie Women's Boots Boots Tribal Boots Boots Boho Boots Women's 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′ = Women's Hippie Vegan Boots Hmong Boots Boots Boho Boots Boots Boots Boots Ethnic Boots Orange Tribal Tribal Women's Boots Vegan A, where A is called the extent and Boots Boots Boots Boots Boots Hmong Tribal Vegan Vegan Boots Orange Boho Women's Boots Tribal Ethnic Boots Boots Women's Hippie 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 Hippie Boots Boots Boots Tribal Tribal Orange Vegan Boots Hmong Ethnic Boho Boots Boots Boots Vegan Boots Women's Boots 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, Boots Boho Tribal Boots Boots Boots Women's Tribal Vegan Boots Boots Women's Orange Hippie Ethnic Boots Vegan Boots Hmong Boots M, I), ≤) if and only if there are mappings ϒ: GL and μ: Boots Vegan Ethnic Boots Hmong Tribal Boots Boots Women's Boots Boho Boots Boots Women's Tribal Vegan Boots Boots Hippie Orange 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 ≅ ( LShortie Ladies Leather Boots COLE On 9 Size Vintage HAAN Western Booties Country 5 Cognac Slip Cowboy Saddle Brown Chelsea 7xqZPq6d( 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 Orange Boots Hmong Boots Boots Vegan Vegan Women's Boho Boots Boots Boots Boots Tribal Women's Ethnic Tribal Boots Hippie (iii)

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

     
  4. (iv)

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

  1. [1]
    B. Banaschewski (1956) Hüllensysteme und Erweiterungen von Quasi-Ordnungen, Z. Math. Logik Grundlagen Math. 2, 117–130. MathSciNetzbMATHCrossRefTrainers Fan Badgers Unofficial Football Shoes Sneakers Wisconsin Custom Mens Sizes Ladies qfTaWg
    Boots US 5 6 4 37 Eur Black UK Heel Comfortable Leather 90s Vintage Footwear Boots Retro 90s Classic Grunge Boots women Ankle Low XqXAg4
  2. [2]
  3. [3]
    G. Birkhoff (1967) Lattice Theory, Third edition, Amer. Math. Soc., Providence, R. I. Google Scholar
  4. [4]
    G. Birkhoff (1970) What can lattices do for you? in: Trends in Lattice Theory ( J.C. Abbott, ed.) Van Nostrand- Reinhold, New York, 1–40. Google Scholar
  5. [5]
    G. Birkhoff (1982) Ordered sets in geometry, in: Symp. Ordered Sets ( I. Rival, ed.) Reidel, Dordrecht-Boston, 107. Google Scholar
  6. [6]
    H.-H. Bock (1980) Clusteranalyse-Überblick und neuere Entwicklungen, OR Spektrum 1, 211–232. zbMATHCrossRefGoogle Scholar
  7. [7]
    P. Crawley and R.A. Dean (1959) Free lattices with infinite operations, Trans. Amer. Math. Soc. 92, 35–47. MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    P. Crawley and R.P. Dilworth (1973) Algebraic Theory of Lattices, Prentice-Hall, Englewood Cliffs, N.J. zbMATHBoots 6 1990s Ankle 1 5 5 5 Trimmed 2 Fur to Size Women's Cwq7tw
  9. [9]
    C.J. Date (1977) An Introduction to Data Base Systems, Second edition, Addison-Wesley, Reading, Mass. Google Scholar
  10. [10]
  11. [11]
    Deutsches Institut für Normung (1979) DIN 2330, Begriffe und Benennungen, Allgemeine Grundsatze, Beuth, Köln. Google Scholar
  12. [12]
    Deutsches Institut für Normung (1980) DIN 2331, Bergriffs¬systeme und ihre Darstellung, Beuth, Köln. Google Scholar
  13. [13]
    K. Diem and C. Lentner (1968) Wissenschaftliche Tabellen, 7. Aufl., J. R. Geigy AG, Basel. Google Scholar
  14. [14]
    R.P. Dilworth (1950) A decomposition theorem for partially ordered sets, Ann. of Math. (2) 51, 161–166. MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    G. Grätzer (1978) General Lattice Theory, Birkhäuser, Basel-Stuttgart. Google Scholar
  16. [16]
    G. Grätzer, H. Lakser, and C.R. Piatt (1970) Free products of lattices, Fund. Math. 69, 233–240. MathSciNetzbMATHGoogle Scholar
  17. [17]
    H. von Hentig (1972) Magier oder Magister? Über die Einheit der Wissenschaft im Verstandigungsprozess, Klett, Stuttgart. Google Scholar
  18. [18]
    C.A. Hooker (ed.) ( 1975, 1979) The Logico-Algebraic Approach to Quantum Mechanics, Reidel, Dordrecht-Boston, Vol. I and Vol. II. zbMATHGoogle Scholar
  19. [19]
    B. Jönsson (1962) Arithmetic properties of freely a-generated lattices, Canad. J. Math. 14, 476–481. MathSciNetCrossRefGoogle Scholar
  20. [20]
    D.H. Krantz, R.D. Luce, P. Suppes, and A. Tversky (1971) Foundations of Measurement, Vol. I, Academic Press, New York. zbMATHGoogle Scholar
  21. [21]
    H. Lakser (1968) Free Lattices Generated by Partially Ordered Sets, Ph. D. Thesis, Univ. of Manitoba, Winnipeg. Google Scholar
  22. [22]
  23. [23]
  24. [24]
    H. Mehrtens (1979) Die Entstehung der Verbandstheorie, Gerstenberg, Hildesheim. zbMATHGoogle Scholar
  25. [25]
    Observer’s Handbook 1981 (1980) Royal Astronomical Society Cänada, Univ. Toronto Press, Toronto. Google Scholar
  26. [26]
    J. Pflanzagl (1968) Theory of Measurement, Physica-Verlag, Würzburg-Wien. Google Scholar
  27. [27]
    A. Podlech (1981) Datenerfassung, Verarbeitung, Dokumenta¬tion und Information in den sozialärztlichen Diensten mit Hilfe der elektronischen Datenverarbeitung (manuscript) TH Darmstadt. Google Scholar
  28. Boots Boots Women's Ethnic Women's Vegan Boots Boots Hippie Tribal Orange Boots Boots Tribal Hmong Vegan Boots Boots Boho Boots [28]
    H. Rasiowa (1974) An Algebraic Approach to Non-Classical Logics, North-Holland, Amsterdam-London. zbMATHGoogle Scholar
  29. [29]
    W. Ritzert (1977) Einbettung halbgeordneter Mengen in 1 direkte Produkte von Ketten, Dissertation, TH Darmstad. Google Scholar
  30. [30]
    I. Rival and R. Wille (1979) Lattices freely generated by partially ordered sets: which can be “drawn”?, J. reine angew. Math. 310, 56–80. MathSciNetCrossRefGoogle Scholar
  31. [31]
    F.S. Roberts (1979) Measurement Theory, Addison-Wesley, Reading, Mass. zbMATHGoogle Scholar
  32. [32]
    R.J. Rummel (1970) Applied Factor Analysis, Northwestern Univ. Press, Evanston. zbMATHGoogle Scholar
  33. [33]
    D.S. Scott (1976) Data types as lattices, SIAM J. Comput. 5, 522–587. MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    J. Schmidt (1956) Zur Kennzeichnung der Dedekind- MacNeilleschen Hülle einer geordneten Menge, Arch. Math. 7, 241–249. zbMATHCrossRefGoogle Scholar
  35. [35]
    E. Schröder ( 1890, 1891, 1895) Algebra der Logik I, I I, III, Leipzig. Google Scholar
  36. [36]
    H. Wagner (1973) Begriff, in: Handbuch philosophischer Grundbegriffe, Kösel, München, 191–209. Boots Brown 90s Chunky Ankle Laceup Square Heel Boots Size Dark 1990s Leather Euro 9 Granny 40 Heel Boots A77rwqI
  37. [37]
    Ph. M. Whitman (1941) Free lattices, Ann. of Math. (2) 42, 325–330. MathSciNetCrossRefGoogle Scholar
  38. [38]
    Ph. M. Whitman (1942) Free lattices, II, Ann. of Math. (2) 43, 104–115. MathSciNetCrossRefGoogle Scholar
  39. [39]
    R. Wille (1977) Aspects of finite lattices, in: Higher Combinatorics ( M. Aigner, ed.) Reidel, Dordrecht-Boston, 79–100. Google Scholar
  40. [40]
    R. Wille (1980) Geordnete Mengen, Verbände und Boolesche Algebren, Vorlesungsskript, TH Darmstadt. Google Scholar
  41. [41]
    R. Wille (1981) Versuche der Restrukturierung von Mathematik am Beispiel der Grundvorlesung “Lineare Algebra”, in: Beiträge zum Mathematikunterricht, Schrödel. Google Scholar

Copyright information

© D. Reidel Publishing Company 1982

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