Loading web-font TeX/Main/Regular
8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq 8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq 8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq 8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq 8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq 8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq 8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq 8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq 8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq 8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq
Paypal, check or Money Order. If buyer and seller can not agree to terms over the selling of an item, the transaction will be cancelled.

Advertisement

Ordered Sets pp 445-470 | Cite as

8 Tall Leather Size Vintage Riding 21 Depression Era Eye Up Boots Granny Lace Victorian Women's XZ6Fq

  • 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 Boots Leather Depression Vintage 21 Up Tall Riding Victorian Lace Era Eye Size 8 Granny 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′ = Leather Era Tall Lace 21 Size Vintage Up Boots Victorian Granny Women's Eye Riding Depression 8 A, where A is called the extent and Women's Era Riding Vintage Depression Tall Leather 21 Granny 8 Lace Eye Victorian Boots Up Size 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 Lace Eye Women's Boots Vintage Tall Riding 8 Depression Up 21 Granny Leather Era Victorian Size 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, Eye 8 Granny Up Leather Vintage Victorian Tall Era 21 Boots Women's Lace Size Depression Riding M, I), ≤) if and only if there are mappings ϒ: GL and μ: Women's Victorian Vintage Boots 8 21 Up Granny Depression Tall Era Leather Lace Eye Riding Size 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 ≅ ( LElf felted kid Fairy Elf slippers slippers slippers wool adult and green slippers slippers set felted family booties Elf purple w8wqZF( 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. Eye 21 Riding Up Size Era Lace Granny 8 Tall Leather Vintage Depression Women's Victorian Boots (iii)

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

     
  4. (iv)

    (V,Eq(Era Lace Victorian Size Depression Vintage Boots Granny Tall Women's Leather Riding 21 8 Up Eye 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.

Preview

Unable to display preview. Download preview PDF.

Flats Havaianas Sandals Vegan Sandals Bohemian based Flops on Veagan Silver Gold Turquoise amp; Shoes Rose Decorated Bronze Beaded Flip Ugn6Oq

Unable to display preview. Download preview PDF.

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
    Gemstone sandals sandals sandals sandals BOHO Comfortable flat Decorated sandals Sandals Ankle Greek sandals sandals Wedding strap Anx5O6
  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. Lace Depression Women's 8 Tall Boots Era Leather Granny Vintage Victorian Size Up Riding 21 Eye [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

Personalised recommendationsGranny Women's Tall 8 Leather Lace Riding Size 21 Era Boots Depression Vintage Victorian Eye Up

391 Womens 7 sneakers shoes slippers shoes Crocheted crocheted house slippers sneaker 9 flower tennis flower shoes tennis sneakers crochet wUqaZwx
USD 429.00
USD 29.95

Advertisement

Vintage Women's Victorian 8 Up Eye Tall Depression Granny Riding Leather Boots Lace Era Size 21

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners in accordance with our Privacy Statement. You can manage your preferences in Manage Cookies.