# Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts

• Conference paper

Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 83))

• 697 Accesses

• 1217

## 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 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′ = A, where A is called the extent and 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 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,M,I), ≤) if and only if there are mappings ϒ: GL and μ: ML such that ϒG is supremum-dense in L, μM is infimum-dense in L, and gIm is equivalent to ϒg ≤ μm for all gG and mM; in particular, L ≅ (L(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. (iii)

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

4. (iv)

(V,Eq(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.

This is a preview of subscription content, log in via an institution to check access.

## Subscribe and save

Springer+ Basic
\$34.99 /Month
• Get 10 units per month
• 1 Unit = 1 Article or 1 Chapter
• Cancel anytime

Chapter
USD 29.95
Price excludes VAT (USA)
• Available as PDF
• Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
• Available as PDF
• Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
• Compact, lightweight edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
• Durable hardcover edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

## References

1. B. Banaschewski (1956) Hüllensysteme und Erweiterungen von Quasi-Ordnungen, Z. Math. Logik Grundlagen Math. 2, 117–130.

2. G. Birkhoff (1938) Lattices and their applications, Bull. Amer. Math. Soc. 44, 793–800.

3. G. Birkhoff (1967) Lattice Theory, Third edition, Amer. Math. Soc., Providence, R. I.

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.

5. G. Birkhoff (1982) Ordered sets in geometry, in: Symp. Ordered Sets ( I. Rival, ed.) Reidel, Dordrecht-Boston, 107.

6. H.-H. Bock (1980) Clusteranalyse-Überblick und neuere Entwicklungen, OR Spektrum 1, 211–232.

7. P. Crawley and R.A. Dean (1959) Free lattices with infinite operations, Trans. Amer. Math. Soc. 92, 35–47.

8. P. Crawley and R.P. Dilworth (1973) Algebraic Theory of Lattices, Prentice-Hall, Englewood Cliffs, N.J.

9. C.J. Date (1977) An Introduction to Data Base Systems, Second edition, Addison-Wesley, Reading, Mass.

10. R.A. Dean (1956) Completely free lattices generated by partially ordered sets, Trans. Amer. Math. Soc. 83, 238–249.

11. Deutsches Institut für Normung (1979) DIN 2330, Begriffe und Benennungen, Allgemeine Grundsatze, Beuth, Köln.

12. Deutsches Institut für Normung (1980) DIN 2331, Bergriffs¬systeme und ihre Darstellung, Beuth, Köln.

13. K. Diem and C. Lentner (1968) Wissenschaftliche Tabellen, 7. Aufl., J. R. Geigy AG, Basel.

14. R.P. Dilworth (1950) A decomposition theorem for partially ordered sets, Ann. of Math. (2) 51, 161–166.

15. G. Grätzer (1978) General Lattice Theory, Birkhäuser, Basel-Stuttgart.

16. G. Grätzer, H. Lakser, and C.R. Piatt (1970) Free products of lattices, Fund. Math. 69, 233–240.

17. H. von Hentig (1972) Magier oder Magister? Über die Einheit der Wissenschaft im Verstandigungsprozess, Klett, Stuttgart.

18. C.A. Hooker (ed.) ( 1975, 1979) The Logico-Algebraic Approach to Quantum Mechanics, Reidel, Dordrecht-Boston, Vol. I and Vol. II.

19. B. Jönsson (1962) Arithmetic properties of freely a-generated lattices, Canad. J. Math. 14, 476–481.

20. D.H. Krantz, R.D. Luce, P. Suppes, and A. Tversky (1971) Foundations of Measurement, Vol. I, Academic Press, New York.

21. H. Lakser (1968) Free Lattices Generated by Partially Ordered Sets, Ph. D. Thesis, Univ. of Manitoba, Winnipeg.

22. J.W. Lea (1972) An embedding theorem for compact semi- lattices, Proc. Amer. Math. Soc. 34, 325–331.

23. H.M. MacNeille (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42, 416–460.

24. H. Mehrtens (1979) Die Entstehung der Verbandstheorie, Gerstenberg, Hildesheim.

25. Observer’s Handbook 1981 (1980) Royal Astronomical Society Cänada, Univ. Toronto Press, Toronto.

26. J. Pflanzagl (1968) Theory of Measurement, Physica-Verlag, Würzburg-Wien.

27. A. Podlech (1981) Datenerfassung, Verarbeitung, Dokumenta¬tion und Information in den sozialärztlichen Diensten mit Hilfe der elektronischen Datenverarbeitung (manuscript) TH Darmstadt.

28. H. Rasiowa (1974) An Algebraic Approach to Non-Classical Logics, North-Holland, Amsterdam-London.

29. W. Ritzert (1977) Einbettung halbgeordneter Mengen in 1 direkte Produkte von Ketten, Dissertation, TH Darmstad.

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.

32. R.J. Rummel (1970) Applied Factor Analysis, Northwestern Univ. Press, Evanston.

33. D.S. Scott (1976) Data types as lattices, SIAM J. Comput. 5, 522–587.

34. J. Schmidt (1956) Zur Kennzeichnung der Dedekind- MacNeilleschen Hülle einer geordneten Menge, Arch. Math. 7, 241–249.

35. E. Schröder ( 1890, 1891, 1895) Algebra der Logik I, I I, III, Leipzig.

36. H. Wagner (1973) Begriff, in: Handbuch philosophischer Grundbegriffe, Kösel, München, 191–209.

37. Ph. M. Whitman (1941) Free lattices, Ann. of Math. (2) 42, 325–330.

38. Ph. M. Whitman (1942) Free lattices, II, Ann. of Math. (2) 43, 104–115.

39. R. Wille (1977) Aspects of finite lattices, in: Higher Combinatorics ( M. Aigner, ed.) Reidel, Dordrecht-Boston, 79–100.

40. R. Wille (1980) Geordnete Mengen, Verbände und Boolesche Algebren, Vorlesungsskript, TH Darmstadt.

41. R. Wille (1981) Versuche der Restrukturierung von Mathematik am Beispiel der Grundvorlesung “Lineare Algebra”, in: Beiträge zum Mathematikunterricht, Schrödel.

Authors

## Rights and permissions

Reprints and permissions

© 1982 D. Reidel Publishing Company

### Cite this paper

Wille, R. (1982). Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts. In: Rival, I. (eds) Ordered Sets. NATO Advanced Study Institutes Series, vol 83. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7798-3_15

• DOI: https://doi.org/10.1007/978-94-009-7798-3_15

• Publisher Name: Springer, Dordrecht

• Print ISBN: 978-94-009-7800-3

• Online ISBN: 978-94-009-7798-3

• eBook Packages: Springer Book Archive