, Volume 24, Issue 1, pp 15–29 | Cite as

Dedekind–MacNeille Completion of n-ordered Sets

  • George Voutsadakis


A completion of an n-ordered set \({\mathbf{P}}{\text{ = }}{\left\langle {P, \lesssim _{1} , \ldots , \lesssim _{n} } \right\rangle }\) is defined, by analogy with the case of posets (2-ordered sets), as a pair \({\left\langle {e,{\mathbf{Q}}} \right\rangle }\), where Q is a complete n-lattice and \(e{\text{:}}{\mathbf{P}} \to {\mathbf{Q}}\) is an n-order embedding. The Basic Theorem of Polyadic Concept Analysis is exploited to construct a completion of an arbitrary n-ordered set. The completion reduces to the Dedekind–MacNeille completion in the dyadic case, the case of posets. A characterization theorem is provided, analogous to the well-known dyadic one, for the case of joined n-ordered sets. The condition of joinedness is trivial in the dyadic case and, therefore, this characterization theorem generalizes the uniqueness theorem for the Dedekind–MacNeille completion of an arbitrary poset.


Formal concept analysis Formal contexts Triadic concept analysis Triordered sets Complete trilattices Trilattices Polyadic concept analysis n-ordered sets n-lattices Completions 

Mathematics Subject Classifications (2000)

Primary: 06A23 Secondary: 62-07 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Biedermann, K.: An equational theory for trilattices. Algebra Univers. 42, 253–268 (1999)MATHCrossRefGoogle Scholar
  2. 2.
    Birkhoff, G.: Lattice Theory, 3rd edn. American Mathematical Society, Providence, RI (1973)Google Scholar
  3. 3.
    Davey, B. A., Priestley, H. A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  4. 4.
    Ganter, B., Wille, R.: Formal Concept Analysis, Mathematical Foundations. Springer, Berlin (1999)MATHGoogle Scholar
  5. 5.
    Koppelberg, S.: Handbook of Boolean Algebras, vol. 1. North-Holland, Amsterdam (1989)Google Scholar
  6. 6.
    MacNeille, H. M.: Partially ordered sets. Trans. Am. Math. Soc. 42, 416–460 (1937)MATHCrossRefGoogle Scholar
  7. 7.
    Voutsadakis, G.: Polyadic concept analysis. Order 19, 295–304 (2002)MATHCrossRefGoogle Scholar
  8. 8.
    Voutsadakis, G.: An equational theory of n-lattices. Submitted to Algebra Universalis. Preprint available at
  9. 9.
    Wille, R.: Restructuring lattice theory: an approach based on hierarchies of concepts. In: Rival, I. (ed.), Ordered Sets, pp. 445–470. Reidel, Dordrecht (1982)Google Scholar
  10. 10.
    Wille, R.: Concept lattices and conceptual knowledge systems. Comput. Math. Appl. 23, 493–515 (1992)MATHCrossRefGoogle Scholar
  11. 11.
    Wille, R.: The basic theorem of triadic concept analysis. Order 12, 149–158 (1995)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceLake Superior State UniversitySault Sainte MarieUSA

Personalised recommendations