Order

, Volume 24, Issue 1, pp 15–29

Dedekind–MacNeille Completion of n-ordered Sets

Article

Abstract

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.

Keywords

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

References

1. 1.
Biedermann, K.: An equational theory for trilattices. Algebra Univers. 42, 253–268 (1999)
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)
4. 4.
Ganter, B., Wille, R.: Formal Concept Analysis, Mathematical Foundations. Springer, Berlin (1999)
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)
7. 7.
8. 8.
Voutsadakis, G.: An equational theory of n-lattices. Submitted to Algebra Universalis. Preprint available at http://www.voutsadakis.com/RESEARCH/papers.html
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)
11. 11.
Wille, R.: The basic theorem of triadic concept analysis. Order 12, 149–158 (1995)