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

Dedekind–MacNeille Completion of n-ordered Sets

  • George Voutsadakis
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 

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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

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