Dedekind-MacNeille Completion of n -ordered Sets

A completion of an n-ordered set P = 〈P, 1, . . . , n〉 is defined, by analogy with the case of posets (2-ordered sets), as a pair 〈e,Q〉, where Q is a complete n-lattice and e : P → Q is an n-order embedding. The Basic Theorem of Polyadic Concept Analysis is exploited to construct a completion of an arbitrary nordered 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.