Stepwise Construction of the Dedekind-MacNeille Completion

form a complete lattice, the D e d e k i n d M a c N e i l l e c o m p l e t i o n (or short comp l e t i o n ) of (P, <). It is the smallest complete lattice containing a subset orderisomorphic with (P, <). The size of the completion may be exponential in [PI. The completion can be computed in steps: first complete a small part of (P, <), then add another element, complete again, et cetera. Each such step increases the size of the completion only moderately and is moreover easy to perform. We shall demonstrate this by describing an elementary algorithm that, given a (finite) ordered set (P, <) and its completion (L, <), constructs the completion of any one-element extension of (P, <) in O(ILI. IP[. w(P)) steps, where w(P) denotes the width of (P, <). The special case that (P, <) is itself a complete lattice and thus isomorphic to its completion, has been considered as the problem of m i n i m a l i n s e r t i o n of an element into a lattice, see e.g. Valtchev [4]. We obtain that the complexity of inserting an element into a lattice (L, <) and then forming its completion is bounded by