Stepwise Construction of the Dedekind-MacNeille Completion
نویسنده
چکیده
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
منابع مشابه
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...
متن کاملDedekind-MacNeille Completion of n-Lattices
A completion of an n-lattice L = 〈L,.1, . . . , .n〉 is defined, by analogy with lattices, as a pair 〈e,C〉, where C is a complete n-lattice and e : L → C is an n-lattice embedding. The Basic Theorem of Polyadic Concept Analysis is exploited to construct a completion of an arbitrary n-lattice. The completion reduces to the Dedekind-MacNeille completion in the dyadic case, which was first formulat...
متن کاملDedekind-MacNeille Completion and Multi-adjoint Lattices
Among other applications, multi-adjoint lattices have been successfully used for modeling flexible notions of truth-degrees in the fuzzy extension of logic programming called MALP (Multi-Adjoint Logic Programming). In this paper we focus in the completion of such mathematical construct by adapting the classical notion of Dedekind-MacNeille in order to relax this usual hypothesis on such kind of...
متن کاملPosets of Finite Functions
The symmetric group S(n) is partially ordered by Bruhat order. This order is extended by L. Renner to the set of partial injective functions of {1, 2, . . . , n} (see, Linear Algebraic Monoids, Springer, 2005). This poset is investigated by M. Fortin in his paper The MacNeille Completion of the Poset of Partial Injective Functions [Electron. J. Combin., 15, R62, 2008]. In this paper we show tha...
متن کاملEquivalent fuzzy sets
Necessary and sufficient conditions under which two fuzzy sets (in the most general, poset valued setting) with the same domain have equal families of cut sets are given. The corresponding equivalence relation on the related fuzzy power set is investigated. Rela tionship of poset valued fuzzy sets and fuzzy sets for which the co-domain is Dedekind-MacNeille completion of that posets is deduced.
متن کامل