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ArticleOriginal scientific text

Title

Finite orders and their minimal strict completion lattices

Authors 1, 2

Affiliations

  1. Departamento de Matematica, Faculdade de Ciencias e Centro de Algebra, Universidade de Lisboa, R. Prof. Gama Pinto, 2; 1699 Lisboa, Portugal
  2. Centre de Recherche en Mathématiques, Statistique et Économie Mathématique (CERMSEM), Université de Paris I (Panthéon Sorbonne), Maison des Sciences Économiques, 106-112 bd de l'Hopital; 75647 Paris Cédex 13, France

Abstract

Whereas the Dedekind-MacNeille completion D(P) of a poset P is the minimal lattice L such that every element of L is a join of elements of P, the minimal strict completion D(P)∗ is the minimal lattice L such that the poset of join-irreducible elements of L is isomorphic to P. (These two completions are the same if every element of P is join-irreducible). In this paper we study lattices which are minimal strict completions of finite orders. Such lattices are in one-to-one correspondence with finite posets. Among other results we show that, for every finite poset P, D(P)∗ is always generated by its doubly-irreducible elements. Furthermore, we characterize the posets P for which D(P)∗ is a lower semimodular lattice and, equivalently, a modular lattice.

Keywords

ENG
atomistic lattice, join-irreducible element, distributive lattice, modular lattice, lower semimodular lattice, Dedekind-MacNeille completion, strict completion, weak order.

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Discussiones Mathematicae - General Algebra and Applications
2003/23/2
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Pages:
85-100
Main language of publication
English
Received
2002-09-24
Accepted
2003-06-27
Published
2003

DOI: 10.7151/dmgaa.1065

Exact and natural sciences