Finite orders and their minimal strict completion lattices

• Autorzy: Gabriela Hauser Bordalo , Bernard Monjardet
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 06B23: Complete lattices, completions
• 06C10: Semimodular lattices, geometric lattices
• 06C05: Modular lattices, Desarguesian lattices
• 06A11: Algebraic aspects of posets
• 06A06: Partial order, general

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2003
• Tom: 23
• Numer: 2
• Strony: 85-100

Daty

wydano

2003

otrzymano

2002-09-24

poprawiono

2003-06-27

poprawiono

2003-09-24

Twórcy

autor

Gabriela Hauser Bordalo

• Departamento de Matematica, Faculdade de Ciencias e Centro de Algebra, Universidade de Lisboa, R. Prof. Gama Pinto, 2; 1699 Lisboa, Portugal

autor

Bernard Monjardet

• 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

Bibliografia

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Identyfikatory

DOI

10.7151/dmgaa.1065