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## Discussiones Mathematicae - General Algebra and Applications

2003 | 23 | 2 | 85-100
Tytuł artykułu

### Finite orders and their minimal strict completion lattices

Autorzy
Treść / Zawartość
Warianty tytułu
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
Kategorie tematyczne
Rocznik
Tom
Numer
Strony
85-100
Opis fizyczny
Daty
wydano
2003
otrzymano
2002-09-24
poprawiono
2003-06-27
poprawiono
2003-09-24
Twórcy
• Departamento de Matematica, Faculdade de Ciencias e Centro de Algebra, Universidade de Lisboa, R. Prof. Gama Pinto, 2; 1699 Lisboa, Portugal
autor
• 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
• [1] G.H. Bordalo, A note on N-free modular lattices, manuscript (2000).
• [2] G.H. Bordalo and B. Monjardet, Reducible classes of finite lattices, Order 13 (1996), 379-390.
• [3] G.H. Bordalo and B. Monjardet, The lattice of strict completions of a finite poset, Algebra Universalis 47 (2002), 183-200.
• [4] N. Caspard and B. Monjardet, The lattice of closure systems, closure operators and implicational systems on a finite set: a survey, Discrete Appl. Math. 127 (2003), 241-269.
• [5] J. Dalík, Lattices of generating systems, Arch. Math. (Brno) 16 (1980), 137-151.
• [6] J. Dalík, On semimodular lattices of generating systems, Arch. Math. (Brno) 18 (1982), 1-7.
• [7] K. Deiters and M. Erné, Negations and contrapositions of complete lattices, Discrete Math. 181 (1995), 91-111.
• [8] R. Freese, K. Jezek. and J.B. Nation, Free lattices, American Mathematical Society, Providence, RI, 1995.
• [9] B. Leclerc and B. Monjardet, Ordres 'C.A.C.', and Corrections, Fund. Math. 79 (1973), 11-22, and 85 (1974), 97.
• [10] B. Monjardet and R. Wille, On finite lattices generated by their doubly irreducible elements, Discrete Math. 73 (1989), 163-164.
• [11] J.B. Nation and A. Pogel, The lattice of completions of an ordered set, Order 14 (1997) 1-7.
• [12] L. Nourine, Private communication (2000).
• [13] G. Robinson and E. Wolk, The embedding operators on a partially ordered set, Proc. Amer. Math. Soc. 8 (1957), 551-559.
• [14] B. Seselja and A. Tepavcević, Collection of finite lattices generated by a poset, Order 17 (2000), 129-139.
Typ dokumentu
Bibliografia
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