Finite orders and their minimal strict completion lattices

Gabriela Hauser Bordalo; Bernard Monjardet

Discussiones Mathematicae - General Algebra and Applications (2003)

  • Volume: 23, Issue: 2, page 85-100
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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Gabriela Hauser Bordalo, and Bernard Monjardet. "Finite orders and their minimal strict completion lattices." Discussiones Mathematicae - General Algebra and Applications 23.2 (2003): 85-100. <http://eudml.org/doc/287638>.

@article{GabrielaHauserBordalo2003,
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.},
author = {Gabriela Hauser Bordalo, Bernard Monjardet},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {atomistic lattice; join-irreducible element; distributive lattice; modular lattice; lower semimodular lattice; Dedekind-MacNeille completion; strict completion; weak order.; completion; weak order},
language = {eng},
number = {2},
pages = {85-100},
title = {Finite orders and their minimal strict completion lattices},
url = {http://eudml.org/doc/287638},
volume = {23},
year = {2003},
}

TY - JOUR
AU - Gabriela Hauser Bordalo
AU - Bernard Monjardet
TI - Finite orders and their minimal strict completion lattices
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2003
VL - 23
IS - 2
SP - 85
EP - 100
AB - 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.
LA - eng
KW - atomistic lattice; join-irreducible element; distributive lattice; modular lattice; lower semimodular lattice; Dedekind-MacNeille completion; strict completion; weak order.; completion; weak order
UR - http://eudml.org/doc/287638
ER -

References

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  9. [9] B. Leclerc and B. Monjardet, Ordres 'C.A.C.', and Corrections, Fund. Math. 79 (1973), 11-22, and 85 (1974), 97. 
  10. [10] B. Monjardet and R. Wille, On finite lattices generated by their doubly irreducible elements, Discrete Math. 73 (1989), 163-164. Zbl0663.06008
  11. [11] J.B. Nation and A. Pogel, The lattice of completions of an ordered set, Order 14 (1997) 1-7. 
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