PDF

Discussiones Mathematicae General Algebra and Applications 23(2) (2003) 85-100
DOI: https://doi.org/10.7151/dmgaa.1065

FINITE ORDERS AND THEIR MINIMAL STRICT COMPLETION LATTICES

Gabriela Hauser Bordalo Departamento de Matematica, Faculdade de Ciencias e Centro de Algebra Universidade de Lisboa R. Prof. Gama Pinto, 2; 1699 Lisboa, Portugal e-mail: mchauser@ptmat.lmc.fc.ul.pt

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 e-mail: monjarde@univ-paris1.fr

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

2000 Mathematics Subject Classification: 06A06, 06A11, 06B23, 06C05, 06C10.

References

[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. Tepavcevi\'c, Collection of finite lattices generated by a poset, Order 17 (2000), 129-139.

Received 24 September 2002
Revised 27 June 2003
Revised 24 September 2003