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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1065,
author = {Gabriela Hauser Bordalo and Bernard Monjardet},
title = {Finite orders and their minimal strict completion lattices},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {23},
year = {2003},
pages = {85-100},
zbl = {1057.06001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1065}
}
Gabriela Hauser Bordalo; Bernard Monjardet. Finite orders and their minimal strict completion lattices. Discussiones Mathematicae - General Algebra and Applications, Tome 23 (2003) pp. 85-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1065/
[000] [1] G.H. Bordalo, A note on N-free modular lattices, manuscript (2000).
[001] [2] G.H. Bordalo and B. Monjardet, Reducible classes of finite lattices, Order 13 (1996), 379-390. | Zbl 0891.06001
[002] [3] G.H. Bordalo and B. Monjardet, The lattice of strict completions of a finite poset, Algebra Universalis 47 (2002), 183-200. | Zbl 1058.06001
[003] [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. | Zbl 1026.06008
[004] [5] J. Dalík, Lattices of generating systems, Arch. Math. (Brno) 16 (1980), 137-151. | Zbl 0455.06001
[005] [6] J. Dalík, On semimodular lattices of generating systems, Arch. Math. (Brno) 18 (1982), 1-7. | Zbl 0512.06008
[006] [7] K. Deiters and M. Erné, Negations and contrapositions of complete lattices, Discrete Math. 181 (1995), 91-111. | Zbl 0898.06003
[007] [8] R. Freese, K. Jezek. and J.B. Nation, Free lattices, American Mathematical Society, Providence, RI, 1995. | Zbl 0839.06005
[008] [9] B. Leclerc and B. Monjardet, Ordres 'C.A.C.', and Corrections, Fund. Math. 79 (1973), 11-22, and 85 (1974), 97.
[009] [10] B. Monjardet and R. Wille, On finite lattices generated by their doubly irreducible elements, Discrete Math. 73 (1989), 163-164. | Zbl 0663.06008
[010] [11] J.B. Nation and A. Pogel, The lattice of completions of an ordered set, Order 14 (1997) 1-7.
[011] [12] L. Nourine, Private communication (2000).
[012] [13] G. Robinson and E. Wolk, The embedding operators on a partially ordered set, Proc. Amer. Math. Soc. 8 (1957), 551-559. | Zbl 0078.01902
[013] [14] B. Seselja and A. Tepavcević, Collection of finite lattices generated by a poset, Order 17 (2000), 129-139. | Zbl 0963.06004