Distributive ordered sets and relative pseudocomplements

Josef Niederle

Discussiones Mathematicae - General Algebra and Applications (2006)

  • Volume: 26, Issue: 2, page 163-181
  • ISSN: 1509-9415

Abstract

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Brouwerian ordered sets generalize Brouwerian lattices. The aim of this paper is to characterize (α)-complete Brouwerian ordered sets in a manner similar to that used previously for pseudocomplemented, Stone, Boolean and distributive ordered sets. The sublattice (G(P)) in the Dedekind-Mac~Neille completion (DM(P)) of an ordered set (P) generated by (P) is said to be the characteristic lattice of (P). We can define a stronger notion of Brouwerianicity by demanding that both (P) and (G(P)) be Brouwerian. It turns out that the two concepts are the same for finite ordered sets. Further, the so-called antiblocking property of distributive lattices is generalized to distributive ordered sets.

How to cite

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Josef Niederle. "Distributive ordered sets and relative pseudocomplements." Discussiones Mathematicae - General Algebra and Applications 26.2 (2006): 163-181. <http://eudml.org/doc/276862>.

@article{JosefNiederle2006,
abstract = {Brouwerian ordered sets generalize Brouwerian lattices. The aim of this paper is to characterize (α)-complete Brouwerian ordered sets in a manner similar to that used previously for pseudocomplemented, Stone, Boolean and distributive ordered sets. The sublattice (G(P)) in the Dedekind-Mac~Neille completion (DM(P)) of an ordered set (P) generated by (P) is said to be the characteristic lattice of (P). We can define a stronger notion of Brouwerianicity by demanding that both (P) and (G(P)) be Brouwerian. It turns out that the two concepts are the same for finite ordered sets. Further, the so-called antiblocking property of distributive lattices is generalized to distributive ordered sets.},
author = {Josef Niederle},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {Brouwerian ordered set; distributive ordered set; relative pseudocomplement; characteristic lattice; antiblocking},
language = {eng},
number = {2},
pages = {163-181},
title = {Distributive ordered sets and relative pseudocomplements},
url = {http://eudml.org/doc/276862},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Josef Niederle
TI - Distributive ordered sets and relative pseudocomplements
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2006
VL - 26
IS - 2
SP - 163
EP - 181
AB - Brouwerian ordered sets generalize Brouwerian lattices. The aim of this paper is to characterize (α)-complete Brouwerian ordered sets in a manner similar to that used previously for pseudocomplemented, Stone, Boolean and distributive ordered sets. The sublattice (G(P)) in the Dedekind-Mac~Neille completion (DM(P)) of an ordered set (P) generated by (P) is said to be the characteristic lattice of (P). We can define a stronger notion of Brouwerianicity by demanding that both (P) and (G(P)) be Brouwerian. It turns out that the two concepts are the same for finite ordered sets. Further, the so-called antiblocking property of distributive lattices is generalized to distributive ordered sets.
LA - eng
KW - Brouwerian ordered set; distributive ordered set; relative pseudocomplement; characteristic lattice; antiblocking
UR - http://eudml.org/doc/276862
ER -

References

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  1. [1] B.A. Davey and H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge 1990. Zbl0701.06001
  2. [2] M. Erné, Distributive laws for concept lattices, Algebra Universalis 30 (1993), 538-580 Zbl0795.06006
  3. [3] G. Grätzer, General Lattice Theory, Akademie-Verlag, Berlin 1978. Zbl0436.06001
  4. [4] R. Halaš, Pseudocomplemented ordered sets, Archivum Math. (Brno) 29 (1993), 153-160 Zbl0801.06007
  5. [5] J. Larmerová and J. Rachůnek, Translations of distributive and modular ordered sets, Acta Univ. Palack. Olom., Math. 27 (1988), 13-23 Zbl0693.06003
  6. [6] J. Niederle, Boolean and distributive ordered sets: characterization and representation by sets, Order 12 (1995), 189-210 Zbl0838.06004
  7. [7] J. Niederle, Identities in ordered sets, Order 15 (1999), 271-278 Zbl0940.06001
  8. [8] J. Niederle, Semimodularity and irreducible elements, Acta Sci. Math. (Szeged) 64 (1998), 351-356 Zbl0924.06012
  9. [9] J. Niederle, On pseudocomplemented and Stone ordered sets, Order 18 (2001), 161-170 Zbl0999.06004
  10. [10] J. Niederle, On pseudocomplemented and Stone ordered sets, addendum, Order 20 (2003), 347-349 Zbl1059.06001
  11. [11] J. Niederle, On infinitely distributive ordered sets, Math. Slovaca 55 (2005), 495-502 Zbl1150.06002
  12. [12] G. Szász, Einführung in die Verbandstheorie, Akadémiai Kiadó, Budapest 1962. 

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