Quasiorder lattices are five-generated

Júlia Kulin

Discussiones Mathematicae General Algebra and Applications (2016)

  • Volume: 36, Issue: 1, page 59-70
  • ISSN: 1509-9415

Abstract

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A quasiorder (relation), also known as a preorder, is a reflexive and transitive relation. The quasiorders on a set A form a complete lattice with respect to set inclusion. Assume that A is a set such that there is no inaccessible cardinal less than or equal to |A|; note that in Kuratowski's model of ZFC, all sets A satisfy this assumption. Generalizing the 1996 result of Ivan Chajda and Gábor Czéedli, also Tamás Dolgos' recent achievement, we prove that in this case the lattice of quasiorders on A is five-generated, as a complete lattice.

How to cite

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Júlia Kulin. "Quasiorder lattices are five-generated." Discussiones Mathematicae General Algebra and Applications 36.1 (2016): 59-70. <http://eudml.org/doc/286932>.

@article{JúliaKulin2016,
abstract = {A quasiorder (relation), also known as a preorder, is a reflexive and transitive relation. The quasiorders on a set A form a complete lattice with respect to set inclusion. Assume that A is a set such that there is no inaccessible cardinal less than or equal to |A|; note that in Kuratowski's model of ZFC, all sets A satisfy this assumption. Generalizing the 1996 result of Ivan Chajda and Gábor Czéedli, also Tamás Dolgos' recent achievement, we prove that in this case the lattice of quasiorders on A is five-generated, as a complete lattice.},
author = {Júlia Kulin},
journal = {Discussiones Mathematicae General Algebra and Applications},
keywords = {quasiorder lattice; preorder lattice; accessible cardinal},
language = {eng},
number = {1},
pages = {59-70},
title = {Quasiorder lattices are five-generated},
url = {http://eudml.org/doc/286932},
volume = {36},
year = {2016},
}

TY - JOUR
AU - Júlia Kulin
TI - Quasiorder lattices are five-generated
JO - Discussiones Mathematicae General Algebra and Applications
PY - 2016
VL - 36
IS - 1
SP - 59
EP - 70
AB - A quasiorder (relation), also known as a preorder, is a reflexive and transitive relation. The quasiorders on a set A form a complete lattice with respect to set inclusion. Assume that A is a set such that there is no inaccessible cardinal less than or equal to |A|; note that in Kuratowski's model of ZFC, all sets A satisfy this assumption. Generalizing the 1996 result of Ivan Chajda and Gábor Czéedli, also Tamás Dolgos' recent achievement, we prove that in this case the lattice of quasiorders on A is five-generated, as a complete lattice.
LA - eng
KW - quasiorder lattice; preorder lattice; accessible cardinal
UR - http://eudml.org/doc/286932
ER -

References

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  1. [1] G. Czédli, A Horn sentence for involution lattices of quasiorders, Order 11 (1994), 391-395. doi: 10.1007/BF01108770 Zbl0817.06007
  2. [2] I. Chajda and G. Czédli, How to generate the involution lattice of quasiorders, Studia Sci. Math. Hungar. 32 (1996), 415-427. Zbl0864.06003
  3. [3] G. Czédli, Four-generated large equivalence lattices, Acta Sci. Math. 62 (1996), 47-69. Zbl0860.06009
  4. [4] G. Czédli, Lattice generation of small equivalences of a countable set, Order 13 (1996), 11-16. doi: 10.1007/BF00383964 Zbl0860.06010
  5. [5] G. Czédli, (1+1+2)-generated equivalence lattices, J. Algebra 221 (1999), 439-462. doi: 10.1006/jabr.1999.8003 Zbl0941.06009
  6. [6] T. Dolgos, Generating equivalence and quasiorder lattices over finite sets (in Hungarian) BSc Thesis, University of Szeged (2015). 
  7. [7] K. Kuratowski, Sur l'état actuel de l'axiomatique de la théorie des ensembles, Ann. Soc. Polon. Math. 3 (1925), 146-147. Zbl51.0170.13
  8. [8] A. Levy, Basic Set Theory (Springer-Verlag, Berlin-Heidelberg-New York, 1979). doi: 10.1007/978-3-662-02308-2 Zbl0404.04001
  9. [9] H. Strietz, Finite partition lattices are four-generated, Proc. Lattice Th. Conf. Ulm (1975), 257-259. 
  10. [10] H. Strietz, Über Erzeugendenmengen endlicher Partitionverbände, Studia Sci. Math. Hungar. 12 (1977), 1-17. Zbl0487.06003
  11. [11] G. Takách, Three-generated quasiorder lattices, Discuss. Math. Algebra and Stochastic Methods 16 (1996), 81-98. Zbl0865.06005
  12. [12] J. Tůma, On the structure of quasi-ordering lattices, Acta Universitatis Carolinae, Mathematica et Physica 43 (2002). doi: 65-74 
  13. [13] L. Zádori, Generation of finite partition lattices, Lectures in Universal Algebra, Colloquia Math. Soc. J. Bolyai 43 Proc. Conf. Szeged (1983) 573-586 (North Holland, Amsterdam-Oxfor. 

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