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2016 | 36 | 1 | 59-70
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Quasiorder lattices are five-generated

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Abstrakty
EN
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.
Twórcy
autor
  • University of Szeged, Bolyai Institute, Szeged, Aradi vértanúk tere 1, Hungary 6720
Bibliografia
  • [1] G. Czédli, A Horn sentence for involution lattices of quasiorders, Order 11 (1994), 391-395. doi: 10.1007/BF01108770
  • [2] I. Chajda and G. Czédli, How to generate the involution lattice of quasiorders, Studia Sci. Math. Hungar. 32 (1996), 415-427.
  • [3] G. Czédli, Four-generated large equivalence lattices, Acta Sci. Math. 62 (1996), 47-69.
  • [4] G. Czédli, Lattice generation of small equivalences of a countable set, Order 13 (1996), 11-16. doi: 10.1007/BF00383964
  • [5] G. Czédli, (1+1+2)-generated equivalence lattices, J. Algebra 221 (1999), 439-462. doi: 10.1006/jabr.1999.8003
  • [6] T. Dolgos, Generating equivalence and quasiorder lattices over finite sets (in Hungarian) BSc Thesis, University of Szeged (2015).
  • [7] K. Kuratowski, Sur l'état actuel de l'axiomatique de la théorie des ensembles, Ann. Soc. Polon. Math. 3 (1925), 146-147.
  • [8] A. Levy, Basic Set Theory (Springer-Verlag, Berlin-Heidelberg-New York, 1979). doi: 10.1007/978-3-662-02308-2
  • [9] H. Strietz, Finite partition lattices are four-generated, Proc. Lattice Th. Conf. Ulm (1975), 257-259.
  • [10] H. Strietz, Über Erzeugendenmengen endlicher Partitionverbände, Studia Sci. Math. Hungar. 12 (1977), 1-17.
  • [11] G. Takách, Three-generated quasiorder lattices, Discuss. Math. Algebra and Stochastic Methods 16 (1996), 81-98.
  • [12] J. Tůma, On the structure of quasi-ordering lattices, Acta Universitatis Carolinae, Mathematica et Physica 43 (2002). doi: 65-74
  • [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.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1248
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