Geodesic
Quasiorder lattices are five-generated
Discussiones Mathematicae. General Algebra and Applications, Tome 36 (2016) no. 1, pp. 59-70

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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.
Keywords: quasiorder lattice, preorder lattice, accessible cardinal 
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Kulin, Júlia. Quasiorder lattices are five-generated. Discussiones Mathematicae. General Algebra and Applications, Tome 36 (2016) no. 1, pp. 59-70. http://geodesic.mathdoc.fr/item/DMGAA_2016_36_1_a4/