A concise approach to small generating sets of lattices of quasiorders and transitive relations

By H. Strietz, 1975, and G. Cz'edli, 1996, the complete lattice $Equ(A)$ of all equivalences is four-generated, provided the size $\mathrm{\mid }A\mathrm{\mid }$ is an accessible cardinal. Results of I. Chajda and G. Cz'edli, 1996, G. Tak'ach, 1996, T. Dolgos, 2015, and J.\ Kulin 2016, show that both the lattice $Quo(A)$ of all quasiorders on $A$ and, for $\mathrm{\mid }A\mathrm{\mid }\le {\mathrm{\aleph }}_0$, the lattice $Tran(A)$ of all transitive relations on $A$ have small generating sets. Based on complicated earlier constructions, we derive some new results in a concise but not self-contained way.