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

TL;DR

This paper explores small generating sets for lattices of quasiorders and transitive relations, building on prior results and providing new, concise findings in the field of lattice theory.

Contribution

It presents new, concise results on small generating sets for lattices of quasiorders and transitive relations, simplifying earlier complex constructions.

Findings

Equivalence lattices are four-generated for accessible cardinals.

Quasiorder lattices have small generating sets for all sets.

Transitive relation lattices are small-generated for countable sets.

Abstract

By H. Strietz, 1975, and G. Cz\'edli, 1996, the complete lattice $Equ(A)$ of all equivalences is four-generated, provided the size $|A|$ 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 $|A|\leq \aleph_0$, 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.