Four-generated quasiorder lattices and their atoms in a four-generated sublattice

ABSTRACT

Quasiorders, also known as preorders, on a set A form a lattice Quo(A). We prove that if A is a finite set consisting of 2, 3, 5, 7, 9, or more than 10 elements, then Quo(A) is four-generated but not three-generated. Also, if A is countably infinite, then a four-generated sublattice contains all atoms of Quo(A). These statements improve Ivan Chajda and the present author’s 1996 result, where six generators were constructed, and Tamás Dolgos and Júlia Kulin’s recent results, where five generators were given.

KEYWORDS:

• Four-generated lattice
• lattice of preorders
• minimum-sized generating set
• quasiorder lattice

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 06B99