How to generate the involution lattice of quasiorders. (English) Zbl 0864.06003

Chajda, Ivan; Czédli, Gábor

Stud. Sci. Math. Hung. 32, No. 3-4, 415-427 (1996).

Given a set \(A\), let \(\text{Quord} (A)\) denote the set of all quasiorders (i.e., reflexive and transitive relations) on \(A\). Equipped with meet (intersection), join (transitive hull of union) and involution \((\rho\mapsto \{\langle x,y\rangle: \langle y,x\rangle \in\rho\})\), \(\text{Quord} (A)\) is an involution lattice. When \(A\) is infinite, \(\text{Quord} (A)\) is considered a complete involution lattice. Let \(\kappa_0= \aleph_0\), the smallest infinite cardinal, and define \(\kappa_{n+1} =2^{\kappa_n}\). It is shown that if \(|A|\leq\kappa_n\) for some integer \(n\), then \(\text{Quord} (A)\) has a three-element generating set.

Reviewer: G.Czédli (Szeged)

MSC:

06B99	Lattices
08A02	Relational systems, laws of composition

Keywords:

quasiorders; involution lattice; generating set