Transitive ternary relations and quasiorderings. (English) Zbl 0714.06001

Novák, Vítězslav; Novotný, Miroslav

Arch. Math., Brno 25, No. 1-2, 5-12 (1989).

A ternary structure is a pair \({\mathcal G}=(G,R)\), where G is a nonempty set and R is a ternary relation on G. A ternary structure \({\mathcal G}=(G,R)\) is transitive if the relation R is transitive. A quasiordered set is a pair \({\mathcal G}=(G,R)\), where G is a nonempty set and R is a binary relation on G that is reflexive and transitive. Given a transitive ternary structure \({\mathcal G}\), the authors show how to define a quasiordered set \({\mathcal Q}({\mathcal G})\). Conversely, given a quasiordered set \({\mathcal G}\), they show how to define a ternary structure \({\mathcal T}({\mathcal G})\) in such a way that \({\mathcal G}\) and \({\mathcal Q}({\mathcal T}({\mathcal G}))\) are isomorphic. The authors also show that if \({\mathcal G}\) is a transitive ternary structure that satisfies two technical conditions, then there is a strong homomorphism of \({\mathcal Q}({\mathcal T}({\mathcal G}))\) onto \({\mathcal G}\), but \({\mathcal G}\) and \({\mathcal Q}({\mathcal T}({\mathcal G}))\) need not be isomorphic.

Reviewer: T.B.Muenzenberger

MSC:

06A06	Partial orders, general
03E20	Other classical set theory (including functions, relations, and set algebra)

Keywords:

transitive binary relation; ternary structure; ternary relation; quasiordered set