Automorphisms, isotone self-maps and cycle-free orders

✍ Scribed by Wei-Ping Liu; Ivan Rival; Nejib Zaguia

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 318 KB
• Volume: 144
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)00286-r

✦ Synopsis

For cycle-free ordered sets, the ratio IAut(P)l/IEnd(P)] converges to zero as IPI goes to infinity.

Although considerable attention has focussed, over the past two decades, on the fixed point property for ordered sets, little is known about the number of all isotone self-maps of an ordered set, or even about the ratio of the number of automorphisms to the number of all isotone self-maps. For an ordered set P let End(P) stand for the set of all isotone maps, that is, all f: P ~ P such that x ~< y implies f(x) <~ f(y), and let Aut(P) stand for all of its automorphisms, that is, 0~: P ~ P which are one-to-one and onto, such that ~ and ~-1 both are isotone, that is, belong to End(P). In a recent article, Rival and Rutkowski I-4] propose this conjecture.