Posets isomorphic to their extensions

verfasst von

Marcel Erné

Abstract

A standard extension for a poset P is a system Q of lower ends ('descending subsets') of P containing all principal ideals of P. An isomorphism φ{symbol} between P and Q is called recycling if ∪φ{symbol}[Y]∈Q for all Y∈Q. The existence of such an isomorphism has rather restrictive consequences for the system Q in question. For example, if Q contains all lower ends generated by chains then a recycling isomorphism between P and Q forces Q to be precisely the system of all principal ideals. For certain standard extensions Q, it turns out that every isomorphism between P and Q (if there is any) must be recycling. Our results include the well-known fact that a poset cannot be isomorphic to the system of all lower ends, as well as the fact that a poset is isomorphic to the system of all ideals (i.e., directed lower ends) only if every ideal is principal.

Organisationseinheit(en)

Institut für Algebra, Zahlentheorie und Diskrete Mathematik

Typ

Artikel

Journal

ORDER

Band

2

Seiten

199-210

Anzahl der Seiten

12

ISSN

0167-8094

Publikationsdatum

06.1985

Publikationsstatus

Veröffentlicht

Peer-reviewed

Ja

ASJC Scopus Sachgebiete

Algebra und Zahlentheorie, Geometrie und Topologie, Theoretische Informatik und Mathematik

Elektronische Version(en)

https://doi.org/10.1007/BF00334857 (Zugang: Geschlossen)