Posets isomorphic to their extensions

authored by: 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.
Organisation(s): Institute of Algebra, Number Theory and Discrete Mathematics
Type: Article
Journal: ORDER
Volume: 2
Pages: 199-210
No. of pages: 12
ISSN: 0167-8094
Publication date: 06.1985
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Algebra and Number Theory, Geometry and Topology, Computational Theory and Mathematics
Electronic version(s): https://doi.org/10.1007/BF00334857 (Access: Closed)