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 between P and Q is called recycling if ∪ϕ[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.
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Communicated by K. Keimel
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Erné, M. Posets isomorphic to their extensions. Order 2, 199–210 (1985). https://doi.org/10.1007/BF00334857
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DOI: https://doi.org/10.1007/BF00334857