Posets isomorphic to their extensions (Q1064328)

An isomorphism \(f: P\to Q\) of a poset P onto a collection Q of its lower ends, or, down-closed subsets of P (ordered by inclusion) is said to be recycling if \(Y\in Q\) implies \(\cup_{y\in Y}f(y)\in Q\). Consequences of the existence of recycling isomorphisms are investigated. For example, if Q contains all lower ends of well-ordered subsets of P, then P satisfies the ascending-chain condition, and Q is the set of all principle ideals of P.