Abstract
The formation of a perfect sequence for a chain-complete poset generalizes the process of dismantling a finite poset by irreducibles. In the finite case, according to a theorem of Duffus and Rival, the end result, or ‘core’, is unique up to isomorphism, no matter how the poset is dismantled. For chain-complete posets with no infinite antichains, the core is unique up to isomorphism, finite, and every perfect sequence has finite length, by an important and difficult theorem of Li and Milner. Li has asked if cores of chain-complete posets with no one-way infinite fence Fω and no tower are all isomorphic. He has also asked if the number of steps in the dismantling process, the length of the perfect sequence, is uniquely determined. The following results are obtained. (1) An example refuting the length conjecture is presented. (2) If at least one perfect sequence of a chain-complete poset has length λ < ω 2, then they all have length less than λ + ω, and their cores are isomorphic. (3) Both the isomorphism class of the core and the length of a perfect sequence are unique for posets with no F ω and no infinite chains; at every step of a perfect sequence, the corresponding poset is unique up to isomorphism. (4) A new, quick proof, perhaps yielding new insights, is presented of the theorem of Li and Milner.
Published Version
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