Width two posets are reconstructible
Given a poset \(P\) with vertex set \(V(P)\), the collection of subposets \(D(P)= \{V(P)-X\mid X\in V(P)\}\) with the order inherited from \(P\) is the deck of \(P\). For posets ``the reconstruction problem'' is to determine whether \(D(P)= D(Q)\) implies \(P=Q\) (up to isomorphism). The reconstruction problem can be viewed in more general settings but is probably most interesting for classes of finite posets. Thus counterexamples exist for Ulam's unrestricted version. The authors of this very useful addition look at the class of width two posets and in their constructive proof are able to take advantage of known results (classes where the reconstruction conjecture holds) to construct a proof which also uses earlier technical results by these same authors. It is observed that at width 4 there may exist counterexamples since ``weak counterexamples'' do exist and also that the methods used here are width 2 specific. Thus a width \(n\to \text{width } (n-1)\) reduction method may be unavailable and the search for some other ``general parameter'' by means of which a ``most general method'' for handling this problem may be constructed is still wide open. Nevertheless, the class of width 2 posets includes many new examples not handled in other ways.
scientific article; zbMATH DE number 1019042