Partial orders and their convex subsets
Let \(P=(P,<)\) and \(Q=(Q,<)\) be (partially) ordered sets. Then \(P\) is called to be representable by a visibility relation of convex connected subsets of \(Q\) if there exists a mapping \(\varphi\) of \(P\) into the set of convex connected subsets of \(Q\) such that, for any \(x, y\in P\), \(x<y\) if and only if (i) \(\varphi(x) \cap\varphi (y)= \emptyset\), and (ii) there exist elements \(e_x\in \varphi(x)\) and \(e_y\in \varphi(y)\) with \(e_x<e_y\). If this is the case then \(Q\) is a host order for \(P\). The authors study and characterize some classes of ordered sets which can have host orders belonging to given classes of ordered sets. Especially, they describe the classes of ordered sets with in-tree and grid order as hosts. Note that the paper provides generalizations of interval orders since they are exactly those orders representable by visibility in total orders. This method is more general than another generalization of interval orders used, e.g., by \textit{J. Mitas} [Discrete Math. 144, 75-95 (1995; Zbl 0837.06001)].
scientific article; zbMATH DE number 1019119