Partial orders on weak orders convex subsets (Q1590175)

Let \(P\) and \(H\) be finite partially ordered sets (orders, for short) and \(\varphi \) be a mapping of \(P\) into the set of all nonempty convex connected subsets of \(H\). Then the pair \((H,\varphi)\) is a visibility model for \(P\) if for \(x,y\in P\), \(x<_Py\) if and only if \(\varphi (x)\) and \(\varphi (y)\) are disjoint and there exist \(a\in \varphi (x)\) and \(b\in (x)\) with \(a<_Hb\). The order \(H\) is called the host order and the subsets \(\varphi (x)\) (\(x\in P\)) are called guests. The authors study orders having visibility models on weak order hosts. Moreover, they characterize orders such that the corresponding subsets \(\varphi (x)\) (i.e. the guests) of the weak orders are total orders or they are mutually isomorphic total orders.