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Diagrams and Covering Graphs
Summary. We make some general remarks on the Hasse diagram and the covering graph of a finite poset. Then we turn to the question of orientations and reorientations of a covering graph. Birkhoff [1948] asked for necessary and sufficient conditions on a lattice L in order that every lattice M whose covering graph is isomorphic with the covering graph of L be lattice isomorphic to L. Solutions to this problem will be given in the subsequent sections.
The Hasse diagram (briefly: the diagram) of a finite poset P = (P, ≤) is an oriented graph with the circles of P as its vertices and an edge x → y if y covers x in P. Usually the arrows on the edges are omitted and the graph is arranged so that all edges point upwards on the page. The diagram of a finite poset is the most common tool for representing the poset graphically. For example, the usual diagram of the lattice 23 of all subsets of a three-element set, ordered by set inclusion, is shown in Figure 1.3(c) (Section 1.2). The diagram of a finite poset P determines P up to isomorphism. The combinatorial interest in posets is largely due to two unoriented graphs associated with a given poset: the comparability graph (which we shall not consider here) and the covering graph. In this chapter we shall have a closer look at the covering graph of certain lattices.
For a finite poset P, its covering graph G(P) was introduced in Section 1.9. As for diagrams, it is common to identify a pictorial representation of the covering graph with the covering graph itself.
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