Aside from ordered sets, the fixed point property has been investigated in other settings. On one hand, the fixed point property is most likely originated in topology. (See Exercise 6-1 for the topological fixed point property.) On the other hand, in any branch of mathematics in which the underlying structures have a natural type of morphism, we can define a fixed point property as “every endomorphism has a fixed point.” Hence, graphs are natural discrete structures for which to investigate the fixed point property. However, we must be careful: For graphs, the natural analogues of order-preserving maps would be the simplicial homomorphisms from Definition 6.1. Nonetheless, graph theorists prefer the notion of a (graph) homomorphism from Definition 6.10. It would be fruitless to argue whether simplicial homomorphisms or graph homomorphisms are “more natural.” Each notion is natural in certain settings: For example, simplicial homomorphisms are in one-to-one correspondence with the simplicial maps of the clique complex of a graph, as we will see in Exercise 9-19 when we investigate simplicial complexes, whereas we will see in this chapter that graph homomorphisms are a good framework to investigate natural graph theoretical topics, such as, for example, colorings.
KeywordsComplete Graph Simplicial Complex Comparability Graph Point Property Fixed Point Property
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