Lexicographic Shellability

Lexicographic shellability is an important tool for studying the topological properties of the order complexes of partially ordered sets. Although, as we shall see in Remark 12.4, discrete Morse theory is more powerful as a method, shellability may still be useful in concrete applications. We take a detailed look at this concept in this section. 

Our presentation centers on the lexshellable posets, as the most general form of lexicographic shellability. We start with the classical situation of order complexes of posets, and then proceed to describe how this generalizes to... 

For the sake of simplicity we have so far discussed lexicographic shellability in the context of order complexes of posets. It turns out that working in the generality of nerves of acyclic categories does not cause any substantial problems, and essentially everything can be extended to this context. 

Lexicographic shellability for the nerves of acychc categories appears in... 

Ko97] D. N. Kozlov, General lexicographic shellability and orbit arrangements. Annals Comb. 1 (1997), no. 1, 67-90. 

Proposition 12.9. Let P be an EL-shellahle poset. Then the simplicial complex A P) is shellable. Moreover, the spanning simplices corresponding to the induced lexicographic shelling order are indexed by the weakly decreasing chains. 

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