Ascending closure operator

Theorem 13.12. LetP be a poset, and let p he a descending closure operator. Then A P) collapses onto A(p P)). By symmetry, the same is true for an ascending closure operator. 

Proof. Set Q =. F(DGn). Define the map f Q Q, taking each graph to its transitive closure. It is easily checked that f is an order-preserving map, that ifP = if, and that G < f G), for any graph G. We conclude that 99 is an ascending closure operator. 

Ascending and descending closure operators induce strong deformation retractions of Z (P) onto A p P)). Here we give a short and self-contained inductive proof of the following stronger fact. 

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