Monotone Poset Maps

The following proposition shows that monotone map s have a canonical decomposition in terms of increasing and decreasing map s. 

Clearly, = ao (3, and Fix a U Fix/ = P. To see that a is an increasing map, we just need to see that it is order-preserving. Since a either fixes an element or maps it to a larger one, the only situation that needs to be considered is x,y P, X y, and a x) = (p x), a y) = y. However, under these conditions we must have y (r/) y thus we get a y) = y f y) f x) = a x), and so a is order-preserving. That / is a decreasing map can be seen analogously. Finally, the uniqueness follows from the fact that each x e P must be fixed by either a or / , and the value (p x) determines which one will fix x.  

The proof of Theorem 13.22 follows the general lines of the proof of Theorem 13.12. However, there are some further technicalities to be dealt with. 

We can assume that P Q. The proof is by induction, and to start with, some explanation is in order. For finite P the proof is by induction on P, and both statements are proved in parallel, with statement (a) being proved first. For infinite P, we can first prove statement (a) using statement (b) for finite posets, and then prove statement (b) by induction on P — (5, which is assumed to be finite. 

Let us now proceed with the induction. We start with statement (a). Since the expression Z (P jj) Z (P jj) is S3mimetric with respect to inverting the partial order of P, without loss of generality, it is enough to consider only the case (x) X. Let us show that in this case, A P x) is nonevasive. 

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Ko06a] D.N. Kozlov, A simple proof for folds on both sides in complexes of graph homomorphisms, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1265-1270. [Ko06b] D.N. Kozlov, Collapsing along monotone poset maps. International J. Math, and Math. Sciences 2006 (2006). 

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