Universal object associated to an acyclic matching

It turns out that for any poset P and any acyclic matching on P, there exists a universal object a poset whose linear extensions enumerate all allowed collapsing orders. 

Of course, the first natural question is whether this new order is actually well-defined. The next proposition answers that question and also explains in what sense U P, M) is a universal object. 

To prove (1) we need to check the three axioms of partial orders. The reflexivity is obvious, and the transitivity is automatic, since we have taken the transitive closure. The only property that needs to be proved is antisymmetry. So assume that it does not hold, and take X,Y U(P,M) such that X uY,Y u X, and X Y. Choose a sequence 

Without loss of generality we may now assume that either p + q 2, or y = 2 andp + q 1. In the first case. 

3delds a cycle, contradicting the assumption that our matching was acyclic in the second case such a cycle is given by 

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