Linear extensions of partial orders preserving monotonicity (Q581425)

The authors show that for every acyclic order-preserving function \(f\) on a poset \((A,r)\), there exists a linear extension \(R\) of \(r\) on \(A\times A\), such that \(f\) is order-preserving on \((A,R)\). This result cannot be generalized to a collection \(\mathcal F\) of order-preserving functions on \((A,r)\) with the property that all finite compositions of elements of \(\mathcal F\) are acyclic. Unfortunately the counterexample, due to one of the referees, is not given.