On the intersection of monotonicity preserving linear extensions (Q922563)

An acyclic partially ordered algebra (A,f,r) is a partial order (A,r) together with an r-monotone function f: \(A\to A\) with the property: For each element \(a\in A\) there is an \(0\leq N=N(A)\leq \infty\) such tha \(a,f(a),...,f^ N(a)\) are different elements in A, and \(f^{N+1}(a)=f^ N(a)\). Let \(L_ f(r)\) be the set of all f-linear extensions of (A,f,r), i.e., \(L_ f(r)=\{R:\) \(r\subseteq R\subseteq A\times A\), R is a linear order, f is R-monotone\(\}\). In Order 4, 31-55 (1987; Zbl 0627.06004), the author and \textit{B. Nagy} proved that \(L_ f(r)\) is not empty. In this paper he gives a description of the intersection of all f-linear extensions, i.e. of \(\cap L_ f(r)\).