On discrete properties of monotone mappings

Let h : A -> A and epsilon be a partial order on A. We deal with properties of oriented graphs which correspond to the algebra (A,h) in the case that h is monotone with respect to epsilon. We derive that every mono-unary algebra except connected one with a cycle of odd length has the property that there exists a nontrivial partial order such that (A,h) is monotone with respect to it. All mono-unary algebras such that there exists a linear order such that h is monotone with respect to this order will be described; if the number of components of (A,h) is infinite, then the number of such orders is equal to the cardinality of the power set of A.