On discrete properties of monotone mappings

Abstract

Let [Formula: see text] and [Formula: see text] be a partial order on [Formula: see text]. We deal with properties of oriented graphs which correspond to the algebra [Formula: see text] in the case that [Formula: see text] is monotone with respect to [Formula: see text]. 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 [Formula: see text] is monotone with respect to it. All mono-unary algebras such that there exists a linear order such that [Formula: see text] is monotone with respect to this order will be described; if the number of components of [Formula: see text] is infinite, then the number of such orders is equal to the cardinality of the power set of [Formula: see text].