  We define several sorts of mappings on a poset like monotone, strictly
monotone, upper cone preserving and variants of these. Our aim is to
characterize posets in which some of these mappings coincide. We define special
mappings determined by two elements and investigate when these are strictly
monotone or upper cone preserving. If the considered poset is a semilattice
then its monotone mappings coincide with semilattice homomorphisms if and only
if the poset is a chain. Similarly, we study posets which need not be
semilattices but whose upper cones have a minimal element. We extend this
investigation to posets that are direct products of chains or an ordinal sum of
an antichain and a finite chain. We characterize equivalence relations induced
by strongly monotone mappings and show that the quotient set of a poset by such
an equivalence relation is a poset again.
