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We define several sorts of mappings on a poset like monotone, strictlymonotone, upper cone preserving and variants of these. Our aim is tocharacterize posets in which some of these mappings coincide. We define specialmappings determined by two elements and investigate when these are strictlymonotone or upper cone preserving. If the considered poset is a semilatticethen its monotone mappings coincide with semilattice homomorphisms if and onlyif the poset is a chain. Similarly, we study posets which need not besemilattices but whose upper cones have a minimal element. We extend thisinvestigation to posets that are direct products of chains or an ordinal sum ofan antichain and a finite chain. We characterize equivalence relations inducedby strongly monotone mappings and show that the quotient set of a poset by suchan equivalence relation is a poset again.

Monotone and cone preserving mappings on posets