Monotone and cone preserving mappings on posets

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