# order-preserving map

Order-preserving map from a poset $L$ to a poset $M$ is a function $f$ such that

 $\forall x,y\in L:(x\geq y\implies f(x)\geq f(y)).$

Order-preserving maps are also called monotone functions or monotonic functions or order homomorphisms or isotone functions or isotonic functions.

Order-reversing map from a poset $L$ to a poset $M$ is a function $f$ such that

 $\forall x,y\in L:(x\geq y\implies f(x)\leq f(y)).$

Order-reversing maps are also called antitone functions.

 Title order-preserving map Canonical name OrderpreservingMap Date of creation 2013-03-22 17:44:43 Last modified on 2013-03-22 17:44:43 Owner porton (9363) Last modified by porton (9363) Numerical id 10 Author porton (9363) Entry type Definition Classification msc 06A06 Synonym monotone function Synonym monotonic function Synonym order homomorphism Synonym isotone function Synonym isotonic function Synonym order-preserving Synonym isotone Synonym isotonic Synonym order-reversing Synonym antitonic Synonym antitone Related topic Poset Related topic LatticeHomomorphism Defines monotonicity