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# 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*.

Defines:

monotonicity

Keywords:

partial order

Related:

Poset,LatticeHomomorphism

Synonym:

monotone function,monotonic function,order homomorphism,isotone function,isotonic function,order-preserving, isotone, isotonic,order-reversing,antitonic,antitone

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

06A06*no label found*

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## Comments

## Order homomorphism

Order homomorphism is the same as order-preserving map?

If yes, I will add it as a synonym to the definition of order-preserving map.

--

Victor Porton - http://www.mathematics21.org

* Algebraic General Topology and Math Synthesis

* 21 Century Math Method (post axiomatic math logic)

* Category Theory - new concepts

## Re: Order homomorphism

Yes, they are the same.