order-preserving map

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

$$\forall x,y\in L:(x\ge y⟹f(x)\ge f(y)).$$

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

$$\forall x,y\in L:(x\ge y⟹f(x)\le f(y)).$$

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