example of non-complete lattice homomorphism

The real number line [-,]={-,} is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath in its usual orderingMathworldPlanetmath of numbers. Furthermore, the meet of a subset S of is the infimumMathworldPlanetmathPlanetmath of the set S.

Now define the map f:[-,][-,] as


First notice that if xy then f(x)f(y), for either xy0 in which case f(x)=0=f(y), or x0<y which gives f(x)=0<1=f(y) or 0<xy so f(x)=1=f(y).

In the second place, if S is a finite subset of then S contains a minimum element sS. So f(s)f(S) and f(s)f(t) for all tS, so f(minS)=f(s)=minf(S). Hence f is a lattice homomorphismMathworldPlanetmath.

However, f is not a complete lattice homomorphism. To see this let S={x:0<x}. Then infS=0. However, f(infS)=f(0)=0 while inff(S)=inf{1}=1.

Title example of non-complete lattice homomorphism
Canonical name ExampleOfNoncompleteLatticeHomomorphism
Date of creation 2013-03-22 16:58:36
Last modified on 2013-03-22 16:58:36
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 4
Author Algeboy (12884)
Entry type Example
Classification msc 06B05
Classification msc 06B99
Related topic ExtendedRealNumbers