# complete lattice homomorphism

Complete lattice homomorphism is a function from one lattice to an other lattice, which preserves arbitrary (not only finite) meets and joins.

If $\phi:L\to M$ is lattice homomorphism between complete lattices $L$ and $M$ such that

• $\phi(\bigvee\{a_{i}\mid i\in I\})=\bigvee\{\phi(a_{i})\mid i\in I\}$, and

• $\phi(\bigwedge\{a_{i}\mid i\in I\})=\bigwedge\{\phi(a_{i})\mid i\in I\}$,

then $\phi$ is called a complete lattice homomorphism.

Most often are considered complete lattice homomorphisms from one complete lattice to an other complete lattice (that is when all meets and joins are defined).

Complete lattice homomorphism is a special case of lattice homomorphism.

Title complete lattice homomorphism CompleteLatticeHomomorphism 2013-03-22 16:58:02 2013-03-22 16:58:02 porton (9363) porton (9363) 6 porton (9363) Definition msc 06B23 CompleteLattice