# lattice homomorphism

Let $L$ and $M$ be lattices. A map $\phi$ from $L$ to $M$ is called a lattice homomorphism  if $\phi$ respects meet and join. That is, for $a,b\in L$,

• $\phi(a\land b)=\phi(a)\land\phi(b)$, and

• $\phi(a\lor b)=\phi(a)\lor\phi(b)$.

If in addition $L$ is a bounded lattice  with top $1$ and bottom $0$, with $\phi$ and $M$ defined as above, then $\phi(a)=\phi(1\wedge a)=\phi(1)\wedge\phi(a)$, and $\phi(a)=\phi(0\vee a)=\phi(0)\vee\phi(a)$ for all $a\in L$. Thus $L$ is mapped onto a bounded    sublattice $\phi(L)$ of $M$, with top $\phi(1)$ and bottom $\phi(0)$.

If both $L$ and $M$ are bounded with lattice homomorphism $\phi:L\to M$, then $\phi$ is said to be a $\{0,1\}$-lattice homomorphism if $\phi(1)$ and $\phi(0)$ are top and bottom of $M$. In other words,

 $\phi(1_{L})=1_{M}\qquad\mbox{ and }\qquad\phi(0_{L})=0_{M},$

where $1_{L},1_{M},0_{L},0_{M}$ are top and bottom elements of $L$ and $M$ respectively.

Remarks.

 Title lattice homomorphism Canonical name LatticeHomomorphism Date of creation 2013-03-22 15:41:31 Last modified on 2013-03-22 15:41:31 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 13 Author CWoo (3771) Entry type Definition Classification msc 06B05 Classification msc 06B99 Related topic OrderPreservingMap Related topic RepresentingABooleanLatticeByFieldOfSets Defines lattice isomorphism Defines lattice endomorphism Defines lattice automorphism Defines Defines 1}-latticehomomorphism