lattice homomorphism (original) (raw)
Let L and M be lattices. A map ϕ from L to M is called a lattice homomorphism
if ϕ respects meet and join. That is, for a,b∈L,
- •
ϕ(a∧b)=ϕ(a)∧ϕ(b), and - •
ϕ(a∨b)=ϕ(a)∨ϕ(b).
If in addition L is a bounded lattice
with top 1 and bottom 0, with ϕ and M defined as above, then ϕ(a)=ϕ(1∧a)=ϕ(1)∧ϕ(a), and ϕ(a)=ϕ(0∨a)=ϕ(0)∨ϕ(a) for all a∈L. Thus L is mapped onto a bounded
sublattice ϕ(L) of M, with top ϕ(1) and bottom ϕ(0).
If both L and M are bounded with lattice homomorphism ϕ:L→M, then ϕ is said to be a {0,1}-lattice homomorphism if ϕ(1) and ϕ(0) are top and bottom of M. In other words,
where 1L,1M,0L,0M are top and bottom elements of L and M respectively.