  We define and prove the existence of free Banach lattices in the category of
Banach lattices and contractive lattice homomorphisms and establish some of
their fundamental properties. We give much more detailed results about their
structure in the case that there are only a finite number of generators and
give several Banach lattice characterizations of the number of generators
being, respectively, one, finite or countable. We define a Banach lattice $P$
to be projective if whenever $X$ is a Banach lattice, $J$ a closed ideal in
$X$, $Q:X\to X/J$ the quotient map, $T:P\to X/J$ a linear lattice homomorphism
and $\epsilon>0$ there is a linear lattice homomorphism $\hat{T}:P\to X$ such
that (i) $T=Q\circ \hat{T}$ and (ii) $\|\hat{T}\|\le (1+\epsilon)\|T\|$. We
establish the connection between projective Banach lattices and free Banach
lattices and describe several families of Banach lattices that are projective
as well as proving that some are not.

Functional Analysis