  In this paper we study the structure of the set $\mbox{Hom}(X,\mathbb{R})$ of
all lattice homomorphisms from a Banach lattice $X$ into $\mathbb{R}$. Using
the relation among lattice homomorphisms and disjoint families, we prove that
the topological dual of the free Banach lattice $FBL(A)$ generated by a set $A$
contains a disjoint family of cardinality $2^{|A|}$, answering a question of B.
de Pagter and A.W. Wickstead. We also deal with norm-attaining lattice
homomorphisms. For classical Banach lattices, as $c_0$, $L_p$-, and
$C(K)$-spaces, every lattice homomorphism on it attains its norm, which shows,
in particular, that there is no James theorem for this class of functions. We
prove that, indeed, every lattice homomorphism on $X$ and $C(K,X)$ attains its
norm whenever $X$ has order continuous norm. On the other hand, we provide what
seems to be the first example in the literature of a lattice homomorphism which
does not attain its norm. In general, we study the existence and
characterization of lattice homomorphisms not attaining their norm in free
Banach lattices. As a consequence, it is shown that no Bishop-Phelps type
theorem holds true in the Banach lattice setting, i.e. not every lattice
homomorphism can be approximated by norm-attaining lattice homomorphisms.

Functional Analysis