  In this article we deal with the free Banach lattice generated by a lattice
and its behavior with respect to subspaces. In general, any lattice embedding
$i\colon \mathbb{L} \longrightarrow \mathbb{M}$ between two lattices
$\mathbb{L} \subseteq \mathbb{M}$ induces a Banach lattice homomorphism $\hat
\imath\colon FBL \langle \mathbb{L} \rangle \longrightarrow FBL \langle
\mathbb{M}\rangle$ between the corresponding free Banach lattices. We show that
this mapping $\hat \imath$ might not be an isometric embedding neither an
isomorphic embedding. In order to provide sufficient conditions for $\hat
\imath$ to be an isometric embedding we define the notion of locally
complemented lattices and prove that, if $\mathbb L$ is locally complemented in
$\mathbb M$, then $\hat \imath$ yields an isometric lattice embedding from
$FBL\langle\mathbb L\rangle$ into $FBL\langle\mathbb M\rangle$. We provide a
wide number of examples of locally complemented sublattices and, as an
application, we obtain that every free Banach lattice generated by a lattice is
lattice isomorphic to an AM-space or, equivalently, to a sublattice of a
$C(K)$-space.
  Furthermore, we prove that $\hat \imath$ is an isomorphic embedding if and
only if it is injective, which in turn is equivalent to the fact that every
lattice homomorphism $x^*\colon \mathbb{L} \longrightarrow [-1,1]$ extends to a
lattice homomorphism $\hat x^*\colon \mathbb{M} \longrightarrow [-1,1]$. Using
this characterization we provide an example of lattices $\mathbb{L} \subseteq
\mathbb{M}$ for which $\hat \imath$ is an isomorphic not isometric embedding.
