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In this article we deal with the free Banach lattice generated by a latticeand 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 thatthis mapping $\hat \imath$ might not be an isometric embedding neither anisomorphic embedding. In order to provide sufficient conditions for $\hat\imath$ to be an isometric embedding we define the notion of locallycomplemented 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 awide number of examples of locally complemented sublattices and, as anapplication, we obtain that every free Banach lattice generated by a lattice islattice 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 andonly if it is injective, which in turn is equivalent to the fact that everylattice homomorphism $x^*\colon \mathbb{L} \longrightarrow [-1,1]$ extends to alattice homomorphism $\hat x^*\colon \mathbb{M} \longrightarrow [-1,1]$. Usingthis characterization we provide an example of lattices $\mathbb{L} \subseteq\mathbb{M}$ for which $\hat \imath$ is an isomorphic not isometric embedding.

Lattice embeddings in free Banach lattices over lattices