Free Banach lattices generated by a lattice and projectivity

In this article we deal with the free Banach lattice $FBL\langle $\mathbb{L}$ \rangle$generatedbyalattice$$\mathbb{L}$$\mathrm{.}Weprovethatif$FBL\langle $\mathbb{L}$ \rangle$isprojectivethen$$\mathbb{L}$$ has a maximum and a minimum. On the other hand, we show that if $\mathbb{L}$ has maximum and minimum then $FBL⟨\mathbb{L}⟩$ is $2$-lattice isomorphic to a $C(K)$-space. As a consequence, $FBL⟨\mathbb{L}⟩$ is projective if and only if it is lattice isomorphic to a $C(K)$-space with $K$ being an absolute neighborhood retract. As an application, we characterize those linearly ordered sets and Boolean algebras for which the corresponding free Banach lattice is projective.