On the free Banach lattice generated by a lattice

We study structural properties of the free Banach lattice $FBL⟨L⟩$ generated by a distributive lattice $L$. We characterize when $FBL⟨L⟩$ has a strong unit, compute its density character, analyze the density character of order intervals and study when is $FVL⟨L⟩$ order dense in $FBL⟨L⟩$. We also study projection bands, quasi-interior points, and Banach lattice homomorphisms induced by lattice homomorphisms. Finally, we show that $FBL⟨L⟩$ is lattice isometric to $FBL⟨L^{\mathrm{o}\mathrm{p}}⟩$, where $L^{\mathrm{o}\mathrm{p}}$ denotes the opposite lattice.