On the free Banach lattice generated by a lattice

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