Free Banach lattices generated by a lattice and projectivity

Abstract

In this article we deal with the free Banach lattice $FBL⟨\mathbb{L}⟩$ generated by a lattice $\mathbb{L}$. We prove that if $FBL⟨\mathbb{L}⟩$ is projective then $\mathbb{L}$ 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.

12 pages