Free Banach lattices generated by a lattice and projectivity

TL;DR

This paper characterizes when free Banach lattices generated by a lattice are projective, showing they are equivalent to being isomorphic to certain $C(K)$-spaces with specific topological properties.

Contribution

It establishes a precise criterion linking the projectivity of free Banach lattices to the lattice having extremal elements and being isomorphic to $C(K)$-spaces.

Findings

Projectivity implies the lattice has a maximum and minimum.

Lattices with extremal elements lead to free Banach lattices isomorphic to $C(K)$-spaces.

Characterization of projective free Banach lattices in terms of $C(K)$-space isomorphisms.

Abstract

In this article we deal with the free Banach lattice $FBL\langle \mathbb{L} \rangle$ generated by a lattice $\mathbb{L}$. We prove that if $FBL\langle \mathbb{L} \rangle$ 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 $FBL\langle \mathbb{L} \rangle$ is $2$-lattice isomorphic to a $C(K)$-space. As a consequence, $FBL\langle \mathbb{L} \rangle$ 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.