Free Banach lattices generated by a lattice and projectivity
Antonio Avil\'es, Gonzalo Mart\'inez-Cervantes, Jos\'e David, Rodr\'iguez Abell\'an, Abraham Rueda Zoca

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 generated by a lattice . We prove that if is projective then has a maximum and a minimum. On the other hand, we show that if has maximum and minimum then is -lattice isomorphic to a -space. As a consequence, is projective if and only if it is lattice isomorphic to a -space with 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Advanced Algebra and Logic
