# Projectivity of the free Banach lattice generated by a lattice

## TL;DR

This paper investigates the conditions under which the free Banach lattice generated by a lattice is projective, showing it is projective for finite lattices but not for certain infinite linearly ordered sets.

## Contribution

It characterizes the projectivity of free Banach lattices generated by lattices, distinguishing between finite and infinite cases with specific order properties.

## Key findings

- Finite lattices generate projective free Banach lattices.
- Infinite linearly ordered sets with unbounded sequences do not generate projective lattices.
- Conditions for non-projectivity depend on the presence of unbounded increasing or decreasing sequences.

## Abstract

We study the projectivity of the free Banach lattice generated by a lattice $\mathbb{L}$ in two cases: when the lattice is finite, and when the lattice is an infinite linearly ordered set. We prove that in the first case it is projective while in the second case, if the linear order contains either an increasing sequence without upper bounds or a decreasing sequence without lower bounds, then it is not projective.